the zero–vector because the norm of a|niis ka|nik= √ n>0. Therefore, 0

Quantenmechanik (Theoretische Physik II)

TU Berlin, WS 2007/2008

Prof. Dr. T. Brandes

25. Januar 2008

INHALTSVERZEICHNIS

3. Einteilchen-Quantenmechanik . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.1 Harmonic Oscillator: Ladder Operators, Phonons and Photons . . . . . . 29

3.1.1 The Ladder Operators a and a† . . . . . . . . . . . . . . . . . . . . 293.1.2 The Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . 313.1.3 Phonons and Photons . . . . . . . . . . . . . . . . . . . . . . . . . 313.1.4 Koharente Zustande . . . . . . . . . . . . . . . . . . . . . . . . . . 323.1.5 Verschobener harmonischer Oszillator . . . . . . . . . . . . . . . . 33

3.2 Der Drehimpuls: Einfuhrung . . . . . . . . . . . . . . . . . . . . . . . . . . 343.2.1 Radial Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.3 Erganzung I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.3.1 Erhaltungsgrossen . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.3.2 Gemeinsame Eigenfunktionen . . . . . . . . . . . . . . . . . . . . . 373.3.3 Knotensatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.4 Symmetrien . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.4.1 Translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.4.2 Rotationen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.5 Drehimpuls-Vertauschungsrelationen (VR) . . . . . . . . . . . . . . . . . . 403.6 Der Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.6.1 Empirische Hinweise auf den Spin . . . . . . . . . . . . . . . . . . 413.6.2 Operator des Spin-Drehimpulses (Spin 1/2) . . . . . . . . . . . . . 413.6.3 ‘Drehung der Stern-Gerlach-Apparatur’ . . . . . . . . . . . . . . . 413.6.4 Anwendung: Zeeman-Aufspaltung . . . . . . . . . . . . . . . . . . 42

3.7 Drehimpulsaddition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.7.1 Rekursion fur Clebsch-Gordan-Koeffizienten . . . . . . . . . . . . . 443.7.2 Fall j1 = 1

2 , j2 = 12 (Zwei Spin-1

2-Teilchen) . . . . . . . . . . . . . . 463.7.3 Fall j1 = l (Bahndrehimpuls), j2 = 1

2 (Spin) . . . . . . . . . . . . . 463.8 Spin-Orbit Coupling and Fine Structure of the Hydrogen Atom . . . . . . 47

3.8.1 Kinetic Energy and Darwin Term . . . . . . . . . . . . . . . . . . . 473.8.2 Spin-Orbit Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . 483.8.3 Fine Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4. Der Zustandsbegriff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.1 Zustande in der QM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.1.1 Reine Zustande . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

Inhaltsverzeichnis iii

4.1.2 Gemischte Zustande . . . . . . . . . . . . . . . . . . . . . . . . . . 534.2 Eigenschaften des Dichteoperators . . . . . . . . . . . . . . . . . . . . . . 54

4.2.1 Charakterisierung des Dichteoperators . . . . . . . . . . . . . . . . 544.2.2 Entropie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.2.3 Thermische Zustande . . . . . . . . . . . . . . . . . . . . . . . . . 554.2.4 Zeitentwicklung, Liouville-von-Neumann-Gleichung . . . . . . . . . 554.2.5 Spezialfall Spin 1/2: die Bloch-Sphare . . . . . . . . . . . . . . . . 56

4.3 Zusammengesetzte Systeme . . . . . . . . . . . . . . . . . . . . . . . . . . 574.3.1 Bipartite Systeme . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.3.2 Reduzierte Dichtematrix . . . . . . . . . . . . . . . . . . . . . . . . 574.3.3 Reine und verschrankte Zustande . . . . . . . . . . . . . . . . . . . 58

4.4 Erganzung: Die Schmidt-Zerlegung . . . . . . . . . . . . . . . . . . . . . . 594.5 Verschrankung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.5.1 Korrelationen in Spin-Singlett-Zustanden . . . . . . . . . . . . . . 604.5.2 Bellsche Ungleichungen . . . . . . . . . . . . . . . . . . . . . . . . 61

5. Storungstheorie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645.1 Zeitunabhangige Storungstheorie . . . . . . . . . . . . . . . . . . . . . . . 64

5.1.1 Projektor-Methode . . . . . . . . . . . . . . . . . . . . . . . . . . . 645.1.2 Auswertung fur die Eigenwerte . . . . . . . . . . . . . . . . . . . . 655.1.3 Auswertung fur die Zustande . . . . . . . . . . . . . . . . . . . . . 685.1.4 Parametrische Abhangigkeit von Spektren . . . . . . . . . . . . . . 69

6. Introduction into Many-Particle Systems . . . . . . . . . . . . . . . . . . . . . . . 716.1 Indistinguishable Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

6.1.1 Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716.1.2 Basis vectors for Fermi and Bose systems . . . . . . . . . . . . . . 72

6.2 2-Fermion Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 756.2.1 Two Electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 766.2.2 Properties of Spin-Singlets and Triplets . . . . . . . . . . . . . . . 776.2.3 The Exchange Interaction . . . . . . . . . . . . . . . . . . . . . . . 78

6.3 Two-electron Atoms and Ions . . . . . . . . . . . . . . . . . . . . . . . . . 806.3.1 Perturbation theory in U . . . . . . . . . . . . . . . . . . . . . . . 80

7. The Hartree-Fock Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 827.1 The Hartree Equations, Atoms, and the Periodic Table . . . . . . . . . . . 82

7.1.1 Effective Average Potential . . . . . . . . . . . . . . . . . . . . . . 827.1.2 Angular Average, Shells, and Periodic Table . . . . . . . . . . . . . 83

7.2 Hamiltonian for N Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . 847.2.1 Expectation value of H0 . . . . . . . . . . . . . . . . . . . . . . . . 857.2.2 Expectation value of U . . . . . . . . . . . . . . . . . . . . . . . . 85

7.3 Hartree-Fock Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 877.3.1 The Variational Principle . . . . . . . . . . . . . . . . . . . . . . . 877.3.2 The Variational Principle for Many-Electron Systems . . . . . . . 89

Inhaltsverzeichnis iv

7.3.3 Hartree-Fock Equations . . . . . . . . . . . . . . . . . . . . . . . . 92

8. Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 958.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

8.1.1 Model Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . 958.2 The Born-Oppenheimer Approximation . . . . . . . . . . . . . . . . . . . 96

8.2.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 968.2.2 Discussion of the Born-Oppenheimer Approximation . . . . . . . . 978.2.3 Adiabaticity and Geometric Phases . . . . . . . . . . . . . . . . . . 988.2.4 Breakdown of the Born-Oppenheimer Approximation . . . . . . . 100

8.3 The Hydrogen Molecule Ion H+2 . . . . . . . . . . . . . . . . . . . . . . . 100

8.3.1 Hamiltonian for H+2 . . . . . . . . . . . . . . . . . . . . . . . . . . 100

8.3.2 The Rayleigh-Ritz Variational Method . . . . . . . . . . . . . . . . 1008.3.3 Bonding and Antibonding . . . . . . . . . . . . . . . . . . . . . . . 102

8.4 Hartree-Fock for Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . 1078.4.1 Roothan Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

9. Time-Dependent Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1109.1 Time-Dependence in Quantum Mechanics . . . . . . . . . . . . . . . . . . 110

9.1.1 Time-evolution with time-independent H . . . . . . . . . . . . . . 1109.1.2 Example: Two-Level System . . . . . . . . . . . . . . . . . . . . . 111

9.2 Time-dependent Hamiltonians . . . . . . . . . . . . . . . . . . . . . . . . . 1139.2.1 Spin 1

2 in Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . 1139.2.2 Landau-Zener-Rosen problem . . . . . . . . . . . . . . . . . . . . . 116

9.3 Time-Dependent Perturbation Theory . . . . . . . . . . . . . . . . . . . . 1169.3.1 Model Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . 1169.3.2 The Interaction Picture . . . . . . . . . . . . . . . . . . . . . . . . 1169.3.3 First Order Perturbation Theory . . . . . . . . . . . . . . . . . . . 1179.3.4 Higher Order Perturbation Theory . . . . . . . . . . . . . . . . . . 1189.3.5 States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

10. Rotations and Vibrations of Molecules . . . . . . . . . . . . . . . . . . . . . . . . 12110.1 Vibrations and Rotations in Diatomic Molecules . . . . . . . . . . . . . . 121

10.1.1 Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12110.1.2 Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . 12210.1.3 Radial SE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12410.1.4 Spin S > 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12510.1.5 Beyond the Harmonic Approximation . . . . . . . . . . . . . . . . 126

10.2 Selection Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12610.2.1 Dipole Approximation . . . . . . . . . . . . . . . . . . . . . . . . . 12610.2.2 Pure Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12610.2.3 Pure Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12710.2.4 Vibration-Rotation Spectra . . . . . . . . . . . . . . . . . . . . . . 129

10.3 Electronic Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

Inhaltsverzeichnis v

10.3.1 The Franck-Condon Principle . . . . . . . . . . . . . . . . . . . . . 129

c©T. Brandes 2005, 2007

3. EINTEILCHEN-QUANTENMECHANIK

3.1 Harmonic Oscillator: Ladder Operators, Phonons and Photons

In this section, we solve the one–dimensional harmonic oscillator

H =p2

2m+

1

2mω2x2, (3.1)

with a powerful operator method that does not rely on complicated differential equationbut on simple algebraic manipulations. We recall the result for the stationary eigenstatesand eigenvalues of the energy,

ψn(x) =(mω

π~

)1/4 1√n!2n

Hn

(

x

√mω

~

)

e−mω2~

x2

En = ~ω

(

n+1

2

)

, n = 0, 1, 2, 3, ..., (3.2)

where Hn is the n-the Hermite polynomial. These solutions are usually obtained bysolving a differential equation via asymptotic analysis and a polynomial ansatz for thewave function coefficients.

AUFGABE: Geben Sie die Energieeigenwerte fur das eindimensionale Potential V (x)mit V (x ≤ 0) =∞, V (x > 0) = 1

2mω20x

2 an.

3.1.1 The Ladder Operators a and a†

We define the two operators

a ≡√mω

2~x+

i√2m~ω

p, a† ≡√mω

2~x− i√

2m~ωp. (3.3)

You have showed in the problems that if two operators A and B are hermitian, A = A†,B = B† the linear combination C := A+iB is not hermitian but C† = A−iB (rememberthe analogy to a complex number z = x + iy, z∗ = x − iy). We know that x andp are hermitian, therefore a+ (‘a dagger’) is the hermitian conjugate of a. From thecommutator of x and p we easily find

[x, p] = i~ [a, a†] = 1. (3.4)

We define the number operator

N := a†a (3.5)

3. Einteilchen-Quantenmechanik 30

which is a hermitian operator because N † = (a†a)† = a†(a†)† = N . The eigenvalues ofN must be real therefore. We denote the eigenvalues of N by n and show that the n arenon–negative integers: First, we denote the corresponding (normalized) eigenvectors ofN by |n〉,

N |n〉 = n|n〉. (3.6)

STEP 1: We show n ≥ 0: remember the scalar product of two states |ψ〉 and |φ〉 isdenoted as 〈φ|ψ〉.

0 ≤ ‖a|n〉‖2 = 〈n|a†a|n〉 = 〈n|N |n〉 = n〈n|n〉 = n. (3.7)

STEP 2: We step down the ladder: if |n〉 is an eigenvector of N with eigenvalue n, thena|n〉 is an eigenvector of N with eigenvalue n − 1, a2|n〉 is an eigenvector of N witheigenvalue n− 2, a3|n〉 is an eigenvector of N with eigenvalue n− 3,...

Na = a†aa =(

aa† − [a, a†])

a =(

aa† − 1)

a = a(

N − 1)

Na|n〉 = a(

N − 1)

|n〉 = (n− 1)a|n〉

Na2|n〉 = (Na)a|n〉 = a(

N − 1)

a|n〉 = a(n − 1− 1)a|n〉 = (n− 2)a2|n〉... (3.8)

The state a|n〉 is an eigenstate of N with eigenvalue n − 1 and therefore must be pro-portional to |n− 1〉,

a|n〉 = Cn|n− 1〉 n = 〈na†an〉 = |Cn|2〈n − 1|n − 1〉 = |Cn|2

a|n〉 =√n|n− 1〉. (3.9)

The operator a takes us from one eigenstate with eigenvalue n to a lower eigenstate witheigenvalue nSTEP 3: We show that n must be an integer: For any n with 0 < n < 1, the eigenvalueequation Na|n〉 = (n − 1)a|n〉 can only be true if a|n〉 = 0 is the zero–vector. It thenbecomes the trivial equation 0 = 0 that contains no contradictions. But a|n〉 cannot bethe zero–vector because the norm of a|n〉 is ‖a|n〉‖ =

√n > 0. Therefore, 0 < n < 1

leads to a contradition. In the same way, there can’t be values of n with 1 < n < 2(application of a leads us to the case 0 < n < 1 which is already excluded. As a result,n is an integer. In particular, the lowest possible n is n = 0, for which the equation

a|0〉 = 0 (3.10)

defines the ground state ket |0〉 which we assumed to be normalized.Step 4: The normalized state a†|n〉 is (AUFGABE)

a†|n〉 =√n+ 1|n+ 1〉. (3.11)


Therefore, a† takes us up the ladder from one eigenstate |n〉 to the next higher |n+ 1〉.All the normalized eigenstates |n〉 can be created from the ground state |0〉 by successiveapplication of the ladder operator a†:

|n〉 = (a†)n√n!|0〉. (3.12)

3.1.2 The Harmonic Oscillator

The connection of the above algebraic tour de force with the harmonic oscillator is verysimple: The Hamiltonian (10.42) can be written as

H =p2

2m+

1

2mω2x2 = ~ω

(

a†a+1

2

)

= ~ω

(

N +1

2

)

(3.13)

which you can check by inserting the definitions of a and a†. The eigenvectors of H arethe eigenvectors of N :

H|n〉 = ~ω

(

N +1

2

)

|n〉 = ~ω

(

n+1

2

)

|n〉, (3.14)

from which we can read off the eigenvalues of the harmonic oscillator, En = ~ω(n+1/2).The corresponding eigenfunctions are, of course, the eigenfunctions of the harmonicoscillator,

|n〉 ↔ ψn(x) =(mω

π~

)1/4 1√n!2n

Hn

(√mω

~

)

e−mω2~

x2. (3.15)

This is not so easy to see directly; it is proofed for the ground state |0〉 in the problems.AUFGABE: Leiten Sie mit Hilfe von a† die explizite Form der Grundzustandwellen-

funktion Ψ0(x) des harmonischen Oszillators in 1d her.

3.1.3 Phonons and Photons

We call the state |n〉 of the harmonic oscillator with energy ~ω(n+ 1/2) a state with nquanta ~ω of energy plus the zero point energy ~ω/2. These quanta are called phononsfor systems where massive particles have oscillatory degrees of freedom, the state |n〉 isa n–phonon state.

|n〉 ←→ n–phonon state. (3.16)

The ladder operator a+ operates as

a+|n〉 =√n+ 1|n+ 1〉 (3.17)

and creates a state with one more phonon which is why it is called a creation operator.In the same way, the operator a,

a|n〉 =√n|n− 1〉 (3.18)


leads to a state with one phonon less (it destroys one phonon) and is called a anni-hilation operator.

In a similar manner, the oscillatory degrees of freedom of the electromagnetic field(light) lead to a Hamiltonian like the one of the harmonic oscillator. The correspondingstates are called n–photon states. This is one of the topics of Quantum Mechanics II,the theory of light, and many–body theory. It is there where operators like the a and a+

show their full versatility and power.

3.1.4 Koharente Zustande

Wegen der grossen Bedeutung der Leiteroperatoren a und a† kann man nach derenEigenzustanden fragen. Wir beginnen mit dem Absteigeoperator a und fordern

a|z〉 = z|z〉 (3.19)

mit dem zu bestimmenden Zustand |z〉 und dem Eigenwert z, der i.a. komplex sein kann,denn a ist ja nicht hermitesch. Wir setzen |z〉 als Linearkombination der Fock-Zustande|n〉 (Eigenzustande der Energie H) an,

|z〉 =

∞∑

n=0

cn|n〉. (3.20)

Iteration und die Forderung nach Normierung, 〈z|z〉 liefert (AUFGABE) den koharentenZustand (Glauber-Zustand 1)

|z〉 = e−12|z|2

∞∑

n=0

zn

√n!|n〉. (3.21)

Diese Iteration funktioniert allerdings nur bei den Eigenzustanden |z〉 des Absteigeopera-tors a: Es gibt keine Eigenzustande des Aufsteigeoperators a† (AUFGABE). Wir konnenallerdings das hermitesch konjugierte der Definitionsgleichung a|z〉 = z|z〉 schreiben als

〈z|a† = z∗〈z| (3.22)

im Sinne des mit den Bra-Vektoren (Funktionalen) definierten Skalarproduktes: DasSkalarprodukt von a|z〉 mit einem beliebigen Ket |f〉 des Hilbertraums ist

(a|z〉, |f〉) = 2(|z〉, a†|f〉) = 3〈z|a†|f〉.

Man bezeichnet 〈z| dann als linker Eigenvektor von a†. Beachte, dass das z in 〈z| nichtkonjugiert komplex geschrieben wird, sondern explizit

〈z| = e−12|z|2

∞∑

n=0

(z∗)n√n!〈n|. (3.23)

1 Roy J. Glauber, ∗1925, Nobelpreis 2005 fur die Entwicklung der Theorie der koharenten Zustandein der Quantenoptik

2 Definition des adjungierten Operators3 ‘Mathematiker-Skalarprodukt’ in ‘Physiker-Skalarprodukt’ umschreiben


Es gilt also z.B. in Skalarprodukten

〈z|a|z〉 = 〈z|z〉z = z, 〈z|a†|z〉 = z∗. (3.24)

Die koharente Zustande haben ein minimales Produkt der quantenmechanischen Unscharfein Ort x und Impuls p: wir schreiben

x =

√

~

2mω

(

a+ a†)

, p = i

√

m~ω

2

(

a† − a)

(3.25)

und berechnen (AUFGABE)

∆2|z〉(x)∆

2|z〉(p) =

~2

4, (3.26)

wobei fur einen Operator A und einen Zustand Ψ

∆2|Ψ〉(A) ≡ 〈Ψ|A2|Ψ〉 − 〈Ψ|A|Ψ〉2 (3.27)

definiert wird.AUFGABE: Beweis der Heisenbergschen Unscharferelation fur ein Operator-

paar A, B.AUFGABE: Berechnen Sie fur den 1d harmonischen Oszillator die Zeitentwicklung

|Ψ(t > 0)〉 eines koharenten Zustandes |Ψ(t = 0)〉 = |z〉 fur ein gegebenes komplexes z.Skizzieren Sie die Zeitentwicklung in der komplexen Ebene.

AUFGABE: 1. Zeigen Sie, dass die koharenten Zustande |z〉 eine vollstandige Basisim Hilbertraum des 1d harmonischen Oszillators sind. Zeigen Sie hierzu

∫d2z

π|z〉〈z| =

∞∑

n=0

|n〉〈n|, (3.28)

wobei das Integral uber die gesamte komplexe Ebene lauft (Mass d2z ≡ dxdy fur z =x+ iy) und

∑∞n=0 |n〉〈n| die vollstandige Eins in der Basis der Fockzustande ist.

2. Zeigen Sie, dass die koharenten Zustande |z〉 keine Orthogonalbasis bilden.

3.1.5 Verschobener harmonischer Oszillator

We shift the harmonic oscillator potential,

H =p2

2m+

1

2mω2x2 → Hλ ≡

p2

2m+

1

2mω2 (x+ λx0)

2

x0 ≡√

2~

mω, (3.29)


where x0 is a length and λ ∈ R a dimensionless real number. Using the ladder operatorsa and a†, we can write Hλ as

Hλ = H + λω0

(

a+ a†)

+ ~ω0λ2 (3.30)

= ~ω0

((

a† + λ)

(a+ λ) +1

2

)

(3.31)

= ~ω0

(

b†b+1

2

)

, (3.32)

where in the last line we introduced shifted ladder operators according to

b ≡ a+ λ. (3.33)

The shifted Hamiltonian has eigenstates

Hλ|n〉λ = En|n〉λ, En = ~ω0

(

n+1

2

)

(3.34)

with eigenstates |n〉λ that refer to the new shifted ladder operators b, b† as usual, i.e.b|n〉λ =

√n|n− 1〉λ etc. The eigenvalues of the energy En are the same as before.

How are the shifted eigenstates |n〉λ related to the unshifted ones? The groundstate|n = 0〉λ of Hλ is defined as

b|0〉λ = 0 (a+ λ)|0〉λ = 0 a|0〉λ = −λ|0〉λ. (3.35)

The last equation, however, is just the eigenvalue equation of a coherent state, a|z〉 =z|z〉, of the unshifted oscillator. By comparison we therefore have up to a phase factor

|0〉λ = |z = −λ〉 = e−12|λ|2

∞∑

n=0

(−λ)n√n!|n〉. (3.36)

In the basis of the unshifted oscillator, the ground state of the shifted harmonic oscillatoris a coherent state.

3.2 Der Drehimpuls: Einfuhrung

Wir studieren den Drehimpuls, um Physik in 3d beschreiben zu konnen (z.B. die Fein-struktur der Atome) und diskutieren bei dieser Gelegenheit gleich eine Reihe wichtigerallgemeiner Konzepte in der QM: Erhaltungsgrossen, Symmetrien, Storungstheorie undeiniges mehr.

Wir betrachten einen Einteilchen-Hamiltonian H in einem radialsymmetrischen Po-tential V (r) in 3d,

H = − ~2

2m∆ + V (r). (3.37)


Wegen der Radialsymmetrie spielt der Bahn-Drehimpulsoperator L,

L ≡ r× p (3.38)

eine wichtige Rolle (r ist der Ortsoperator, p der Impulsoperator). In Kugelkoordinatenhat man

Lx = −i~(

− sinϕ∂

∂θ− cosϕ cot θ

∂

∂ϕ

)

Ly = −i~(

cosϕ∂

∂θ− sinϕ cot θ

∂

∂ϕ

)

Lz = −i~ ∂

∂ϕ, (3.39)

sowie fur das Quadrat

L2 = L2x + L2

y + L2z = −~

2

[1

sin θ

∂

∂θ

(

sin θ∂

∂θ

)

+1

sin2 θ

∂2

∂ϕ2

]

. (3.40)

Damit folgt fur den Laplace-Operator in Kugelkoordinaten

∆ =∂2

∂r2+

2

r

∂

∂r− L2

~2r2, (3.41)

und Losungen der stationaren SG, HΨ = EΨ, lassen sich durch den Separationsansatz

Ψ(r, θ, ϕ) = Ylm(θ, ϕ)R(r) (3.42)

auf ein 1d-Problem (in r) zuruckfuhren. Hierbei sind die Funktionen Ylm(θ, ϕ) die Ku-gelflachenfunktionen (engl. spherical harmonics), die durch die Eigenwertgleichungenvon L2 und Lz definiert sind,

L2Ylm(θ, ϕ) = ~2l(l + 1)Ylm(θ, ϕ), l = 0, 1, 2, 3, ... (3.43)

LzYlm(θ, ϕ) = ~mYlm(θ, ϕ), (3.44)

Die Ylm haben Quantenzahlen l und m und die explizite Form

Ylm(θ, ϕ) = (−1)(m+|m|)/2il[2l + 1

4π

(l − |m|)!(l + |m|)!

]1/2

P|m|l (cos θ)eimϕ

P|m|l (x) :=

1

2ll!(1− x2)|m|/2 d

l+|m|

dxl+|m| (x2 − 1)l

l = 0, 1, 2, 3, ...; m = −l,−l + 1,−l + 2, ..., l − 1, l. (3.45)

The P|m|l are called associated Legendre polynomials. The spherical harmonics

are an orthonormal function system on the surface of the unit sphere |x| = 1. We writethe orthonormality relation both in our abstract bra –ket and in explicit form:

|lm〉 ←→ Ylm(θ, ϕ) (3.46)

〈l′m′|lm〉 = δll′δmm′ ←→∫ 2π

0

∫ π

0Y ∗

l′m′(θ, ϕ)Ylm(θ, ϕ) sin θdθdϕ = δll′δmm′ .


Abbildung 3.1: Absolute squares of various spherical harmonics. Fromhttp://mathworld.wolfram.com/SphericalHarmonic.html

The spherical harmonics with l = 0, 1, 2, 3, 4, ... are denoted as s-, p-, d-, f -, g–,... func-tions which you might know already from chemistry (‘orbitals’). The explicit forms forsome of the first sphericals are

Y00 =1√4π, Y10 = i

√

3

4πcos θ, Y1±1 = ∓i

√

3

8πsin θ · e±iϕ. (3.47)

The Spherical harmonics are used in many areas of science, ranging from nuclear physicsup to computer vision tasks.

3.2.1 Radial Solutions

The solutions of the hydrogen problem are separated into radial partRnl(r) and sphericalpart Ylm(θ, ϕ),

Ψnlm(r, θ, ϕ) = Rnl(r)Ylm(θ, ϕ), (3.48)

where radial eigenfunctions for the bound states are characterised by the two integerquantum numbers n ≥ l + 1 and l,

Rnl(r) = − 2

n2

√

(n − l − 1)!

[(n+ l)!]3e−Zr/na0

(2Zr

na0

)l

L2l+1n+l

(2Zr

na0

)

, l = 0, 1, ..., n − 1 (3.49)

Lmn (x) = (−1)m

n!

(n−m)!exx−m dn−m

dxn−me−xxn generalized Laguerre polynomials.


The radial wave functions Rnl(r) have n− l nodes. For these states, the possible eigen-values only depend on n, E = En with

En = −1

2

Z2e2

4πε0a0

1

n2, n = 1, 2, 3, ... Lyman Formula

a0 ≡ 4πε0~2

me2Bohr Radius. (3.50)

In Dirac notation, we write the stationary states as |nlm〉 with the correspondence

|nlm〉 ↔ 〈r|nlm〉 ≡ Ψnlm(r). (3.51)

The ground state is |GS〉 = |100〉 with energy E0 = −13.6 eV. The degree of degeneracyof the energy level En, i.e. the number of linearly independent stationary states withquantum number n, is

n−1∑

l=0

(2l + 1) = n2. (3.52)

3.3 Erganzung I

3.3.1 Erhaltungsgrossen

Definition Eine Erhaltungsgrosse A ist eine Observable, die mit dem HamiltonoperatorH vertauscht,

[A,H] = 0. (3.53)

Aus den Heisenbergschen Bewegungsgleichungen folgt dann namlich

A(t) = 0, (3.54)

im Heisenberg-Bild ist A dann also eine Konstante der Bewegung.AUFGABE: Sei der Einteilchen-Hamiltonian H = − ~2

2m∆+V (r) in 3d. a) Zeige, dassL2 und Lz Erhaltungsgrossen sind und miteinander vertauschen. b) Zeige, dass L eineErhaltungsgrosse ist.

3.3.2 Gemeinsame Eigenfunktionen

Satz 1. Zwei miteinander kommutierende Observablen A und B haben gemeinsameEigenfunktionen.

Beweis als AUFGABE. Im obigen Fall H = − ~2

2m∆ + V (r) sind die Eigenfunktio-nen Ylm von L2 und Lz also auch Eigenfunktionen von H - deshalb funktioniert derSeparationsansatz.


3.3.3 Knotensatz

Wir haben das radialsymmetrische Problem also auf ein 1d-Problem (in r) zuruckgefuhrt,damit lasst sich der Knotensatz anwenden. Es handelt sich dabei um eine Aussage uberdie Nullstellen der Losungen Ψ(x) von Sturm-Liouville-Problemen, d.h. Eigenwertpro-blemen der Form

h(x)Ψ′′(x) + u(x)Ψ′(x) + v(x)Ψ(x) = EΨ(x), x ∈ [a, b] (3.55)

auf einem Intervall [a, b] mit bestimmten Randbedingungen fur Ψ(x) und/oder Ψ′(x) anden Randern x = a, b. Beispiele: Freies Teilchen im Kasten [0, L]; 1d harmonischer Os-zillator v(x) ∝ x2 auf [−∞,∞]; Wasserstoff-Radial-Wellenfunktionen auf [0,∞]. Unterbestimmten Voraussetzungen gilt (s. z.B. W. Walter, ‘Gewohnliche Differentialgleichun-gen’)

Satz 2. Das Sturm-Liouville Eigenwertproblem hat unendlich viele reelle EigenwerteE0 < E1 < E2... < En < ... mit Eigenfunktionen Ψn(x), die in (a, b) genau n Nullstel-len haben. Insbesondere hat der Grundzustand Ψ0(x) keine Nullstelle. Zwischen je zweiNullstellen von Ψn(x) liegt eine Nullstelle von Ψn+1(x).

UBERPRUFEN anhand der obigen Beispiele.

3.4 Symmetrien

3.4.1 Translation

Sei |Ψ〉 ein Zustand mit Wellenfunktion Ψ(r). Wir verschieben die Wellenfunktion Ψ(r)raumlich um den Vektor a (aktive Transformation) und erhalten den neuen Zustand |Ψ′〉mit Wellenfunktion Ψ′(r) = Ψ(r − a). Wenn wir mehrere Zustande verschieben, sollensolche Translationen die Norm und das Skalarprodukt nicht andern,

T : Ψ(r) → Ψ′(r) = Ψ(r− a)

Φ(r) → Φ′(r) = Φ(r− a), 〈Φ|Ψ〉 = 〈Φ′|Ψ′〉. (3.56)

Nach einem Satz von E. Wigner funktioniert das genau dann, wenn die Zustande alletransformiert werden wie

|Ψ′〉 = U |Ψ〉, |Φ′〉 = U |Φ〉 (3.57)

mit demselben unitaren oder anti-unitaren Hilbertraum-Operator U . Fur kontinuierlicheTransformationen wie die obige Translation ist U unitar (Spiegelungen fuhren z.B. zuantiunitarem U). Damit haben wir fur die obige Translation

UaΦ(r) = Φ(r− a). (3.58)

Wir bestimmen Ua durch Entwickeln fur infinitesimal kleines a,

Φ(r− a) = Φ(r)− a∇Φ(r) + ... Ua = 1− a∇+ ...

Ua = 1− iap

~+ ... (3.59)


mit dem Impulsoperator p = (~/i)∇ als Erzeuger raumlicher Translationen. Furendliches a durch Exponentieren

Ua = exp

(

− iap~

)

. (3.60)

AUFGABE: Betrachte den unverschobenen Oszillator, H, und den verschobenen har-monischen Oszillator Hλ in 1d, Eq.(3.29).

1. Zeige durch raumliche Translation, dass verschobene und unverschobene Eigen-zustande zusammenhangen gemass

|n〉λ = Xλ|n〉, Xλ ≡ eλ(a−a†) (3.61)

mit dem unitaren Verschiebeoperator (displacement operator) Xλ.2. Transformiere mit Hilfe von Xλ die Eigenwertgleichung des unverschobenen Os-

zillators, H|n〉 = En|n〉, in die des verschobenen Oszillators, Hλ|n〉λ = En|n〉λ um.

3. Beweise a+ λ = XλaX†λ. Hierbei ist die nested commutator expansion

eSOe−S = O + [S,O] +1

2![S, [S,O]] +

1

3![S, [S, [S,O]]] + ... (3.62)

nutzlich, die durch Herleitung einer DGL erster Ordnung fur f(x) ≡ exSOe−xS undTaylor-Entwicklung in x bewiesen wird.

3.4.2 Rotationen

Jetzt das gleiche fur Rotationen: Wir rotieren eine Wellenfunktion Ψ(r) raumlich um dieAchse n und den Winkel θ (aktive Transformation) und erhalten den neuen Zustand |Ψ′〉mit Wellenfunktion Ψ′(r) = Ψ(R−1r), wobei R−1 die Ruckrotation darstellt. Wiederumfordern wir

URΨ(r) = Ψ(R−1r), UR unitar. (3.63)

Eine infinitesimale Rotation R z.B. um die z-Achse (gegen den Uhrzeigersinn, positiverinfinitesimaler Winkel θ) lautet

x→ x− θy, y → y + θx, z → z, (3.64)

und damit

Ψ(R−1r) = Ψ(x+ θy, y − θx, z) = Ψ(r) + θ (y∂x − x∂y)Ψ(r) + ...

UR = 1− iθLz

~+ ... (3.65)

mit der z-Komponente Lz des Bahn-Drehimpulsoperators L! Fur Drehungen um belie-bige Achsen n und Winkel θ hat man

UR = exp

(

− iθnL

~

)

, (3.66)

d.h. der Bahn-Drehimpulsoperator ist der Erzeuger raumlicher Rotationen.


3.5 Drehimpuls-Vertauschungsrelationen (VR)

Die Bahn-Drehimpuls-VR folgen aus der Definition L ≡ r× p und lauten

[Lj, Lk] = iǫjklLl, (3.67)

wobei die Indizes j etc. die Werte 1, 2, 3(x, y, z) annehmen. Alternativ folgen sie aus derNichtvertauschbarkeit von raumlichen Rotationen um die x,y, z-Achse, die sich auf dieentsprechenden Erzeuger dieser Rotationen ubertragt (AUFGABE).

Allgemeiner kann man jetzt Drehimpuls (nicht notwendig Bahndrehimpuls) -VR

[Jj , Jk] = iǫjklJl (3.68)

an den Anfang stellen und nach moglichen Darstellungen dieser VR durch Hilbertraum-Operatoren fragen. Dabei stosst man auch auf halbzahlige Drehimpulse, z.B. den Spin1/2. Wir machen das so: Zunachst gilt (NACHRECHNEN)

[J2, J3] = 0, (3.69)

damit haben J2 und J3 ein gemeinsames System von Eigenfunktionen,

J2|λν〉 = λ|λν〉, J3|λν〉 = mν |λν〉. (3.70)

Jetzt definiert man Schiebeoperatoren

J± ≡ J1 ± J2, [J3, J±] = ±J±, (3.71)

Es gilt (AUFGABE)

J−J+|λν〉 = (λ−m2ν −mν)|λν〉, λ−m2

ν −mν ≥ 0

J3J+|λν〉 = (mν + 1)|λν〉. (3.72)

Mehrfaches Anwenden von J+ auf |λν〉 gibt immer hohere Werte mν , obwohl mν be-schrankt sein muss. Es existiert also ein maximales mν = j, fur das man

j ≥ mν , λ = j(j + 1) (3.73)

erhalt. Analog findet man durch Anwenden des Schiebeoperators J− ein minimales mν =−j. Es gilt also

− j ≤ mν ≤ j. (3.74)

Von −j gelangt man nach j in ganzzahligen Schritten nur, falls j entweder ganz oderhalbzahlig ist! Die Quantenzahl j des Drehimpulses hat also nur die moglichen Werte

j = 0,1

2, 1,

3

2, ... (3.75)

Fur gegebenes j gibt es 2j + 1 Eigenwerte m von J3, wir haben also

J2|jm〉 = j(j + 1)|jm〉, J3|jm〉 = m|jm〉. (3.76)

Bis auf einen beliebigen Phasenfaktor hat man weiterhin (AUFGABE)

J±|jm〉 =√

(j ∓m)(j ±m+ 1)|jm± 1〉. (3.77)


3.6 Der Spin

3.6.1 Empirische Hinweise auf den Spin

Stern-Gerlach-Versuch: Atomstrahl mit Ag-Atomen durch inhomogenes Magnetfeld Bin z-Richtung. Magnetisches Moment µ gibt Potential −µB und Kraft F in z-Richtung

F = (0, 0, µz∂zBz). (3.78)

Wurde klassisch kontinuierliche Ablenkung der Atome in z-Richtung geben. Beobachtetwerden aber nur zwei Flecke Elektron hat Eigendrehimpuls (Spin) vom Betrag ~/2mit zwei Projektionen ±~/2, der mit einem magnetischen Moment verknupft ist. Dassieht man an der Pauli-Gleichung (nicht-relativistischer Grenzfall der Dirac-Gleichung):Term (vgl. Skript S. 9, 1.35 i)

HσB ≡ −gee~

2mσB, ge = 1 in Dirac-Theorie (3.79)

µB ≡ e~

2mBohr-Magneton, (3.80)

wobei ge der g-Faktor des Elektrons ist. Das magnetische Moment wird in der QM alsoein Operator µ = geµBσ. Proton: gp = 2.79.., Neutron gn = −1.91...

3.6.2 Operator des Spin-Drehimpulses (Spin 1/2)

Der Operator des Spin-Drehimpulses S erfullt wie alle Drehimpulse

[Sj, Sk] = iǫjklSl. (3.81)

Es gilt weiterhin

S2|sm〉 = s(s+ 1)|sm〉, S3|sm〉 = m|sm〉, s =1

2, m = ±1

2. (3.82)

Die Darstellung mit den Pauli-Matrizen σk haben wir bereits kennen gelernt,

Sk ≡~

2σk, [σj , σk] = 2iǫjklσl. (3.83)

(CHECK die Faktoren 2 und 1/2!). Die Spin-Zustande |sm〉 (s = 1/2 ist fest) bezeichnetman als Spinoren.

3.6.3 ‘Drehung der Stern-Gerlach-Apparatur’

Die zwei Spinoren

| ↑, ez〉, | ↓, ez〉 (3.84)

mit m = ±1/2 sind Eigenzustande von Sz und damit Eigenzustande des Zeeman-TermsHσB ≡ −ge

e2mc~σB in der Pauli-Gleichung fur Magnetfeld B = (0, 0, B) in z-Richtung.


Jetzt drehen wir das Magnetfeld in eine beliebige Richtung n. Dann brauchen wir dieEigenzustande von

nσ, n = (sin θ cosφ, sin θ sinφ, cos θ), (3.85)

die wir in AUFGABE 8.2 berechnet hatten:

| ↑,n〉 =

(cos θ

2e−iφ/2

sin θ2e

iφ/2

)

, | ↓,n〉 =

(− sin θ

2e−iφ/2

cos θ2e

iφ/2

)

. (3.86)

Die Eigenwertgleichung

S3|m, ez〉 = m|m, ez〉 (3.87)

gilt auch in einem System mit Magnetfeld in n-Richtung, in das wir uns von der z-Achsedurch Drehung um die Achse

n′ = (− sinφ, cos φ, 0) (3.88)

und den Winkel θ drehen (Bild!). Dazu multiplizieren wir die Eigenwertgleichung mit(zu bestimmenden) unitaren Operatoren (2 mal 2 Matrizen) Un′(θ),

Un′(θ)S3U†n′(θ)Un′(θ)|m, ez〉 = mUn′(θ)|m, ez〉

↔ 1

2nσ|m,n〉 = m|m,n〉, m = ±1

2(3.89)

Un′(θ)S3U†n′(θ) ≡

1

2nσ, Un′(θ)|m, ez〉 ≡ |m,n〉. (3.90)

Direkter Vergleich liefert (AUFGABE, eindeutig bis auf eine Phase)

Un′(θ) = cos

(θ

2

)

− i sin(θ

2

)

n′σ = exp

(

−iθ2n′

σ

)

. (3.91)

Hier ist Un′(θ) nur fur spezielle Rotationen konstruiert, es gilt aber allgemein

Un(θ) = exp

(

−iθ2nσ

)

. (3.92)

AUFGABE: a) Zeige Un(2π) = −1: man sagt, die Darstellung von Rotationen mitden Spin-Matrizen ist doppelwertig b) Berechne die Mittelwerte von Sx, Sy, und Sz inden Zustanden |m,n〉,m =↑, ↓.

3.6.4 Anwendung: Zeeman-Aufspaltung

Wir betrachten die Pauli-Gleichung mit konstantem Magnetfeld B und skalarem Poten-tial Φ(r),

i∂tΨ = HΨ, H =(p− eA2)

2m− e~

2mσB + eΦ(r). (3.93)


Der Hilbertraum der Wellenfunktionen mit Spin ist

H = L2(R3)⊗ C2, (3.94)

wobei L2(R3) den Bahnanteil und C2 den Spinanteil beinhaltet.Wir wahlen das Vektorpotential A als

A =1

2B× r (3.95)

und erhalten (AUFGABE)

H =p2

2m+ eΦ(r) +

µB

~(L + ~σ)B +

e2

8m(B× r)2 (3.96)

mit dem Drehimpuls L. Fur kleine Magnetfelder schreiben wir genahert

H = H0 + V, H0 ≡p2

2m+ eΦ(r), V ≡ µB

~(L + ~σ)B. (3.97)

Seien |nlm〉 die Eigenzustande von H0 mit Eigenenergien E0n fur ein Coulombpotential

Φ(r) (Wasserstoff-Atom). Wir schreiben die Eigenzustande von H mit Spin als Produkt-zustande,

|nlmσ〉 ≡ |nlm〉 ⊗ |σ〉, (3.98)

wobei |σ〉 ein Spinor-Eigenzustand von σn, n = B/B ist. Dann sind die zugehorigenEigenenergien Enlmσ

Enlmσ = E0n + µBB(m+ σ), (3.99)

d.h. die ursprunglichen E0n werden aufgespalten (AUFGABE: Aufspaltung im Term-

schema skizzieren!)

3.7 Drehimpulsaddition

(MERZBACHER). Addition wie im Wasserstoffatom oben vom Typ L + ~σ, mathema-tisch genauer L⊗ 1 + 1⊗ ~σ als Operator im Produktraum

H = H1 ⊗H2, (3.100)

im obigen Beispiel H = L2(R3)⊗C2. Wir untersuchen jetzt allgemein die Eigenzustande|jm〉 von

J = J1 + J2, (3.101)

wenn J1 und J2 zwei miteinander vertauschende Drehimpulsoperatoren sind. Eigen-zustande gemass

J2|jm〉 = j(j + 1)|jm〉, Jz|jm〉 = m|jm〉J2

1 |j1m1〉 = j1(j1 + 1)|j1m1〉, J22 |j2m2〉 = j2(j2 + 1)|j2m2〉. (3.102)


Wir wollen jetzt j1 und j2 als fest gegeben annehmen (z.B. als zwei Spins j1 = s1,j2 = s2).

AUFGABE: Zeige, dass die Eigenzustande |jm〉 auch Eigenzustande von J21 und J2

2

sind.Man kann also schreiben

|jm〉 =∑

m1m2

cm1m2 |j1m1〉 ⊗ |j2m2〉. (3.103)

Weil j1 und j2 fest sind, schreiben wir verkurzt

|jm〉 =∑

m1m2

〈m1m2|jm〉|m1m2〉. (3.104)

d.h. wir entwickeln die Eigenzustande des Gesamtdrehimpulses nach den Produktzustandender Einzeldrehimpulse. Zu bestimmen sind dabei die Clebsch-Gordan-Koeffizienten〈m1m2|jm〉.

3.7.1 Rekursion fur Clebsch-Gordan-Koeffizienten

SCHRITT 1: Anwendung von Jz = J1,z + J2,z liefert

Jz|jm〉 = m|jm〉 =∑

m1m2

〈m1m2|jm〉(m1 +m2)|m1m2〉

=∑

m1m2

〈m1m2|jm〉m|m1m2〉

〈m1m2|jm〉(m1 +m2) = 〈m1m2|jm〉m〈m1m2|jm〉 = δm1+m2,m〈m1m2|jm〉, (3.105)

d.h. die z-Quantenzahlen mi addieren sich einfach.SCHRITT 2: Anwendung von J± liefert

J±|jm〉 =√

(j ∓m)(j ±m+ 1)|jm± 1〉=

∑

m1m2

〈m1m2|jm〉√

(j1 ∓m1)(j1 ±m1 + 1)|m1 ± 1,m2〉

+∑

m1m2

〈m1m2|jm〉√

(j2 ∓m2)(j2 ±m2 + 1)|m1,m2 ± 1〉. (3.106)

Daraus folgt direkt durch Projektion (AUFGABE) die Rekursionsformel

√

(j ∓m)(j ±m+ 1)〈m1m2|jm± 1〉 = 〈m1 ∓ 1m2|jm〉√

(j1 ±m1)(j1 ∓m1 + 1)

+ 〈m1m2 ∓ 1|jm〉√

(j2 ±m2)(j2 ∓m2 + 1)

(3.107)


SCHRITT 3: Wir fixieren j, dann sind in die CG-Koeffizienten nur Funktionen Am1m

von m1 und m (auffassen als Matrix oder Gitter) und die Rekursionsformel hat dieStruktur

aAm1,m + bAm1∓1,m + cAm1m±1 = 0. (3.108)

Gitter mit m1 als Zeilenindex undm als Spaltenindex: ausgehend von Aj1j konnen durchzwei unterschiedliche Operationen (entsprechend dem ±) alle Am1m generiert werden.

AUFGABE: Skizzieren Sie die Rekursionsformel auf einem Gitter (m1,m) und fuhrenSie schematisch (z.B. mit Hilfe von Pfeildiagrammen) vor, wie die Rekursion funktioniert.

Der CG-Koeffizient Aj1j ist ausgeschrieben

〈j1m2 = j − j1|jj〉 −j2 ≤ m2 = j − j1 ≤ j2 (3.109)

oder

j1 − j2 ≤ j ≤ j1 + j2 (3.110)

als Bedingung fur j. Jetzt genau dieselbe Konstruktion wie oben, aber die Rolle von m1

und m2 vertauscht: Fuhrt auf

j2 − j1 ≤ j ≤ j2 + j1, (3.111)

insgesamt also zur Dreiecks-Bedingung

|j1 − j2| ≤ j ≤ j1 + j2. (3.112)

Der Drehimpuls j lauft also uber die Werte

j = j1 + j2, j1 + j2 − 1, ..., |j1 − j2|. (3.113)

Die Anzahl der Basis-Kets |jm〉 muss gleich der Anzahl der Basis-Kets |m1m2〉 sein(AUFGABE)

j1+j2∑

j=|j1−j2|(2j + 1) = (2j1 + 1)(2j2 + 1). (3.114)

CG- Koeffizienten sind Koeffizienten einer unitaren Matrix C(m1m2),(jm), wobei µ ≡(m1m2) und ν ≡ (jm) als Multi-Indizes aufgefasst werden. Die CG-Beziehung lautet ja

|jm〉 =∑

m1m2

〈m1m2|jm〉|m1m2〉

|ν〉 =∑

µ

〈µ|ν〉|µ〉. (3.115)

Das ist eine Trafo zwischen orthogonalen Eigenvektoren, die zugehorige Matrix ist alsounitar, sie ist sogar orthogonal (reell), da wegen der Rekursion alle CG reell gewahltwerden konnen. Es gilt also

∑

m1m2

〈m1m2|jm〉〈m1m2|j′m′〉 = δmm′δjj′ . (3.116)


3.7.2 Fall j1 = 12 , j2 = 1

2 (Zwei Spin-12-Teilchen)

Wir haben

j = 1, 0. (3.117)

FALL j = m = 0 oben in√

(j ∓m)(j ±m+ 1)〈m1m2|jm± 1〉 = 〈m1 ∓ 1m2|jm〉√

(j1 ±m1)(j1 ∓m1 + 1)

+ 〈m1m2 ∓ 1|jm〉√

(j2 ±m2)(j2 ∓m2 + 1)

m1 = m2 =1

2 0 = 〈−1

2

1

2|00〉 + 〈1

2− 1

2|00〉. (3.118)

Per Definition (m1 = j1, j = m = 0) ist 〈12− 12 |00〉 = cmit konstantem c, also 〈−1

212 |00〉 =

−c.FALL j = 1 als analog. Insgesamt erhalt man (AUFGABE)

|j = 0m = 0〉 =1√2

(

|12− 1

2〉 − | − 1

2

1

2〉)

|j = 1m = 1〉 = |12

1

2〉

|j = 1m = 0〉 =1√2

(

|12− 1

2〉+ | − 1

2

1

2〉)

|j = 1m = −1〉 = | − 1

2− 1

2〉

(3.119)

Ublicherweise schreibt man hier j = S und m = Sz. Die Zustande zu S = 1 heissenTriplets, die zu S = 0 Singlets. Etwas verkurzt schreibt man die Basis-Zustande

|S〉 =1√2

(| ↑↓〉 − | ↓↑〉)

|T1〉 = | ↑↑〉

|T0〉 =1√2

(| ↑↓〉 + | ↑↓〉)

|T−1〉 = | ↓↓〉. (3.120)

Diese vier Zustande sollte man sich einpragen, sie spielen auch weiter unten eine grosseRolle bei der Diskussion von Zwei-Elektronen-Systemen.

3.7.3 Fall j1 = l (Bahndrehimpuls), j2 = 12 (Spin)

Es ist l = 0, 1, 2, .... Wir haben zunachst

j =1

2, l = 0

j = l +1

2, l − 1

2. (3.121)

CG als AUFGABE.


3.8 Spin-Orbit Coupling and Fine Structure of the Hydrogen Atom

The fine structure is a result of relativistic corrections to the Schrodinger equation,derived from the relativistic Dirac equation for an electron of mass m and charge −e < 0in an external electrical field −∇Φ(r). Performing an expansion in v/c, where v is thevelocity of the electron and c is the speed of light, the result for the Hamiltonian H canbe written as H = H0 + H1, where

H0 = − ~2

2m∆− Ze2

4πε0r(3.122)

is the non-relativistic Hydrogen atom, (Z = 1), and H1 is treated as a perturbation toH0, using perturbation theory. H1 consists of three terms: the kinetic energy correction,the Darwin term, and the Spin-Orbit coupling,

H1 = HKE + HDarwin + HSO. (3.123)

Literature: Gasiorowicz [1] cp. 12 (Kinetic Energy Correction, Spin-Orbit coupling);Weissbluth [2] (Dirac equation, Darwin term); Landau Lifshitz Vol IV chapter. 33,34.

3.8.1 Kinetic Energy and Darwin Term

3.8.1.1 Kinetic Energy Correction

HKE = − 1

2mc2

(p2

2m

)2

. (3.124)

Exercise: Derive this term.

3.8.1.2 Darwin term

This follows from the Dirac equation and is given by

HDarwin =−e~2

8m2c2∆Φ(r), (3.125)

where ∆ is the Laplacian. For the Coulomb potential Φ(r) = Ze/4πε0r one needs

∆1

r= −4πδ(r) (3.126)

with the Dirac Delta function δ(r) in three dimensions.


3.8.2 Spin-Orbit Coupling

This is the most interesting term as it involves the electron spin. Furthermore, this typeof interaction has found a wide-ranging interest in other areas of physics, for example inthe context of spin-electronics (‘spin-transistor’) in condensed matter systems.

The general derivation of spin-orbit coupling from the Dirac equation for an electronof mass m and charge −e < 0 in an external electrical field E(r) = −∇Φ(r) yields

HSO =e~

4m2c2σ[E(r)× p], (3.127)

where p = mv is the momentum operator and σ is the vector of the Pauli spin matrices,

σx ≡(

0 11 0

)

, σy ≡(

0 −ii 0

)

, σz ≡(

1 00 −1

)

. (3.128)

3.8.2.1 Spin-Orbit Coupling in Atoms

In the hydrogen atom, the magnetic moment µ of the electron interacts with the magneticfield B which the moving electron experiences in the electric field E of the nucleus,

B = −v×E

c2. (3.129)

One has

µ = − e

2mgS, g = 2, (3.130)

where g is the g-factor of the electron and

S =1

2~σ (3.131)

is the electron spin operator. Therefore,

− v ×E = v ×∇ Ze

4πε0r= v× r

r

d

dr

Ze

4πε0r=

1

mL

Ze

4πε0r3, (3.132)

where L is the orbital angular momentum operator. This is reduced by an additionalfactor of 2 (relativistic effects) such that

HSO = −µB =Ze2

4πε0

1

2m2c2SL

r3, (3.133)

which introduces a coupling term between spin and orbital angular momentum. Notethat Eq. (3.133) can directly been derived by inserting Eq. (3.132) as E×v = −v×Einto Eq. (3.127).


3.8.2.2 Spin-Orbit Coupling in Solids

In solids, the spin-orbit coupling effect has shot to prominence recently in the contextof spin-electronics and the attempts to build a spin-transistor. The spin-orbit couplingEq. (3.127),

HSO =e~

4m2c2σ[E(r)× p], (3.134)

leads to a spin-splitting for electrons moving in solids (e.g., semiconductors) even inabsence of any magnetic field. Symmetries of the crystal lattice then play a role (Dres-selhaus effect), and in artificial heterostructures or quantum wells, an internal electricfield E(r) can give rise to a coupling to the electron spin. This latter case is called Rashbaeffect.

For a two-dimensional sheet of electrons in the x-y-plane (two-dimensional electrongas, DEG), the simplest case is a Hamiltonian

HSO = −α~

[p× σ]z , (3.135)

where the index z denotes the z component of the operator in the vector product p×σ andα is the Rashba parameter. In the case of the hydrogen atom, this factor was determinedby the Coulomb potential. In semiconductor structures, it is determined by many factorssuch as the geometry.

The Rashba parameter α can be changed externally by, e.g., applying additional‘back-gate’ voltages to the structure. This change in α then induces a change of thespin-orbit coupling which eventually can be used to manipulate electron spins.

3.8.3 Fine Structure

The calculation of the fine structure of the energies for hydrogen now involves two steps:1. as one has degenerate states of H0, one needs degenerate perturbation theory (seebelow). 2. This is, however, simplified by the fact that the corresponding matrix in thesubspace of the degenerate eigenstates can be made diagonal in a suitable basis, usingthe total angular moment

J = L + S. (3.136)

We thus can do the calculation ‘by hand’, using the correct angular momentum eigen-basis.

Including spin, the level En of hydrogen belongs to the states

|nlsmlms〉, s = 1/2, ms = ±1/2, (3.137)

which are eigenstates of L2, S2, Lz, and Sz (‘uncoupled representation’). With L and Sadding up to the total angular momentum J = L+ S, an alternative basis is the ‘coupledrepresentation’

|nlsjm〉, j = l + s, l + s− 1, ..., |l − s|, m = ml +ms. (3.138)


of eigenfunctions of J2, L2, S2, and Jz. Here, s = 1/2 is the total electron spin which ofcourse is fixed and gives the two possibilities j = l + 1/2 and j = l − 1/2 for l ≥ 1 andj = 1/2 for l = 0 (l runs from 0 to n− 1).

The perturbation HSO, Eq. (3.133), can be written in the |nlsjm〉 basis, using

SL =1

2

(

J2 − L2 − S2)

(3.139)

〈nl′sj′m′|SL|nlsjm〉 =1

2~

2 (j(j + 1)− l(l + 1)− s(s+ 1)) δjj′δll′δmm′ .

For fixed n, l, and m, (s = 1/2 is fixed anyway and therefore a dummy index), there arefor l ≥ 1 two states, |nlsj = l ± 1/2m〉, and the two-by-two matrix H of the spin-orbitpart is diagonal,

H ↔ 〈nlsj′m|HSO|nlsjm〉 =Ze2

4πε0

1

2m2c2

⟨1

r3

⟩

nl

1

2~

2

(l 00 −(l + 1)

)

, (3.140)

where⟨

1r3

⟩

nlindicates that this matrix elements has to be calculated with the radial

parts of the wave functions 〈r|nlsj = l ± 1/2m〉, with the result

⟨1

r3

⟩

nl

=Z3

a30

2

n3l(l + 1)(2l + 1), l 6= 0. (3.141)

The resulting energy shifts E′SO corresponding to the two states with j = l ± 1/2 are

E′SO =

Z4e2~2

2m2c2a304πε0

1

n3l(l + 1)(2l + 1)

l, j = l + 1

2−(l + 1), j = l − 1

2

(3.142)

3.8.3.1 Putting everything together

Apart from the corrections E′SO, one also has to take into account the relativistic cor-

rections dur to HKE and HDarwin from section 3.8.1. It turns out that the final re-sult for the energy eigenvalue in first order perturbation theory with respect to H1 =HKE + HDarwin + HSO, Eq. (3.8.1), is given by the very simple expression

Enlsjm = E(0)n +

(E(0)n )2

2mc2

[

3− 4n

j + 12

]

, j = l ± 1

2. (3.143)

Note: j is always positive, l = 0 has only j = 12 , not j = −1

2 . For a detailed derivationof this final result (though I haven’t checked all details), cf. James Branson’s page,

http://hep.ucsd.edu/ branson/or Weissbluth [2], cf. 16.4. Gasiorowicz [1] 12-16 seems to be incorrect.Final remark: we do not discuss the effects of a magnetic field (anamalous Zeeman

effect) or the spin of the nucleus (hyperfine interaction) here. These lead to furthersplittings in the level scheme.


Abbildung 3.2: Fine-Splitting of the hydrogen level En=2, from Gasiorowicz[1]

4. DER ZUSTANDSBEGRIFF

4.1 Zustande in der QM

Systemzustande haben wir bisher durch (normierte) Vektoren (Kets) |Ψ〉 eines Hilbert-raums beschrieben. Bei der Bildung von Erwartungswerten von Observablen A,

〈A〉 ≡ 〈Ψ|A |Ψ〉 (4.1)

kommt es auf einen Phasenfaktor eiα mit reellem α nicht an. Systemzustande werdendeshalb genauer gesagt durch Aquivalenzklassen von Strahlen

eiαΨ, α reell (4.2)

beschrieben.

4.1.1 Reine Zustande

Man kann diesen Phasenfaktor loswerden, indem man zu Projektionsoperatoren 1 ubergeht.

Definition Ein Projektionsoperator (Projektor) P ist ein Operator mit

P 2 = P. (4.3)

Jetzt definiert man

Definition Ein reiner Zustand eines Hilbertraumes H ist durch einen Projektions-operator

PΨ ≡ |Ψ〉〈Ψ|, Ψ ∈ H (4.4)

definiert.

Hier kurzt sich ein Phasenfaktor eiα in |Ψ〉 jetzt heraus. Erwartungswerte von Observa-blen A im reinen Zustand PΨ definieren wir jetzt als Spur:

Definition Der Erwartungswert der Observablen A im reinen Zustand PΨ ist

〈A〉Ψ = Tr (PΨA) = Tr (|Ψ〉〈Ψ|A) , (4.5)

wobei TrX die Spur des Operators X, gebildet mit einem VOS (vollstandigem Ortho-normalsystem) ist,

TrX =∑

n

〈n|X|n〉. (4.6)

1 Definition nicht nur fur Hilbertraume, sondern auch fur Banachraume, vgl. Skript S. 20.

4. Der Zustandsbegriff 53

Die Spur eines Operators ist basisunabhangig: gegeben seien zwei VOS |n〉 und |α〉,∑

n

|n〉〈n| =∑

α

|α〉〈α| = 1 (4.7)

Tr(X) =∑

n

〈n|X|n〉 =∑

n,α

〈n|α〉〈α|X|n〉 = (4.8)

=∑

n,α

〈α|X|n〉〈n|α〉 =∑

α

〈α|X|1|α〉 =∑

α

〈α|X|α〉. (4.9)

Die Spur ist invariant unter zyklischer Vertauschung (AUFGABE),

Tr(AB) = Tr(BA). (4.10)

AUFGABE: Zeige, dass 〈A〉Ψ mit der ublichen Definition ubereinstimmt, d.h. zeige〈A〉Ψ = 〈Ψ|A |Ψ〉.

4.1.2 Gemischte Zustande

Die QM enthalt durch die Kopenhagener Deutung ein intrinsisches Element an Stochasti-zitat: Voraussagen fur Messungen sind Aussagen uber Wahrscheinlichkeiten, selbst wennder Zustand Ψ des Systems zu einer bestimmten Zeit t exakt bekannt ist.

Es gibt nun Falle, wo ‘durch die Dummheit der Menschen’ selbst der Zustand desSystems zu einer bestimmten Zeit t nicht exakt bekannt ist. Das ist wie in der klassischenMechanik, wo man statt eines Punktes im Phasenraum nur eine gewisse Verteilung imPhasenraum angeben kann. Die Bestimmung dieser Verteilung ist Aufgabe der Statistik(Ensembletheorie, Thermodynamik). Wir definieren:

Definition Uber ein quantenmechanisches System im Hilbertraum H mit VOS (voll-standigem Orthonormalsystem) |n〉 liege nur unzureichende Information vor: Mit Wahr-scheinlichkeit pn befinde es sich im Zustand |n〉. Die Menge (pn, |n〉) heisst Ensemblevon reinen Zustanden. Dann ist der Erwartungswert einer Observablen A gegebendurch

〈A〉 ≡∑

n

pn〈n|A|n〉,∑

n

pn = 1, 0 ≤ pn ≤ 1. (4.11)

Definition Der Dichteoperator (Dichtematrix) ρ eines Ensembles (pn, |n〉) ist

ρ ≡∑

n

pn|n〉〈n|, (4.12)

d.h. eine Summe von Projektionsoperatoren auf die reinen Zustande |n〉. Es gilt

〈A〉 ≡∑

m

pm〈m|A|m〉 =∑

m

∑

n

pm〈m|n〉〈n|A|m〉 (4.13)

=∑

m

〈m(∑

n

pn|n〉〈n|)

A|m〉 = Tr(ρA). (4.14)

Der Erwartungswert druckt sich also wieder mittels der Spur aus.


4.2 Eigenschaften des Dichteoperators

Normierung: Fur Dichteoperatoren ρ gilt

Tr(ρ) =∑

n,m

pn〈m|n〉〈n|m〉 = 1. (4.15)

Hermitizitat: Weiterhin in einer beliebigen Basis gilt fur Dichteoperatoren ρ

〈α|ρ|β〉 =∑

n

pn〈α|n〉〈n|β〉 =∑

n

pn〈β|n〉∗〈n|α〉∗ = 〈β|ρ|α〉∗ (4.16)

ρ = ρ†, ρ ist Hermitesch . (4.17)

Positivitat: Es gilt (AUFGABE)

〈ψ|ρ|ψ〉 ≥ 0 ρ ≥ 0 (4.18)

fur beliebige Zustande ψ.

4.2.1 Charakterisierung des Dichteoperators

Es gilt

Satz 3. Ein Operator ρ ist genau dann Dichteoperator zu einem Ensemble, wenn Tr(ρ) =1 und ρ ≥ 0.

Beweis: Sei ρ ≥ 0, dann ist ρ auch Hermitesch (AUFGABE) und hat deshalb eineZerlegung ρ =

∑

l λl|l〉〈l| mit reellen Eigenwerten, die wegen der Positivitat λl ≥ 0sind, weshalb mit der Normierung Tr(ρ) = 1 =

∑

l λl die Darstellung ρ =∑

l λl|l〉〈l|ein Ensemble (λl, |l〉) beschreibt. Die umgekehrte Richtung erfolgt aus Gl.(4.15), (4.18).QED.

Man beachte, dass ein Dichteoperator ρ ein Ensemble nicht eindeutig festlegt: Ver-schiedene Ensembles (λl, |l〉), (pn, |n〉) konnen zu ein und demselbem ρ gehoren (s.u.)

Es gilt (AUFGABE)

Tr(ρ2) ≤ 1. (4.19)

Insbesondere definiert man

ρ2 = ρ, Tr(ρ2) = 1, reiner Zustand. (4.20)

ρ2 6= ρ, Tr(ρ2) < 1, gemischter Zustand. (4.21)

Ein reiner Zustand hat die Form ρ = |Ψ〉〈Ψ|, d.h. alle Wahrscheinlichkeiten pn sind Nullbis auf die eine, die eins ist und zum Projektor |Ψ〉〈Ψ| gehort.


4.2.2 Entropie

Man definiert die von-Neumann-Entropie eines Zustands ρ als

S = −kBTr (ρ ln ρ) = −kB

∑

n

pn ln pn. (4.22)

Hierbei hat man im letzten Schritt die Diagonaldarstellung von ρ =∑

n pn|n〉〈n| benutzt,sowie die Funktion eines Operators X in Diagonaldarstellung,

X =∑

n

xn|n〉〈n| f(X) =∑

n

f(xn)|n〉〈n|. (4.23)

(uberprufe mit Taylor-Entwicklung von f). Es gilt

S = 0, reiner Zustand (4.24)

S > 0, gemischter Zustand. (4.25)

4.2.3 Thermische Zustande

Fur diese haben wir die pn in der Quantenstatistik bestimmt. Fur das kanonischeEnsemble mit Hamiltonoperator H und Eigenwertgleichung H|n〉 = En|n〉 haben wir

ρ =∑

n

e−βEn

Z|n〉〈n| = 1

Ze−βH , Z = Tre−βH . (4.26)

Hierbei ist die Exponentialfunktion eines Operators einfach uber die Potenzreihedes Operators definiert.

4.2.4 Zeitentwicklung, Liouville-von-Neumann-Gleichung

Jetzt betrachten wir die unitare Zeitentwicklung eines Zustands ρ, die einfach aus derSchrodingergleichung folgt:

i∂

∂t|Ψ(t)〉 = H|Ψ(t)〉 |Ψ(t)〉 = U(t)|Ψ(0)〉 (4.27)

i∂

∂tU(t) = HU(t) (4.28)

U(t) = e−iHt, Zeitentwicklungsoperator. (4.29)

Es folgt nun

ρ(t) =∑

n

pn|n(t)〉〈n(t)| =∑

n

pnU(t)|n(0)〉〈n(0)|U †(t) (4.30)

= U(t)ρ(0)U †(t), (4.31)


und die Zeitentwicklung des Zustands ist

i∂

∂tρ(t) = = HU(t)ρ(0)U †(t)− U(t)ρ(0)U †(t)H (4.32)

= [H, ρ(t)], Liouville-von-Neumann-Gleichung . (4.33)

Da U(t) unitar, ist die Zeitentwicklung von ρ(t) unitar. Das wird sich andern (s.u.), wennwir Information aus ρ(t) ‘herausreduzieren´ (reduzierte Dichtematrix). Die Liouville-von-Neumann-Gleichung ist der Ausgangspunkt der Nichtgleichgewichts-Quantenstatistik.Aus ρ(t) = U(t)ρ(0)U †(t) folgt weiterhin

[ρ(0),H] = 0 ρ(t) = ρ(0) ∀t, (4.34)

d.h. falls der Zustand ρ mit dem Hamiltonian fur eine bestimmte Zeit (t = 0 hier)vertauscht, ist er fur alle Zeiten konstant (auch t < 0)! 2 Deshalb die

Definition Ein Gleichgewichtszustand eines durch einen Hamiltonian H beschrie-benen Systems ist ein Zustand ρ, der mit H vertauscht.

4.2.5 Spezialfall Spin 1/2: die Bloch-Sphare

Der Dichteoperator ρ ist in diesem Fall eine hermitische 2 mal 2 Matrix mit Spur 1, siekann geschrieben werden als

ρ ≡ 1

2

(1 + pσ

)(4.35)

mit einem reellen dreikomponentigem Vektor p und dem Vektor σ der Pauli-Matrizen.Mit den Eigenwerten λ1, λ2 von ρ gilt

λ1λ2 = det ρ =1

4

(1− p2

). (4.36)

Damit gilt: a) ρ ≥ 0 positiv (semi)definit λ1λ2 ≥ 0 |p| ≤ 1. b) |p| ≤ 1 λ1λ2 ≥ 0,wegen λ1 + λ2 konnen nicht beide λi negativ sein λ1 ≥ 0, λ2 ≥ 0 ρ ≥ 0. Also: ρ istDichtematrix ↔ |p| ≤ 1 . Es gilt also

Satz 4. Es gibt eine 1-zu-1-Beziehung zwischen allen Dichtematrizen eines Spin 1/2und den Punkten der Einheitskugel (‘Bloch-Sphare’) |p| ≤ 1.

AUFGABE: 1. Die Erwartungwerte der Spinkomponenten im Zustand ρ erfullen

〈σi〉 = pi, i = 1, 2, 3. (4.37)

2. Die reinen Zustande entsprechen dem Rand der Bloch-Sphare |p| = 1.

2 vgl. mit holomorphen Funktionen: f(z) = const auf einem (kleinen) Gebiet der komplexen Ebene f(z) =const uberall.


4.3 Zusammengesetzte Systeme

Ein System kann aus N Teilsystemen bestehen (Engl. ‘N-partite’) und wird quanten-mechanisch durch das Tensorprodukt der Hilbertraume der Teilsysteme beschrieben,

H = H1 ⊗ ...⊗HN . (4.38)

(großter Bruch mit der klassischen Physik, vgl. oben).

4.3.1 Bipartite Systeme

In der Quantenstatistik betrachtet man haufig bipartite Systeme (N = 2),

H = HA ⊗HB. (4.39)

z.B. aus System und Bad.Wir betrachten den Fall, wo sich das Gesamtsystem in H (z.B. System plus Bad) mit

Wahrscheinlichkeit eins in einem reinen Zustand |Ψ〉 befindet, der zerlegt wird als

|Ψ〉 =∑

ab

cab|a〉 ⊗ |b〉 (4.40)

wobei |a〉 ein VOS in HA und |b〉 ein VOS in HB. Die Matrix c ist i. A. rechteckig,

c↔

c11 ... ... c1N

c21 ... ... c2N

... ... ... ...cM1 ... ... cMN

, dim(HA) = M, dim(HB) = N. (4.41)

4.3.2 Reduzierte Dichtematrix

Wir wollen uns jetzt nur fur (reduzierte) Information uber das System A interessie-ren, d.h. Erwartungswerte aller Observablen A des Systems A, aber nicht von B. DieseErwartungswerte sind

〈A〉 = 〈Ψ|A⊗ 1|Ψ〉, Observable operiert nur in HA (4.42)

=∑

aba′b′

c∗abca′b′〈b| ⊗ 〈a|A⊗ 1|a′〉 ⊗ |b′〉 (4.43)

=∑

aba′

c∗abca′b〈a|A|a′〉 (4.44)

≡ TrA(ρAA), ρA ≡∑

aba′

c∗abca′b|a′〉〈a|. (4.45)

Hiermit wird der reduzierte Dichteoperator ρA des Teilsystems A definiert, dessenKenntnis die Berechnung samtlicher Erwartungswerte in A ermoglicht. Die Spur TrA


wird dabei nur im Teilraum A ausgefuhrt! Es gilt

ρA = TrB (|Ψ〉〈Ψ|) =∑

aba′b′

c∗abca′b′TrB(|a′〉 ⊗ |b′〉〈b| ⊗ 〈a|

)(4.46)

=∑

aba′

c∗abca′b|a′〉〈a| =∑

aa′

(

cc†)

a′a|a′〉〈a| (4.47)

(

cc†)

=

c11 ... ... c1N

c21 ... ... c2N

... ... ... ...cM1 ... ... cMN

c∗11 ... c∗M1

c∗12 ... c∗M2

... ... ...

... ... ...c∗1N ... c∗MN

. (4.48)

Weiterhin gilt (AUFGABE)

ρA = ρ†A, TrAρA = 1, ρA ≥ 0. (4.49)

Deshalb definiert ρA tatsachlich einen Dichteoperator in HA.

4.3.3 Reine und verschrankte Zustande

Definition Zustande |Ψ〉 eines bipartiten Systems H = HA ⊗HB heissen reine Ten-soren (separabel), falls sie sich in der Form

|Ψ〉 = |φ〉A ⊗ |φ′〉B , reiner Tensor (4.50)

mit (normierten) |φ〉A ∈ HA und |φ′〉B ∈ HB schreiben lassen. Zustande |Ψ〉, die sichnicht als reine Tensoren schreiben lassen, heissen verschrankt.

Wenn |Ψ〉 separabel ist, folgt fur die zugehorigen reduzierten Dichteoperatoren

ρA = TrB |φ〉A ⊗ |φ′〉B〈φ′| ⊗ 〈φ| (4.51)

= |φ〉A〈φ| × TrB |φ′〉B〈φ′| = |φ〉A〈φ| (4.52)

ρ2A = ρA, |Ψ〉 separabel ρA rein. (4.53)

Entsprechend ist dann auch ρB rein. Wenn |Ψ〉 verschrankt ist, gilt nicht mehr ρ2A = ρA,

es muss also TrAρ2A < 1 gelten und deshalb

|Ψ〉 verschrankt ρA Gemisch und ρB Gemisch.

Beispiele:a) 2-Qubit

|Ψ〉 ≡ a|00〉+ b|11〉 ≡ a|0〉A ⊗ |0〉B + b|1〉A ⊗ |1〉B (4.54)

ρA = |a|2|0〉〈0|A + |b|2|1〉〈1|A. (4.55)

AUFGABE: Ist |Ψ〉 verschrankt ?b) 2-Qubit

|Ψ〉 ≡ 1√2(|01〉 + |11〉) (4.56)

AUFGABE: Berechne hierfur ρA. Ist |Ψ〉 verschrankt ?


4.4 Erganzung: Die Schmidt-Zerlegung

Ausgehend von einem reinen Zustand |Ψ〉 eines bipartiten Systems H = HA⊗HB erhaltman die reduzierten Dichtematrizen ρA (fur System A) und ρB (fur System B). Wiehangen ρA und ρB miteinander zusammen, und wie lassen sie sich charakterisieren? Wirbeginnen mit einem

Satz 5. Jeder Zustand (Tensor) |Ψ〉 eines bipartiten Systems H = HA⊗HB kann zerlegtwerden als

|Ψ〉 =∑

n

λn|αn〉 ⊗ |βn〉, λn ≥ 0, (4.57)

wobei |α〉 ein VOS in HA und |β〉 ein VOS in HB ist.

Beweis: Zunachst fur dim(HA) = dim(HB) = N (endlichdimensional). Ein Tensorlasst sich immer schreiben als |Ψ〉 =

∑

ab cab|a〉 ⊗ |b〉 mit |a〉 ein VOS in HA und|b〉 ein VOS in HB. Wir zerlegen die quadratische Matrix C (Elemente cab) mit derSingularwertzerlegung (singular value decomposition) (s.u.),

C = UDV, U ,V unitar, D = diag(λ1, ..., λN ), λn ≥ 0 diagonal. (4.58)

Man hat dann

|αn〉 ≡∑

a

Uan|a〉, |βn〉 ≡∑

b

Vnb|b〉 (4.59)

|Ψ〉 =∑

abn

UanDnnVnb|a〉 ⊗ |b〉 =∑

n

λn|αn〉 ⊗ |βn〉, (4.60)

wie behauptet. QED. Hier ist noch das

Satz 6. Singularwertzerlegung: Jede quadratische Matrix A lasst sich zerlegen als

A = UDV, U ,V unitar, D = diag(λ1, ..., λN ), λn ≥ 0 diagonal. (4.61)

Die Diagonalelemente von D heissen die singularen Werte der Matrix A.

Bemerkung: diese Zerlegung ist allgemeiner als die Spektralzerlegung fur hermite-sche Matrizen H, H = UDU † mit diagonalem (aber nicht unbedingt positivem) D undunitarem U und kann auch einfach auf rechteckige Matrizen erweitert werden. Literatur:NIELSEN/CHUANG.

Diskussion der Schmidt-Zerlegung:

• Man hat also statt der doppelten Summe |Ψ〉 =∑

ab cab|a〉 ⊗ |b〉 nur eine einfacheSumme |Ψ〉 =∑n λn|αn〉 ⊗ |βn〉, λn ≥ 0.


• Folgerung fur die reduzierten Dichtematrizen ρA (fur System A) und ρB (fur Sys-tem B):

ρA =∑

n

λ2n|αn〉〈αn|, ρB =

∑

n

λ2n|βn〉〈βn|, (4.62)

d.h. die reduzierten Dichtematrizen in beiden Teilsystemen haben dieselben Eigen-werte λ2

n! Die Kenntniss von ρA und ρB ist allerdings nicht ausreichend, um denZustand |Ψ〉 zu rekonstruieren: die Information uber die Phasen der |αn〉, |βn〉 istin ρA und ρB nicht enthalten!

• Fur zwei verschiedene Ausgangszustande |Ψ〉 und |Ψ′〉 erhalt man i.A. Schmidt-Zerlegungen mit verschiedenen VOS in HA und HB. Die VOS |α〉 und |β〉 inder Schmidt-Zerlegung |Ψ〉 =

∑

n λn|αn〉 ⊗ |βn〉 hangen also ganz vom Ausgangs-zustand |Ψ〉 ab.

• Die von-Neumann-Entropie S = −kB∑

n λ2n lnλ2

n ist dieselbe fur beide ZustandeρA und ρB.

• Fur dim(HA) 6= dim(HB) gibt es in genau der gleichen Weise eine Schmidt-Zerlegung, nur dass einige der λn dann Null sein konnen.

• Die Anzahl der von Null verschiedenen Eigenwerte λn in der Schmidt-Zerlegungvon |Ψ〉 heisst Schmidt-Zahl nS. Fur nS = 1 ist der Zustand |Ψ〉 separabel, furnS > 1 ist er verschrankt.

4.5 Verschrankung

4.5.1 Korrelationen in Spin-Singlett-Zustanden

Wir gehen von zwei Spins A und B aus, die auf zwei Teilchen A und B lokalisiert sind.Die Teilchen werden zusammengefuhrt und die beiden Spins in einem Singlett-Zustand

|S〉 =1√2

(| ↑, z〉| ↓, z〉 − | ↓, z〉| ↑, z〉) (4.63)

prapariert. Die Teilchen A und B werden jetzt getrennt, dabei sollen die Spinfreiheitsgra-de vollstandig unverandert bleiben. Zwei raumartig getrennte Beobachter A (Alice)und B (Bob) haben anschliessend Teilchen A und B bei sich.

Die beiden Spins sind miteinander verschrankt: die reduzierte Dichtematrix in Bob’sSystem z.B. ist ein Gemisch,

ρB = Tr|S〉〈S| = 1

2

(1 00 1

)

. (4.64)

Definition Ein solches Paar zweier verschrankter Spins heisst Einstein-Podolsky-Rosen (EPR)-Paar 3.

3 Phys. Rev. 47, 777 (1935).


1. A misst ihren Spin in z-Richtung. Damit ist Bobs Zustand automatisch festgelegt.Findet A z.B. spin-up | ↑, z〉A, so muss Bob | ↓, z〉B messen.

Bis hierher noch nicht so aufregend, vgl. klassisches Experiment mit einer weissenund einer schwarzen Kugel jeweils in einem Kasten: wenn A ihren Kasten offnet, weisssie, was B hat.

2. A misst ihren Spin in z-Richtung. Bob misst in x-Richtung und bekommt up oderdown mit je Wahrscheinlichkeit 1/2 unabhangig von A’s Resultat.

3. A misst ihren Spin in x-Richtung. Findet A z.B. spin-up | ↑, x〉A, so muss Bob| ↓, x〉B messen.

Bob’s Messergebnisse an seinem Spin hangen davon ab, wie und ob Alice misst,obwohl beide raumartig voneinander getrennt sind. Trotzdem lasst sich dadurch keineInformation mit Uberlichtgeschwindigkeit ubertragen: die Resultate von Bobs Messun-gen werden durch seine Dichtematrix ρB bestimmt, da er die Information uber AlicesMessergebnisse nicht hat. Wenn A und B 10000 Exemplare von |S〉 haben, wird B beiseinen Messungen (egal in welche Richtung) immer nur eine zufallige Folge von up oderdown finden. Erst wenn sich die beiden zusammensetzen und ihre Messergebnisse ver-gleichen, werden sie Korrelationen zwischen ihren Messwerten finden.

Einstein war mit dieser Tatsache unzufrieden und forderte

Definition “Einstein-Lokalitat”: Fur ein EPR-Paar sollte die ‘gesamte Physik’ in B nurlokal durch den Spin B gegeben sein und z.B nicht davon abhangen, was in A passiert(z.B. davon, was und ob A misst). Eine vollstandige Beschreibung der Physik in B sollteergeben, dass der Spin B nicht mehr mit Spin A korreliert ist.

Nach diesem Kriterium ist die Quantenmechanik eine unvollstandige Beschreibung derNatur. Versteckte-Variablen-Theorien versuchen, hier einen Ausweg zu finden:

Definition (Versteckte-Variablen-Theorie fur Spin): Ein Spin ↑ in n-Richtung wird inWirklichkeit durch einen Zustand | ↑ n, λ〉 beschrieben, wobei λ uns noch unbekann-te Parameter sind. Ware λ bekannt, so waren alle Messwerte des Spins deterministisch.Weil λ unbekannt ist, erhalten wir in der QM zufallige Messwerte.

Es gibt jetzt allerdings Ungleichungen, mit denen Versteckte-Variablen-Theorien ex-perimentell getestet werden konnen (Bellsche Ungleichungen).

4.5.2 Bellsche Ungleichungen

Wir diskutieren hier eine Variante fur unser Spin-Singlett |S〉: Alice misst ihren Spin inn-Richtung. Erhalt sie | ↓, n〉A, so ist Bobs Spin im Zustand | ↑, n〉B (AUFGABE), wobei

| ↑, n〉B =

(cos θ

2e−iφ/2

sin θ2e

iφ/2

)

= cosθ

2e−iφ/2| ↑, z〉B + sin

θ

2eiφ/2| ↓, z〉B (4.65)


Abbildung 4.1: (Aus Sakurai, ‘Modern Quantum Mechanics’)

vgl. Gl.(3.86). Misst Bob seinen Spin in z-Richtung, so findet er up mit Wahrscheinlich-keit cos2 θ

2 und down mit Wahrscheinlichkeit sin2 θ2 , es gilt also

W (↓A, ↓B) = W (↑A, ↑B) =1

2sin2 θ

2(4.66)

W (↓A, ↑B) = W (↑A, ↓B) =1

2cos2 θ

2, (4.67)

wobei der Faktor 1/2 die Wahrscheinlichkeit ist, dass Alice up (bzw. down) misst.Wie waren die entsprechenden Wahrscheinlichkeiten in einer Versteckte-Variablen-

Theorie? Alice und Bon mogen ihren Spin in insgesamt einer der drei Richtungen a, b, cmessen. Versteckte Variablen legen fest, ob dabei + (Spin up) oder − (Spin down) her-auskommt. Die Variablen sind unbekannt, man kann aber eine Einteilung in 8 Zustandevornehmen. In jedem dieser 8 Zustnde ist das Messergebnis festgelegt, wir wissen abernicht, welchen der 8 wir messen (siehe Figur). Hier treten 8 relative Haufigkeiten

pi ≡Ni

∑8i=1Ni

, 0 ≤ pi ≤ 1, i = 1, ..., 8. (4.68)

auf. Wenn das Gesamtsystem z.B. im Zustand 3 ist, erhalt Alice (particle 1) bei Messungin einer von ihr gewahlten Richtung immer einen Wert (z.B. + fur a), der unabhangigdavon ist, was Bob macht.

Wir konnen jetzt Wahrscheinlichkeiten P (a+; b+) etc. angeben, z.B.

P (a+; b+) = p3 + p4 (4.69)


durch Nachschauen in der Tabelle. Entsprechend

P (a+; c+) = p2 + p4, P (c+; b+) = p3 + p7. (4.70)

Wegen pi ≥ 0 gilt trivialerweise p3 + p4 ≤ p3 + p4 + p2 + p7 oder

P (a+; b+) ≤ P (a+; c+) + P (c+; b+). (4.71)

Das ist bereits eine Form der Bellschen Ungleichungen. Wir vergleichen nun mit derquantenmechanischen Vorhersage: P (a+; b+) entspricht W (↑ a+; ↑ b+) = 1

2 sin2 θab

2 ,

wobei θab der Winkel zwischen a und b ist. Wir zeigen nun: die quantenmechanischen(gemeinsamen) Wahrscheinlichkeiten W (↑ a+; ↑ b+) verletzen die Bellschen Ungleichun-gen, GL. (4.71). Es reicht, das mit einer bestimmten Wahl der Achsen a, b, c zu zeigen:c symmetrisch zwischen a und b,

θ = θac = θcb, θab = 2θ. (4.72)

Mit P = W hatte man

sin2 θ ≤ sin2 θ

2+ sin2 θ

2= 2 sin2 θ

2Widerspruch (4.73)

Widerspruch fur 0 < θ < π2 . Daraus folgt, das die Quantenmechanik nicht mit den

Bellschen Ungleichungen vereinbar ist. Die Bellschen Ungleichungen konnen andererseitsexperimentell uberpruft werden; bisher fand man stets eine Verletzung der BellschenUngleichungen, also keinen Widerspruch zur Quantenmechanik.

Literatur: J. J. Sakurai “Modern Quantum Mechanics”, Benjamin 1985.

5. STORUNGSTHEORIE

5.1 Zeitunabhangige Storungstheorie

Gegeben sei ein Hamiltonian

H = H0 +H1 (5.1)

in Hilbertraum H mit diskretem Spektrum. Das Eigenwertproblem von H0 sei bereitsgelost,

H0|i, ν〉 = εi|i, ν〉, i = 1, 2, ...; ν = 1, 2, ..., di, (5.2)

wobei di die Entartung des i-ten Eigenwertes sei. Die Kets |i, ν〉 seien ein VOS in H mit

H = H1 ⊕H2 ⊕H3..., (5.3)

also einer Zerlegung in orthogonale Teilraume zu den Eigenwerten εi.Aufgabe: Losung des Eigenwertproblems

H|Ψ〉 = E|Ψ〉. (5.4)

inH. Idee: wir betrachten H1 als ‘kleine Storung’ von H0, die zu einer ‘kleinen Anderung’der εi und |i, ν〉 fuhrt.

5.1.1 Projektor-Methode

Lit.: SCHERZ. Wir wollen die durch H1 verursachte Korrektur der Energie εb 6= 0bestimmen. Wir haben also

H0|bν〉 = εb|bν〉, ν = 1, 2, ..., d (5.5)

mit dem d-fach entarteten Eigenwert εb im Teilraum Hb. Alles Folgende bezieht sich aufdiesen festen Teilraum und die feste Energie εb.

Wir subtrahieren εb in

h ≡ H − εb1, h0 ≡ H0 − εb1, ε = E − εb (5.6)

zum Umschreiben von

H|Ψ〉 = E|Ψ〉 h|Ψ〉 = (h0 +H1)|Ψ〉 = ε|Ψ〉. (5.7)

5. Storungstheorie 65

Jetzt definieren wir einen Projektor P und Q mit folgenden Eigenschaften,

P ≡d∑

ν=1

|bν〉〈bν|, Q ≡ 1− P, PQ = QP = 0 (5.8)

Es gilt weiterhin (AUFGABE)

[H0, P ] = [h0, P ] = [H0, Q] = [h0, Q] = 0. (5.9)

Deshalb folgt

h0Q|Ψ〉 = Qh0|Ψ〉 = Q(ε−H1)|Ψ〉. (5.10)

Der Operator h0 lasst sich jetzt im Teilraum H⊖Hb invertieren,

h−10 =

1

H0 − εb= − 1

εb

1

1− H0εb

(5.11)

(formale Potenzreihe wie bei geometrischer Reihe): wurde man h−10 auf einen Vektor aus

Hb loslassen, wurde man durch Null teilen und die Inverse ware nicht definiert, fur alleanderen Vektoren aus H ⊖Hb ist die Inverse ist definiert - solche Vektoren sind genauvon der Form Q|Ψ〉. Damit hat man

Q|Ψ〉 = h−10 Q(ε−H1)|Ψ〉 Q|Ψ〉 = R(εb)(ε−H1)|Ψ〉 (5.12)

R(εb) ≡ Q1

H0 − εbQ, Resolvente (Pseudo-Inverse). (5.13)

Hierbei haben wir auch von links Q heranmultipliziert, um sicherzustellen, dass auch dieOperation auf Dirac-Bras definiert ist, z.B. 〈Ψ|Qh−1

0 Q.Im letzten Schritt schreiben wir jetzt noch um,

Q|Ψ〉 = R(εb)(ε −H1)|Ψ〉 (5.14)

|Ψ〉 = P |Ψ〉+R(εb)(ε−H1)|Ψ〉 (5.15)

= P |Ψ〉+R(εb)(ε−H1) [P |Ψ〉+R(εb)(ε−H1)|Ψ〉] (5.16)

=∞∑

n=0

[R(εb)(ε−H1)]n P |Ψ〉. (5.17)

Bis hierhin ist alles nur eine formal exakte Umformung.

5.1.2 Auswertung fur die Eigenwerte

Wir wenden den Projektor P auf

(h0 +H1)|Ψ〉 = ε|Ψ〉. (5.18)


an und erhalten

εP |Ψ〉 = P (h0 +H1)|Ψ〉 = h0P |Ψ〉+ PH1|Ψ〉 = PH1|Ψ〉, (5.19)

denn P |Ψ〉 liegt ja in Hb und wird deshalb von h0 ≡ H0− εb1 annulliert. Damit hat man

εP |Ψ〉 = PH1

∞∑

n=0

[R(εb)(ε−H1)]n P |Ψ〉. (5.20)

Die Energie-Korrektur ε schreiben wir jetzt als

ε = ε(1) + ε(2) + ε(3) + ..., (5.21)

wobei der Korrektur-Anteil ε(i) von der Ordnung O(H1) sein soll. Formal kann man aucheine Taylor-Reihe ansetzen

H = H0 + λH1, ε = λε(1) + λ2ε(2) + λ3ε(3) + ... (5.22)

und Koeffizientenvergleich in Potenzen von λ machen.

5.1.2.1 Erste Ordnung Storungstheorie: Energien

In niedrigster (erster) Ordnung in H1 (λ) erhalt man

ε(1)P |Ψ〉 = PH1P |Ψ〉 (5.23)d∑

ν=1

|bν〉〈bν|[

H1

d∑

ν′=1

|bν ′〉〈bν ′| − ε(1)]

|Ψ〉 = 0 (5.24)

d∑

ν′=1

〈bµ|H1|bν ′〉〈bν ′|Ψ〉 − ε(1)〈bµ|Ψ〉 = 0. (5.25)

Das ist eine Eigenwertgleichung im Unterraum Hb fur die Hermitesche d mal d MatrixPH1P , deren Losung d Eigenwerte ε(1) und d Eigenvektoren |Ψ〉 liefert.

Im Spezialfall d = 1 (keine Entartung) hat man einfach

〈b|H1|b〉〈b|Ψ〉 − ε(1)〈b|Ψ〉 = 0, (5.26)

was auf das ausserordentlich wichtige Resultat (Prufung!)

ε(1) = 〈b|H1|b〉, 1. Ordnung Storungstheorie ohne Entartung (5.27)

fuhrt. Die ungestorte Energie εb zum ungestorten Zustand |b〉 verschiebt sich also umdas Diagonalmatrixelement des Storoperators,

εb → εb + 〈b|H1|b〉, (5.28)

der Eigenwert wird verschoben. Im Fall d > 1 wird die d-fache Entartung des Eigenwertsεb ganz oder nur teilweise aufgehoben,

εb → εb + ε(1)ν , ν = 1, 2, ..., d, (5.29)

wobei die Anzahl der Losungen ε(1) von der Stor-Matrix PH1P abhangt: ‘die Liniespaltet auf’.


5.1.2.2 Zweite Ordnung Storungstheorie: Energien

In zweiter Ordnung in H1 (λ) erhalt man

(ε(1) + ε(2) + ...)P |Ψ〉 = PH1P |Ψ〉+ PH1

[

R(εb)(ε(1) + ε(2) + .... −H1)

]

P |Ψ〉

(ε(1) + ε(2))P |Ψ〉 = PH1P |Ψ〉 − PH1R(εb)H1P |Ψ〉. (5.30)

Man beachte, dass man in der letzten Gleichung auf der rechten Seite ε(1)+ε(2) weglassenmuss, da diese Terme von dritter bzw. vierter Ordnung in H1 sind. Man hat also fur diezweite Ordnung analog zur ersten Ordnung

ε(2)P |Ψ〉 = −PH1R(εb)H1P |Ψ〉 (5.31)

−d∑

ν=1

〈bµ|H1R(εb)H1|bν〉〈bν|Ψ〉 − ε(2)〈bµ|Ψ〉 = 0, (5.32)

die zweite Korrektur ist i. A. also wieder aus der Losung eines d mal d Eigenwertproblemszu erhalten. Explizit hat man fur die entsprechend zu diagonalisierende Matrix

−H1R(εb)H1 = −H1Q1

H0 − εbQ∑

i

di∑

ν=1

|iν〉〈iν|H1 (5.33)

= −H1Q1

H0 − εb∑

i6=b

di∑

ν=1

|iν〉〈iν|H1 (5.34)

= −H1Q1

εi − εb∑

i6=b

di∑

ν=1

|iν〉〈iν|H1 (5.35)

=∑

i6=b

di∑

ν=1

H1|iν〉〈iν|H1

εb − εi. (5.36)

(5.37)

Im Fall d = 1 (keine Entartung) wird das wesentlich ubersichtlicher: man bekommt

∑

i6=b

〈b|H1|i〉〈i|H1

εb − εi|b〉〈b|Ψ〉 − ε(2)〈b|Ψ〉 = 0 (5.38)

ε(2) =∑

i6=b

〈b|H1|i〉〈i|H1|b〉εb − εi

=∑

i6=b

|〈b|H1|i〉|2εb − εi

. (5.39)

Dieses ist wiederum ein wichtiges Resultat. Insbesondere sieht man: ist die ungestorteEnergie εb die Grundzustandsenergie, so fuhrt der Term zweiter Ordnung in der Storungstheoriezu einer negativen Korrektur, d.h. zu einer Absenkung der Energie. Das ist insbesonderedann wichtig, wenn z.B. durch Auswahlregeln der Term erster Ordnung Null ist.


5.1.3 Auswertung fur die Zustande

Wir betrachten wiederum einen d-fach entarteten Eigenwert εb von H0, H0|bν〉 = εb|bν〉,mit einer ON-Basis |bν〉 im Teilraum Hb. Wir schreiben

H|Ψ〉 = E|Ψ〉 (5.40)

(H0 +H1)(|Ψ0〉+ |Ψ1〉+ ...) = (εb + ε(1) + ε(2) + ...)(|Ψ0〉+ |Ψ1〉+ ...) (5.41)

5.1.3.1 Erste Ordnung Storungstheorie fur die Zustande, d = 1 (keine Entartung)

Dieser Fall ist der einfachste. Wir haben |Ψ0〉 = |b〉. Wir sortieren alle Beitrage, dieerster Ordnung in H1 sind:

(H0 +H1)(|b〉+ |Ψ1〉) = (εb + ε(1))(|b〉 + |Ψ1〉+ ...) (5.42)

〈i|(H0 +H1)(|b〉+ |Ψ1〉) = 〈i|(εb + ε(1))(|b〉+ |Ψ1〉+ ...), i 6= b (5.43)

〈i|H1|b〉+ εi〈i|Ψ1〉 = εb〈i|Ψ1〉 (5.44)

〈i|Ψ1〉 =〈i|H1|b〉εb − εi

(5.45)

Die Korrektur |Ψ1〉 besteht also aus Komponenten, die orthogonal zu |b〉 sind. Der Zu-stand |Ψ〉 ist also

|Ψ〉 =

|b〉+∑

i6=b

|i〉〈i|Ψ1〉+ ...

=

1 +∑

i6=b

|i〉〈i|εb − εi

H1 + ...

|b〉 (5.46)

5.1.3.2 Erste Ordnung Storungstheorie fur die Zustande, d > 1 (Entartung)

Das ist letztlich auch nicht schwieriger. Statt im eindimensionalen, von |Ψ0〉 = |b〉 auf-gespannten Unterraum arbeitet man jetzt im Unterraum Hb. Als Bais nimmt man die dEigenzustande |Ψ0ρ〉 aus der Bestimmung der Eigenenergien in erster Ordnung,

ε(1)P |Ψ0ρ〉 = PH1P |Ψ0ρ〉. (5.47)

Jetzt benutzen wir unsere allgemeine Gleichung (5.14),

|Ψ〉 =

∞∑

n=0

[R(εb)(ε−H1)]n P |Ψ〉 = P |Ψ〉+R(εb)(ε −H1)P |Ψ〉+ ... (5.48)

und schreiben

Q(|Ψ0ρ〉+ |Ψ1ρ〉+ ...) = QP |Ψ〉+QR(εb)(ε−H1)P |Ψ0ρ〉+ ... (5.49)

Q|Ψ1ρ〉 = −R(εb)H1P |Ψ0ρ〉, (5.50)


wobei wir QP = 0, QR = R und RP = 0 ausgenutzt haben. Zu jedem |Ψ0ρ〉 gibt es alsoKomponenten Q|Ψ1ρ〉, die orthogonal zu Hb sind. Ausgeschrieben und in erster Ordnungin H1 lauten sie

Q|Ψ1ρ〉 =∑

i6=b

di∑

ν=1

|iν〉〈iν|εb − εi

H1P |Ψ0ρ〉. (5.51)

Die Komponenten von |Ψ1ρ〉 in Hb bleiben frei wahlbar - selbst die Kets |Ψ0ρ〉 hangenja bereits als Eigenvektoren von PH1P nicht-linear von den Parametern in H1 ab.Zweckmassigerweise setzt man die Komponenten von |Ψ1ρ〉 in Hb gleich Null und hatdamit

|Ψρ〉 =(1−R(εb)H1

)P |Ψ0ρ〉+ ... (5.52)

=

1−∑

i6=b

di∑

ν=1

|iν〉〈iν|εb − εi

H1

P |Ψ0ρ〉+ ... (5.53)

AUFGABEN: 1. Berechne die Eigenwerte des effektiven Hamiltonoperators des Dop-pelmuldenpotentials,

H ≡ ε

2σz + Tcσx, Tc > 0, (5.54)

a) fur ε 6= 0 storungstheoretisch in Tc bis zur zweiten Ordnung.b) fur ε = 0 mit entarteter Storungstheorie in Tc. Vergleiche mit der exakten Losung(Taylorentwicklung!). Begrunde das Ergebnis von b).c) Berechne die Eigenzustande storungstheoretisch in Tc bis zur ersten Ordnung.2. Berechne die Eigenwerte des anharmonischen Oszillators in d = 1,

H =p2

2M+

1

2Mω2x2 + λx4, λ > 0 (5.55)

a) storungstheoretisch in λ bis zur ersten Ordnung (Hinweis: Verwendung von Leiter-operatoren).

b) (Zusatzaufgabe) zu hoherer Ordnung durch Entwicklung eines Codes (MATHE-MATICA).3. Berechnen Sie die Eigenwerte des Rabi-Hamiltonians

H =∆

2σz + gσx(a+ a†) + Ωa†a. (5.56)

storungstheoretisch in g soweit Sie kommen.

5.1.4 Parametrische Abhangigkeit von Spektren

Wir nehmen einen Hamiltonian an, der von einem Parameter λ abhangt (kann stehenfur Magnetfeld, elektrisches Feld, etc.).


5.1.4.1 Niveauaufspaltung

Wir nehmen folgende Form an:

H(λ) ≡ H0(λ) + V. (5.57)

Als Funktion von λ konnen sich bestimmte Niveaus εa(λ) und εb(λ) des ungestortenH0(λ) an einer Stelle λc kreuzen (SKIZZE). Was ist der Effekt der Storung V auf dieNiveaus in der Nahe von λc? Seien εa(λ) und εb(λ) fur λ 6= λc nichtentartet. Fur λ 6= λc

sagt uns die zweite Ordnung Storungstheorie

εa → εa + 〈a|V |a〉+∑

i6=a

|〈a|V |i〉|2εa − εi

+O(V 3) (5.58)

εb → εb + 〈b|V |b〉+∑

i6=b

|〈b|V |i〉|2εb − εi

+O(V 3) (5.59)

Nahe λc wird die Energiekorrektur durch die kleinsten Nenner des zweite-Ordnung-Termsdominiert,

εa → εa +|〈a|V |b〉|2εa − εb

+ ... (5.60)

εb → εb +|〈b|V |a〉|2εb − εa

+ ... (5.61)

In der Nahe von λc wird also das tiefere Niveau weiter nach unten geschoben, das hohereNiveau weiter nach oben: die Niveaus stossen sich ab (Niveau-Abstossung, engl. level re-pulsion. Fur λ→ λc wird diese Storungstheorie allerdings sehr schlecht, da die Ausdruckewegen der immer kleiner werdenden Energie-Nenner divergieren.

An der Stelle λc muss man entartete Storungstheorie machen. Im von |a〉, |b〉 aufge-spannten zweidimensionalen Unterraum muss man dann

ε(1)P |Ψ〉 = PV P |Ψ〉 (5.62)

losen, d.h. die 2 mal 2 Matrix Vij (Matrixelemente Vaa, Vab, Vba, Vbb) diagonalisieren.

6. INTRODUCTION INTO MANY-PARTICLE

SYSTEMS

6.1 Indistinguishable Particles

In Quantum Mechanics, a system of N particles with internal spin degrees of freedomσi is described by a wave function which in the position representation reads

Ψ(r1, σ1; r2, σ2; ...; rN , σN ). (6.1)

Here, |Ψ(...)|2 is the probability density for finding particle 1 at r1 with spin quantumnumber(s) σ1, particle 2 at r2 with spin quantum number(s) σ2,... etc. Note that for spin

1/2, one would choose for σi one of the spin projections, e.g. σi = σ(z)i = ±1

2 .Remark: Usually, many-particle wave functions and the issue of indistinguishability arediscussed in the position representation.

6.1.1 Permutations

Two particles are called indistinguishable when they have the same ‘elementary’ para-meters such as mass, charge, total spin. As an example, it is believed that all electronsare the same in the sense that they all have the same mass, the same charge, and thesame spin 1/2. The evidence for this comes from experiments.

If some of the N particles described by the wave function Ψ, Eq. (6.1), are indistin-guishable, this restricts the form of Ψ. Let us assume that all N particles are pairwiseindistinguishable. We define the abbreviations ξi ≡ (ri, σi). Since particle j is indistin-guishable from particle k, the N -particle wave functions with ξj and ξk swapped shoulddescribe the same physics: they may only differ by a phase factor,

Ψ(ξ1, ..., ξj , ..., ξk, ..., ξN ) = eiφjkΨ(ξ1, ..., ξk, ..., ξj , ..., ξN ). (6.2)

Swapping j and k a second time must yield the original wave function and therefore

e2iφjk = 1 φjk = 0,±π,±2π,±3π, (6.3)

In fact, the phases 0,±2π etc. are all equivalent: they lead to symmetrical wave functions.The phases ±π,±3π etc. are also all equivalent: they lead to antisymmentrical wavefunctions.

It turns out that this argument (swapping the coordinates) depends on the dimensionof the space in which the particles live, and that there is a connection to the spin of the

6. Introduction into Many-Particle Systems 72

particles. For d ≥ 3, indistinguishable particles with half-integer spin are called Fermionswhich are described by antisymmentrical wave functions. For d ≥ 3, indistinguishableparticles with integer spin are called Bosons which are described by symmentrical wavefunctions. For d = 3, this connection between spin and statistics can be proved inrelativistic quantum field theory (Spin-Statistics-Theorem, W. Pauli 1940).

Ψ(ξ1, .., ξj , ..., ξk, .., ξN ) = −Ψ(ξ1, .., ξk, ..., ξj , .., ξN ), Fermions

Ψ(ξ1, .., ξj , ..., ξk, .., ξN ) = Ψ(ξ1, .., ξk, ..., ξj , .., ξN ), Bosons. (6.4)

In two dimensions, things become more complicated. First of all, the connection withspin (integer, half integer in d = 3) is different in d = 2 because angular momentum ingeneral is no longer quantized: rotations in the x-y plane commuted with each other, i.e.the rotation group SO(2) is abelian and has only one generator which can have arbitraryeigenvalues. Second, topology is different in two dimensions, in particular when discussingwave functions excluding two particles sitting on the same place xk = xj which leads toeffective configuration spaces which are no longer simply connected.

In two dimensions, one obtains a plethora of possibilities with exciting new possibi-lities for ‘fractional spin and statistics’. These are important and have been discoveredrecently in, e.g., the fractional quantum Hall effect. For further literature on this topic,cf. S. Forte, ‘Quantum mechanics and field theory with fractional spin and statistics’,Rev. Mod. Phys. 64, 193.

6.1.2 Basis vectors for Fermi and Bose systems

6.1.2.1 Single Particle

We assume to have a Hilbert space with a complete basis of wave vectors |ν〉 correspon-ding to wave functions 〈rσ|ν〉 including the spin,

|ν〉 ↔ ψν(rσ) = 〈rσ|ν〉. (6.5)

Examples:

• harmonic oscillator, |ν〉 = |n〉 with n = 0, 1, 2, ... and the harmonic oscillator wavefunctions ψn(r).

• two-level system with |ν〉 and ν = + and ν = −.

• hydrogen atom with |ν〉 = |nlsjm〉.

The last example shows that ν is a ‘multi-index’ (index ‘vector’).

6.1.2.2 N -particle system

We have N particles and N quantum numbers ν1,...,νN . A basis consists of all productstates |ν1, ..., νN 〉 corresponding to wave functions ψν1(ξ1)...ψνN

(ξN ), ξ = rσ,

|ν1, ..., νN 〉 ↔ ψν1(ξ1)...ψνN(ξN ) = 〈ξ1|ν1〉...〈ξN |νN 〉. (6.6)


These wave functions still don’t have any particular symmetry with respect to permu-tation of particles. We use them to construct the basis wave functions for Bosons andFermions.

6.1.2.3 Permutations

There are N ! permutations of N particles. We label the permutations by N ! indices pand define a permutation operator Πp, for example

Πp=(1,3)Ψ(ξ1, ξ2, ξ3) = Ψ(ξ3, ξ2, ξ1) (6.7)

Πp=(1,2,3)Ψ(ξ1, ξ2, ξ3) = Πp=(2,3)Ψ(ξ2, ξ1, ξ3) = Ψ(ξ2, ξ3, ξ1) (6.8)

We furthermore define the symmetrization operator S and the anti-symmetrization ope-rator A,

S =1√N !

∑

p

Πp (6.9)

A =1√N !

∑

p

Πpsign(p), (6.10)

where sign(p) is the sign of the permutation which is either −1 or +1, sign(p) = (−1)n(p)

where n(p) is the number of swaps required to achieve the permutation p.

6.1.2.4 N -Boson systems

A basis for symmetric wave functions with N Bosons is constructed in the following way.1. If we just have one possible state |ν1〉 of the system, the symmetric state and the

corresponding wave function is

| ν1, ..., ν1︸︷︷︸

〉S ↔ 〈ξ1, ..., ξN |ν1, ..., ν1〉S ≡ ψν1(ξ1)...ψν1(ξN )

Ntimes(6.11)

This wave function is obviously symmetric.2. If we have two particles (N = 2), the basis is constructed from the states |ν1, ν2〉

with corresponding wave functions ψν1(ξ1)ψν2(ξ2): this product is made symmetric,

|ν1, ν2〉S ↔ 〈ξ1, ξ2|ν1, ν2〉S ≡ 1√2

[ψν1(ξ1)ψν2(ξ2) + ψν1(ξ2)ψν2(ξ1)]

= Sψν1(ξ1)ψν2(ξ2). (6.12)

3. If we just have two possible state |ν1〉 and |ν2〉 for a system with N particles, N1

particles sit in |ν1〉 and N2 particles sit in |ν2〉. We now have to symmetrize the states

|ν1, ..., ν1︸︷︷︸

, ν2, ..., ν2︸︷︷︸

〉 ↔ ψν1(ξ1)...ψν1(ξN1)ψν2(ξN1+1)...ψν2(ξN2)

N1times N2times

N1 +N2 = N. (6.13)


If we apply the symmetrization operator S to this product,

1√N !

∑

p

Πpψν1(ξ1)...ψν1(ξN1)ψν2(ξN1+1)...ψν2(ξN2), (6.14)

we get a sum of N ! terms, each consisting of N products of wave functions. For example,for N1 = 1 and N2 = 2 we get

1√3!

∑

p

Πpψν1(ξ1)ψν2(ξ2)ψν2(ξ3) = (6.15)

=1√N !

[

ψν1(ξ1)ψν2(ξ2)ψν2(ξ3) + ψν1(ξ1)ψν2(ξ3)ψν2(ξ2) (6.16)

+ ψν1(ξ2)ψν2(ξ1)ψν2(ξ3) + ψν1(ξ2)ψν2(ξ3)ψν2(ξ1) (6.17)

+ ψν1(ξ3)ψν2(ξ1)ψν2(ξ2) + ψν1(ξ3)ψν2(ξ2)ψν2(ξ1)]

, (6.18)

where in each line in the above equation we have N2! = 2! identical terms. Had we chosenan example with N1 > 1 and N2 > 1, we would have got N1!N2! identical terms in eachline of the above equation. The symmetrized wave function therefore looks as follows:

1√N !N1!N2!

[

sum ofN !

N1!N2!orthogonal wave functions

]

, (6.19)

which upon squaring and integrating would give[

1√N !N1!N2!

]2 N !

N1!N2!= N1!N2! (6.20)

and not one! We therefore need to divide the whole wave function by 1/√N1!N2! in

order to normalise it to one, and therefore the symmetric state with the correspondingnormalised, symmetrical wave function is

|ν1, ..., ν1, ν2, ..., ν2〉S (6.21)

↔ 1√N !√N1!√N2!

∑

p

Πpψν1(ξ1)...ψν1(ξN1)ψν2(ξN1+1)...ψν2(ξN2).

This is now easily generalised to the case where we have N1 particles in state ν1, N2

particles in state ν2,...,Nr particles in state νr withr∑

i=1

Nr = N. (6.22)

We then have

|ν1, ..., ν1, ν2, ..., ν2, ..., νr , ..., νr〉S (6.23)

↔ 〈ξ1, ..., ξ1, ξ2, ..., ξ2, ..., ξr, ..., ξr|ν1, ..., ν1, ν2, ..., ν2, ..., νr , ..., νr〉S ≡

≡ 1√N !√N1!√N2!...

√Nr!×

∑

p

Πpψν1(ξ1)...ψν1(ξN1)ψν2(ξN1+1)...ψν2(ξN2)...ψνr (ξN−Nr+1)...ψνr (ξN ).


6.1.2.5 N -Fermion systems

In this case, we have to use anti-symmetrized states with anti-symmetric wave functions,

|ν1, ..., νN 〉A = A|ν1, ..., νN 〉 =1√N !

∑

p

Πpsign(p)|ν1, ..., νN 〉

↔ 〈ξ1, ..., ξN |ν1, ..., ν1〉A ≡1√N !

∑

p

Πpsign(p)ψν1(ξ1)...ψνN(ξN )

=1√N !

∣∣∣∣∣∣∣∣

ψν1(ξ1) ψν1(ξ2) ... ψν1(ξN )ψν2(ξ1) ψν2(ξ2) ... ψν2(ξN )...

ψνN(ξ1) ψνN

(ξ2) ... ψνN(ξN )

∣∣∣∣∣∣∣∣

. (6.24)

These determinants are called Slater determinants.

• A permutation of two of the particles here corresponds to a swapping of the cor-responding columns in the determinant and therefore gives a minus sign: the wavefunction is anti-symmetric.

• If two of the quantum numbers ν1, ..., νN are the same, the determinant is zero: ina system with identical Fermions, two or more than two particles can not be in thesame state (in contrast to Bosons). This important fact is called Pauli principle.

Finally, we remark that in Slater determinants we can let the permutations all operateeither on the coordinates ξi, or all on the indices νi:

〈ξ1, ..., ξN |ν1, ..., νN 〉A ≡1√N !

∑

p

sign(p)ψν1(ξp(1))...ψνN(ξp(N))

=1√N !

∑

p

sign(p)ψνp(1)(ξ1)...ψνp(N)

(ξN ). (6.25)

Exercise: Explicitly verify this identity for the case of N = 3 particles.

This is in particular useful when it comes to calculation of matrix elements. The lastform justifies the notation

|ν1, ..., ν1〉A =1√N !

∑

p

sign(p)|νp(1), ..., νp(N)〉. (6.26)

6.2 2-Fermion Systems

In order to get a feeling for how to work with Fermion systems, we start with the simplestcase N = 2. The basis states are the Slater determinants

〈ξ1, ξ2|ν1, ν2〉A =1√2!

∣∣∣∣

ψν1(ξ1) ψν1(ξ2)ψν2(ξ1) ψν2(ξ2)

∣∣∣∣

=1√2

[ψν1(ξ1)ψν2(ξ2)− ψν1(ξ2)ψν2(ξ1)] . (6.27)


6.2.1 Two Electrons

Electrons have spin 12 and we now have to work out how the electron spin enters into

the Slater determinants. The single particle wave functions for particle 1 are productsof orbital wave functions and spin wave functions,

ψ(ξ1) = ψ(r1)|σ1〉(1). (6.28)

For spin-1/2, the spin label σ1 can take the two values σ1 = ±1/2 which by conventionare denoted as ↑ and ↓. The two spinors have the following representation in the two-dimensional complex Hilbert space (spin-space),

| ↑〉(1) =

(10

)

(1)

, | ↓〉(1) =

(01

)

(1)

. (6.29)

Here, the index (1) means that this spin referes to particle (1).We now consider the four possibilities for the spin projections σ1 and σ2 and the

corresponding four sets of basis wave functions,

1√2

[ψν1(r1)ψν2(r2)| ↑↑〉(12) − ψν1(r2)ψν2(r1)| ↑↑〉(12)

]

1√2

[ψν1(r1)ψν2(r2)| ↑↓〉(12) − ψν1(r2)ψν2(r1)| ↓↑〉(12)

]

1√2

[ψν1(r1)ψν2(r2)| ↓↑〉(12) − ψν1(r2)ψν2(r1)| ↑↓〉(12)

]

1√2

[ψν1(r1)ψν2(r2)| ↓↓〉(12) − ψν1(r2)ψν2(r1)| ↓↓〉(12)

]. (6.30)

Here,

| ↑↓〉(12) ≡ | ↑〉(1) ⊗ | ↓〉(2) (6.31)

is a product spinor, i.e. a spin wave function with particle (1) with spin up and particle(2) with spin down, and corresp[ondingly for the other product spinor.

We can now re-write the basis states Eq. (6.30) by forming linear combinations ofthe ‘mixed’ spinors (exercise: check these !),

ψS(ξ1, ξ2) = ψsymν1,ν2

(r1, r2)|S〉 (6.32)

ψT−1(ξ1, ξ2) = ψasymν1,ν2

(r1, r2)|T−1〉 (6.33)

ψT0(ξ1, ξ2) = ψasymν1,ν2

(r1, r2)|T0〉 (6.34)

ψT+1(ξ1, ξ2) = ψasymν1,ν2

(r1, r2)|T+1〉. (6.35)

Here, the symmetric and antisymmetric orbital wave functions are defined as

ψsymν1,ν2

(r1, r2) =1√2

[ψν1(r1)ψν2(r2) + ψν1(r2)ψν2(r1)] (6.36)

ψasymν1,ν2

(r1, r2) =1√2

[ψν1(r1)ψν2(r2)− ψν1(r2)ψν2(r1)] . (6.37)


Furthermore, the spin wave functions are defined as

|S〉 = 1√2[| ↑↓〉−| ↓↑〉] Singlet state

|T−1〉 = | ↓↓〉 Triplet State|T0〉 = 1√

2[| ↑↓〉+| ↓↑〉] , Triplet State

|T+1〉 = | ↑↑〉 Triplet State

. (6.38)

6.2.2 Properties of Spin-Singlets and Triplets

We have another look at the two-particle spin states Eq. (6.38), writing them moreexplicitly as

|S〉 = 1√2[| ↑〉1 ⊗ | ↓〉2−| ↓〉1 ⊗ | ↑〉2] Singlet state

|T−1〉 = | ↓〉1 ⊗ | ↓〉2 Triplet State|T0〉 = 1√

2[| ↑〉1 ⊗ | ↓〉2+| ↓〉1 ⊗ | ↑〉2] Triplet state

|T+1〉 = | ↑〉1 ⊗ | ↑〉2 Triplet State

. (6.39)

6.2.2.1 Total Spin

One advantage of working with singlets and triplets is the fact that they are spin statesof fixed total spin: rthe singlets has total spin S = 0, the three triplets have total spinS = 1 and total spin projections M = −1, 0, 1:

S2|S〉 = ~S(S + 1)|S〉, S = 0, Sz|S〉 = ~M |S〉,M = 0 (6.40)

S2|T−1〉 = ~S(S + 1)|T−1〉, S = 1, Sz|T−1〉 = ~M |T−1〉,M = −1

S2|T0〉 = ~S(S + 1)|T0〉, S = 1, Sz|T0〉 = ~M |T0〉,M = 0

S2|T+1〉 = ~S(S + 1)|T+1〉, S = 1, Sz|T+1〉 = ~M |T+1〉,M = +1.

Often the total spin is conserved when we deal with interacting systems. If , for example,the system is in a state that is a linear combination of the three triplets, it has to stayin the sub-space spanned by the triplets and can’t get out of it. In that case instead ofhaving a four-dimensional space we just have to deal with a three-dimensional space.

6.2.2.2 Entanglement

There is a fundamental difference between the M = ±1 states |T±1〉 on the one side andthe M = 0 states |S〉 and |T0〉 on the other side:

• |T−1〉 = | ↓〉1 ⊗ | ↓〉2 and |T+1〉 = | ↑〉1 ⊗ | ↑〉2 are product states.

• |S〉 = 1√2[| ↑〉1 ⊗ | ↓〉2−| ↓〉1 ⊗ | ↑〉2] and |T0〉 = 1√

2[| ↑〉1 ⊗ | ↓〉2+| ↓〉1 ⊗ | ↑〉2] can

not be written as product states: they are called entangled states.

For product states of two particles 1 and 2 (pure tensors),

|ψ〉1 ⊗ |φ〉2, (6.41)


one can say that particle 1 is in state |ψ〉 and particle 2 is in state |φ〉. States that cannot be written as product states are called entangled states. For example, for the state

|ψ〉1 ⊗ |φ〉2 + |φ〉1 ⊗ |ψ〉2, (6.42)

one can not say which particle is in which state: the two particles are entangled. Entan-glement is the key concept underlying all modern quantum information theory, such asquantum cryptography, quantum teleportation, or quantum computing.

6.2.3 The Exchange Interaction

6.2.3.1 Spin-independent Hamiltonian

We assume a Hamiltonian for two identical electrons of the form

H = − ~2

2m∆1 + V (r1)−

~2

2m∆2 + V (r2) + U (|r1 − r2|) (6.43)

which does not depend on the spin. The Hamiltonian is symmetric with respect to theparticle indices 1 and 2. The solutions of the stationary Schrodinger equation Hψ(r1, r2) =Eψ(r1, r2) for the orbital parts of the wave function can be classified into symmetric andanti-symmetric with respect to swapping r1 and r2: this is because we have

Hψ(r1, r2) = Eψ(r1, r2)↔ Hψ(r2, r1) = Eψ(r2, r1)

↔ HΠ12ψ(r1, r2) = EΠ12ψ(r1, r2) = Π12Eψ(r1, r2) = Π12Hψ(r1, r2)

↔ [H, Π12] = 0, (6.44)

which means that the permutation operator Π12 commutes with the Hamiltonian. Theeigenstates of H can therefore be chosen such they are also simultaneous eigenstates ofΠ12 which are symmetric and antisymmetric wave functions with respect to swappingr1 and r2.

Since the total wave function (orbital times spin) must be antisymmetric, this meansthat for energy levels corresponding to symmetric orbital wave functions lead to spinsinglets with total spin S = 0. Energy levels corresponding to anti-symmetric orbitalwave functions lead to spin triplets with total spin S = 1. Even though there is nospin-dependent interaction term in the Hamiltonian, the spin and the possible energyvalues are not independent of each other!

6.2.3.2 Perturbation Theory

Assume we treat the interaction term V (|r1 − r2|) in the Hamiltonian Eq. (6.43) as aperturbation,

H = H0 + H1, H0 = − ~2

2m∆1 + V (r1)−

~2

2m∆2 + V (r2)

H1 = U (|r1 − r2|) . (6.45)


We seek the first correction to an energy level E(0)αβ of H0,

H0φ±αβ(r1, r2) = E

(0)αβφ

±αβ(r1, r2), E

(0)αβ = E(0)

α +E(0)β

φ±αβ(r1, r2) =1√2

[φα(r1)φβ(r2)± φα(r2)φβ(r1)] , (6.46)

where φα and φβ are two eigenstates with eigenenergies E(0)α and E

(0)β of the (identical)

single particle Hamiltonians − ~2

2m∆ + V (r). We assume α 6= β in the following.We assume the single particle levels to be non-degenerate. Still, the two-electron level

E(0)αβ is degenerate because it corresponds to the two states |φ±αβ〉 (+ for the symmetric

and − for the anti-symmetric state. The corresponding two-by-two matrix of H1 we needdiagonalise for the degenerate first order perturbation theory in the sub-space spannedby |φ±αβ〉 is however diagonal so that things become easy:

H1

=

(

〈φ+αβ |H1|φ+

αβ〉〈φ+αβ |H1|φ−αβ〉

〈φ−αβ |H1|φ+αβ〉〈φ−αβ |H1|φ−αβ〉

)

=

(Aαβ + Jαβ 0

0 Aαβ − Jαβ

)

. (6.47)

Inserting the definitions, we have (i, j,= ±)

〈φiαβ |H1|φj

αβ〉 =

∫ ∫

dr1dr2

[φi

αβ(r1, r2)]∗U (|r1 − r2|)φj

αβ(r1, r2). (6.48)

Exercise: Show that 〈φ+αβ |H1|φ−αβ〉 = 〈φ−αβ |H1|φ+

αβ〉 = 0.

The explicit calculation of the remaining diagonal elements 〈φ+αβ |H1|φ+

αβ〉 and 〈φ−αβ |H1|φ−αβ〉yields

Aαβ =

∫ ∫

dr1dr2|φα(r1)|2U (|r1 − r2|) |φβ(r2)|2

(direct term) (6.49)

Jαβ =

∫ ∫

dr1dr2φ∗α(r2)φ

∗β(r1)U (|r1 − r2|)φα(r1)φβ(r2)

(exchange term, exchange integral) (6.50)

Exercise: Verify these expressions.

The symmetrical orbital wave function (+) belongs to the S = 0 (singlet) spinor,whereas the anti-symmetrical orbital wave function (−) belongs to the T = 0 (triplet)

spinors. Therefore, the unperturbed energy level E(0)αβ splits into two levels

E(1)αβ,S=0 = E

(0)αβ +Aαβ + Jαβ, S = 0 singlet (6.51)

E(1)αβ,S=1 = E

(0)αβ +Aαβ − Jαβ, S = 1 triplet. (6.52)


6.2.3.3 Direct and Exchange Term: Discussion

Extreme examples for the interaction potential:

a) U (|r1 − r2|) = U = const

Aαβ = U, Jαβ = Uδαβ (6.53)

b) U (|r1 − r2|) = U0δ(r1 − r2)

Aαβ = Jαβ = U0

∫

dr|φα(r)|2|φβ(r)|2. (6.54)

6.3 Two-electron Atoms and Ions

Here, we deal with the Helium atom (Z = 2) , Lithium ion Li+ (Z = 3), Beryllium ionBe++(Z = 4) etc. These are two-electron systems as in section 6.2.1 with HamiltonianEq. (6.43),

H = − ~2

2m∆1 + V (r1)−

~2

2m∆2 + V (r2) + U (|r1 − r2|)

V (r) = − Ze2

4πε0r, U (|r1 − r2|) =

e2

4πε0|r1 − r2|. (6.55)

6.3.1 Perturbation theory in U

A problem with perturbation theory here is the fact that the interaction U between thetwo electrons is not small.

The unperturbed states |α〉 and |β〉 for the orbital wave functions (cf. section 6.2.3.2)are the eigenstates of the hydrogen problem, Eq. (3.51),

|α〉 = |nlm〉, |β〉 = |n′l′m′〉. (6.56)

Note that these do not contain a spin index.

6.3.1.1 Ground state

The unperturbed ground state has |nlm〉 = |100〉 and |n′l′m′〉 = |100〉, i.e. α = β witha symmetrical orbital wave function φ+

αα(r1, r2) = φ100(r1)φ100(r2) and a singlet spinor|S〉. The energy to first order in U therefore is

E(1)αα = E(0)

αα +Aαα, α = (100) (6.57)

Aαα =

∫ ∫

dr1dr2|φ100(r1)|2U (|r1 − r2|) |φ100(r2)|2. (6.58)

Calculation of A yields

E(1)100,100 = E

(0)1 + E

(0)1 +A100,100 = 2

(

−1

2

Z2e2

4πε0a0

)

+5

8

Ze2

4πε0a0. (6.59)


For Z = 2, one has 2E(0)1 = −108.8eV and A100,100 = 34eV such that E

(1)100,100 = −74.8eV.

Exercise: Calculate the integral leading to the result Eq. (6.59). Solution hints are givenin Gasiorowicz [1].

6.3.1.2 Excited states

Now our perturbation theory with α 6= β and finite exchange term Jαβ comes into play.Further details: textbooks.

7. THE HARTREE-FOCK METHOD

7.1 The Hartree Equations, Atoms, and the Periodic Table

7.1.1 Effective Average Potential

The basic idea here is to replace the complicated interactions among the electrons by aneffective, average potential energy that each electron i at position ri experiences.

In the Hartree approach one assumes that particle j is described by a wave function(spin orbital) ψνj

(ξj) with orbital part ψνj(rj), and the statistics (anti-symmetrization

of all the total N -particle wave function for Fermions, symmetrizatin for Bosons) isneglected. In the following, we discuss electrons.

For electrons interacting via the Coulomb interaction U(r) = e2/4πε0r, the potentialseen by an electron i at position ri is given by

VH(ri) =−e

4πε0

N∑

j=1(6=i)

∫

drj|ψνj

(rj)|2|rj − ri|

. (7.1)

This is the sum over the potentials generated by all other electrons j 6= i which have acharge density −e|ψj(rj)|2. The corresponding potential energy for electron i is−eVH(ri),and therefore one describes electron i by an effective single particle Hamiltonian,

H(i)Hartree = H

(i)0 + VHartree(ri)

= − ~2

2m∆i + V (ri) +

e2

4πε0

N∑

j=1(6=i)

∫

drj|ψνj

(rj)|2|rj − ri|

, (7.2)

where V (ri) is the usual potential energy due to the interaction with the nucleus. Thecorresponding Schrodinger equations for the orbital wave functions ψνi

for electron i are

− ~2

2m∆i + V (ri) +

e2

4πε0

N∑

j=1(6=i)

∫

drj|ψνj

(rj)|2|rj − ri|

ψνi(ri) = εiψνi

(ri). (7.3)

The total wave function in this Hartree approximation is the simple product

ΨHartree(r1, σ1; ...; rN , σN ) = ψν1(r1, σ1)...ψνN(rN , σN ). (7.4)

Remarks:

7. The Hartree-Fock Method 83

• The Hartree equation Eq. (7.3) is a set of i = 1, ...,N non-linear coupled integro-differential equations.

• As the solutions ψνiof the equations appear again as terms (the Hartree potential)

in the equations, these are called self-consistent equations. One way to solve them

is by iteration: neglect the Hartree term first, find the solutions ψ(0)νi , insert them

in the Hartree potential, solve the new equations for ψ(1)νi , insert these again, and

so on until convergence is reached.

• The Pauli principle is not properly accounted for in this approach, as we do nothave a Slater determinant but only a product wave function. This can be improvedby the Hartree-Fock equations which we derive in the next section.

7.1.2 Angular Average, Shells, and Periodic Table

A further simplification of the Hartree equations, Eq. (7.3), is achieved by replacing theHartree potential by its angular average,

VHartree(r)→ 〈VHartree〉 (r) ≡∫dΩ

4πVHartree(r). (7.5)

This still depends on all the wave functions ψνi, but as the one-particle potential now

is spherically symmetric, we can use the decomposition into spherical harmonics, radialwave functions, and spin,

〈ξ|νi〉 = ψνi(ξ) = Rni,li(r)Yli,mi

(θ, φ)|σi〉, νi = (ni, li,mi, σi). (7.6)

Here, the index νi = (ni, li,mi, σi) indicates that we are back to our usual quantumnumbers nlmσ that we know from the hydrogen atom. In contrast to the latter, theradial functions now depend on n and l because we do not have the simple 1/r Coulombpotential as one-particle potential.

An even cruder approximation to VHartree(r) would be a parametrization of the form

VHartree(r) +e2

4πε0

Z

r→ Veff(r) ≡ e2

4πε0

Z(r)

r(7.7)

Z(r → 0) = Z, Z(r→∞) = 1. (7.8)

by which one loses the self-consistency and ends up with one single Schrodinger equationfor a particle in the potential Veff(r).

Exercise: Give a physical argument for the condition Z(r → 0) = Z, Z(r → ∞) = 1in the above equation.


7.1.2.1 Periodic Table

The ground states of atoms with N = Z electrons in the period table can now beunderstood by forming Slater determinants (‘configurations’) with N spin-orbitals |νi〉 =|nilimiσi〉. The atoms are thus ‘built up’ from these solutions. This is denoted as

H 1s 2S1/2

He (1s)2 1S0

Li (He)(2s) 2S1/2

Be (He)(2s)2 1S0

B (He)(2s)2(2p) 2P1/2

C (He)(2s)2(2p)2 3P0

... ...

(7.9)

These are built up by ‘filling up the levels’ with electrons. For a given (n, l) there are2(2l + 1) orbitals (2 spin states for each given m-value).

The spectroscopic description is given by the quantum numbers S, L, J (total spin,orbital, angular momentum) in the form

2S+1LJ . (7.10)

Carbon is the first case where Hund’s Rules kick in. These ‘rules’ are rules and nostrict theorems, but they seem to work well for the understanding of atoms. Here I citethem after Gasiorowicz (web-supplement)

1. The state with largest S lies lowest: spin-symmetric WFs have anti-symmetricorbital WFs and therefore reduced electron-electron interaction.

2. For a given value of S, the state with maximum L lies lowest: the higher L, themore lobes (and thereby mutual ‘escape routes’ for interacting electrons) there arein the Ylms.

3. L, S given. (i) not more than half-filled incomplete shell: J = |L − S|; (ii) morethan half-filled shell: J = L+ S: due to spin-orbit interaction.

7.2 Hamiltonian for N Fermions

This is a preparation for the new method (Hartree-Fock) we learn in the next sectionwhere we deal with interactions between a large number of Fermions.

The Hamiltonian for N Fermions is given by the generalization of the N = 2 case,Eq. (6.43), and reads

H = H0 + U ≡N∑

i=1

H(i)0 +

1

2

N∑

i6=j

U(ξi, ξj)

H(i)0 = − ~

2

2m∆i + V (ri). (7.11)


7.2.1 Expectation value of H0

Let us consider a N -Fermion state (Slater determinant), cf. Eq. (6.24),

|Ψ〉 = |ν1ν2...νN 〉A =1√N !

∑

p

Πpsign(p)|νp(1)νp(2)...νp(N)〉. (7.12)

We wish to calculate the expectation value 〈Ψ|H0|Ψ〉 with H0 from Eq. (7.11). Consider

for example the free Hamiltonian H(1)0 for the first particle,

〈Ψ|H(1)0 |Ψ〉 =

1

N !

∑

pp′

sign(p)sign(p′)〈νp(N)...νp(2)νp(1)|H(1)0 |νp′(1)νp′(2)...νp′(N)〉

=1

N !

∑

pp′

sign(p)sign(p′)〈νp(N)...νp(2)|νp′(2)...νp′(N)〉〈νp(1)|H(1)0 |νp′(1)〉.

For N−1 numbers we must have p(2) = p′(2),...,p(N) = p′(N) (otherwise the term is ze-ro), but if you have a permutation with N−1 terms fixed, the last term ist automaticallyfixed and we have p = p′, thus (note sign(p)2 = 1)

〈Ψ|H(1)0 |Ψ〉 =

1

N !

∑

p

〈νp(1)|H(1)0 |νp(1)〉. (7.13)

The sum of the single-particle Hamiltonians yields

〈Ψ|H0|Ψ〉 = (7.14)

=1

N !

∑

p

〈νp(1)|H(1)0 |νp(1)〉+ 〈νp(2)|H(2)

0 |νp(2)〉+ ...+ 〈νp(N)|H(N)0 |νp(N)〉,

but all the Hamiltonians H(i)0 have the same form, the sum

∑

p just gives N ! identicalterms, and therefore

〈Ψ|H0|Ψ〉 =

N∑

i=1

〈νi|H0|νi〉, (7.15)

where we can omit the index (i) in H(i)0 and write H0 for the free Hamiltonian of a single

particle (note that H0 in Eq. (7.11) is the total free Hamiltonian; some books use h0

instead of H0 to make this distinction clearer, but small letters are not nice as a notationfor a Hamiltonian).

7.2.2 Expectation value of U

This is only slightly more complicated: consider for example the term U(ξ1, ξ2),

〈Ψ|U(ξ1, ξ2)|Ψ〉 =1

N !

∑

pp′

sign(p)sign(p′)〈νp(N)...νp(2)νp(1)|U(ξ1, ξ2)|νp′(1)νp′(2)...νp′(N)〉

=1

N !

∑

pp′

〈νp(N)...νp(3)|νp′(3)...νp′(N)〉〈νp(2)νp(1)|U(ξ1, ξ2)|νp′(1)νp′(2)〉.


Again, only those terms survive where νp(N) = νp′(N),...,νp(3) = νp′(3). We could have,e.g., p(1) = 4 and p(2) = 7 in which case neither 4 nor 7 can’t be among the p′(3),...,p′(N)(this would yield zero overlap in 〈νp(N)...νp(3)|νp′(3)...νp′(N)〉) and therefore 4 and 7 mustbe among p′(1) and p′(2).

This means we get two possibilities for the permutation pairs p and p′ now: one withνp(1) = νp′(1) and νp(2) = νp′(2), and the other with νp(1) = νp′(2) and νp(2) = νp′(1). Inthe first case νp(1) = νp′(1),νp(2) = νp′(2), νp(3) = νp′(3),...,νp(N) = νp′(N) which meansthe permutaton p′ is the same as p. In the second case, p′ is the same permutation asp apart from one additional swap of p(1) and p(2): this means that sign(p′) = −sign(p)and therefore

〈Ψ|U(ξ1, ξ2)|Ψ〉 =1

N !

∑

p

〈νp(2)νp(1)|U(ξ1, ξ2)|νp(1)νp(2)〉 − 〈νp(1)νp(2)|U(ξ1, ξ2)|νp(1)νp(2)〉.

The sum over all pairs i, j now again yields

〈Ψ|U |Ψ〉 = 1

N !

∑

p

∑

i6=j

1

2

[〈νp(j)νp(i)|U |νp(i)νp(j)〉 − 〈νp(i)νp(j)|U |νp(i)νp(j)〉

]

=1

2

∑

i6=j

[〈νjνi|U |νiνj〉 − 〈νiνj|U |νiνj〉] . (7.16)

7.2.2.1 Spin independent symmetric U

In this case,

U(ξi, ξj) = U (|ri − rj |) . (7.17)

We write this explicitly with wave functions which are products of orbital wavefunctions ψν(r) and spinors |σ〉,

〈ξ|ν〉 = ψν(r)|σ〉, (7.18)

and take advantage of the fact that the interaction U does not depend on spin. Then,Eq. (7.16) becomes

〈Ψ|U |Ψ〉 =1

2

∑

i6=j

[〈νjνi|U |νiνj〉 − 〈νiνj |U |νiνj〉]

=1

2

∑

i6=j

∫ ∫

drdr′[

ψ∗νj

(r′)ψ∗νi

(r)U(|r− r′|

)ψνi

(r)ψνj(r′)〈σi|σi〉〈σj |σj〉

− ψ∗νi

(r′)ψ∗νj

(r)U(|r− r′|

)ψνi

(r)ψνj(r′)〈σj |σi〉〈σi|σj〉

]

=1

2

∑

i6=j

∫ ∫

drdr′[

|ψνj(r′)|2|ψνi

(r)|2U(|r− r′|

)

− ψ∗νi

(r′)ψ∗νj

(r)U(|r− r′|

)ψνi

(r)ψνj(r′)δσiσj

]

. (7.19)


Using our direct and exchange term notation, Eq. (6.49), we can write this in a verysimple form as a sum over direct terms Aνiνj

and exchange terms Jνiνj,

〈Ψ|U |Ψ〉 =1

2

∑

i,j

[Aνiνj

− Jνiνjδσiσj

]. (7.20)

Here, we recognize that we actually don’t need the restriction i 6= j in the double sum:this term is zero anyway.

Note that Eq. (7.19) refers to states |Ψ〉 which are simple Slater determinants.It cannot be used, e.g., for states like the M = 0 singlet or triplet which are linearcombinations

|ψ1↑ψ2↓〉A ± |ψ1↓ψ2↑〉A, (7.21)

because these would lead to mixed terms

〈ψ2↓ψ1↑|U |ψ1↓ψ2↑〉A (7.22)

in the expectation value!

7.3 Hartree-Fock Equations

7.3.1 The Variational Principle

The stationary Schrodinger equation

HΨ = εΨ (7.23)

can be derived from a variational principle. For the ground state of the system, this isformulated as a problem of finding the wave vector Ψ of the system among all possiblewave vectors such that the expectation value of the energy (i.e., the Hamiltonian) isminimized,

〈Ψ|H |Ψ〉 = min, 〈Ψ|Ψ〉 = 1, (7.24)

under the additional condition that Ψ be normalised. We are therefore looking for aminimum of the energy functional

E[Ψ] ≡ 〈Ψ|H|Ψ〉 (7.25)

under the additional condition that Ψ be normalised.

7.3.1.1 Functional Derivates

1. If the Hilbert space belonging to H was finite dimensional (for example in the caseof the two-level system), the energy functional would just be a quadratic form andΨ = (c1, c2)

T would just be a two-component vector.


2. For states |Ψ〉 corresponding to wave functions Ψ(r), the energy functional is a‘function of a (wave) function’. Minimising E[Ψ] means that we have to set its firstfunctional ‘derivative’ to zero (in very much the same way as we set the first derivativeof a function to zero in order to find its minimum).Definition: The derivative of a function f(x) is defined as

df(x)

dx≡ lim

ε→0

f [x+ ε · δx] − f [x]

ε. (7.26)

(δx is a small deviation around the variable x).Definition: The functional derivative of a functional F [Ψ] is defined as

δF [Ψ]

δΨ≡ lim

ε→0

F [Ψ + ε · δΨ]− F [Ψ]

ε. (7.27)

(δΨ is a small deviation around the function Ψ).So we recognise that everything is really quite analogous to ordinary derivative. The

functional derivative of E[Ψ] is obtained from calculating

E[Ψ + ε · δΨ] =

∫

dr Ψ(r) + ε · δΨ(r)∗ H Ψ(r) + ε · δΨ(r)

=

∫

drΨ∗(r)HΨ(r) + ε

∫

dr[

δΨ∗(r)HΨ(r) + Ψ∗(r)HδΨ(r)]

+ ε2∫

drδΨ∗(r)HδΨ(r) (7.28)

and therefore

δE[Ψ]

δΨ=

∫

dr[

δΨ∗(r)HΨ(r) + Ψ∗(r)HδΨ(r)]

≡ 〈δΨ|H |Ψ〉+ 〈Ψ|H|δΨ〉. (7.29)

7.3.1.2 Lagrange Multiplier

The additional condition 〈Ψ|Ψ〉 = 1 can be incorporated into the minimisation procedureby adding a term to the energy functional, introducing a Lagrange multiplier λ, andthereby defining the functional

F [Ψ] ≡ E[Ψ] + λ [〈Ψ|Ψ〉 − 1] . (7.30)

Its functional derivative is

δF [Ψ]

δΨ= 〈δΨ|H |Ψ〉+ 〈Ψ|H |δΨ〉+ λ [〈δΨ|Ψ〉 + 〈Ψ|δΨ〉] . (7.31)

Exercise: Check this equation.


Minimization then means

0 =δF [Ψ]

δΨ 〈δΨ|H + λ|Ψ〉+ 〈Ψ|H + λ|δΨ〉 = 0. (7.32)

As δΨ is arbitrary and complex, this can only be true if

[H + λ]|Ψ〉 = 0, 〈Ψ|[H + λ] = 0 (7.33)

which are two equations which are the conjugate complex to each other. Writing λ = −ε,this means

HΨ = εΨ, (7.34)

which is the stationary Schrodinger equation. However, here ε is the lowest eigenvaluewith corresponding eigenstate Ψ. We thus recognise:

Minimization of the functional F [Ψ] ≡ E[Ψ] − ε [〈Ψ|Ψ〉 − 1] is equivalent to findingthe lowest eigenvalue and eigenstate of the stationary Schrodinger equation HΨ = εΨ.

7.3.2 The Variational Principle for Many-Electron Systems

The basic idea of Hartree-Fock now is to determine the lowest eigenenergy with corre-sponding eigenstate Ψ of an N -electron system not by solving the stationary Schrodingerequation HΨ = εΨ, but by minimizing the functional F [Ψ]. As these two are equiva-lent, nothing would have been gained. However, for N -electron systems either of thesemethods has to be done approximately anyway and the argument is now that the mini-mization procedure is the better starting point.

Idea: do not carry out the minimization of the functional over all possible states Ψ,but just over a certain sub-class of states, i.e., those which can be written as a anti-symmetrized products of some single particle states |νi〉, with the |νi〉 to be determined,i.e. Slater determinants |ν1, ..., νN 〉. The determination of the |νi〉 leads to the Hartree-Fock equations. Note that here and in the following, |νi〉 does not refer to any fixedset of basis states but to the states to be determined from the Hartree-Fock equations.Definition: The single particle states |νi〉 correspond to single particle wave functionsψνi

(r, σ). The label νi includes the spin index. In quantum chemistry, these wave func-tions are sometimes called spin-orbitals, molecular orbitals, or shells.

7.3.2.1 Functional Derivative

We use the Hamiltonian Eq. (7.11),

H = H0 + U ≡N∑

i=1

H(i)0 +

1

2

N∑

i6=j

Uij

H(i)0 = − ~

2

2m∆i + V (ri), Uij = U(ξi, ξj). (7.35)


The energy functional now depends on the N wave functions ψνi(r, σ), i = 1, ...,N ,

F [Ψ] = F [ψν1 , ..., ψνN] = F [ψνi

]. (7.36)

The definition of the functional derivative is not more complicated than in the one-component case,

δF [Ψ]

δΨ≡ lim

ε→0

F [ψνi+ ε · δψi]− F [ψνi

]ε

, (7.37)

where we now have i = 1, ..., N independent ‘deviations’ δψi from the functions ψi. Wefurthermore want to ensure that all single particle states |νi〉 are normalised. Therefore,we introduce our functional F [Ψ] with N Lagrange multipliers λi,

F [Ψ] ≡ 〈νN , ..., ν1|H|ν1, ..., νN 〉A +

N∑

i=1

λi[〈νi|νi〉 − 1]. (7.38)

We have calculated the energy expectation values already in Eq. (7.15) and Eq. (7.19),

F [ψνi] =

N∑

i=1

〈νi|H0|νi〉+1

2

∑

i6=j

[〈νjνi|U |νiνj〉 − 〈νiνj |Uij|νiνj〉]

+

N∑

i=1

λi[〈νi|νi〉 − 1]. (7.39)

The individual terms are simply calculated:

δ

δΨ

N∑

i=1

〈νi|H0|νi〉 =

= limε→0

1

ε

[N∑

i=1

〈νi + εδνi|H0|νi + εδνi〉 −N∑

i=1

〈νi|H0|νi〉]

=

N∑

i=1

[

〈δνi|H0|νi〉+ 〈νi|H0|δνi〉]

. (7.40)

The term from the interaction U yields

δ

δΨ

1

2

∑

ij

[〈νjνi|U |νiνj〉 − 〈νiνj |U |νiνj〉] (7.41)

=1

2

∑

ij

[

〈δνjνi|U |νiνj〉+ 〈νjδνi|U |νiνj〉+ 〈νjνi|U |δνiνj〉+ 〈νjνi|U |νiδνj〉]

− 1

2

∑

ij

[

〈δνiνj|U |νiνj〉+ 〈νiδνj |U |νiνj〉+ 〈νiνj|U |δνiνj〉+ 〈νiνj|U |νiδνj〉]

.


We can use the symmetry property (Exercise: proof!)

〈νiνk|U |νlνm〉 = 〈νkνi|U |νmνl〉 (7.42)

to simplify things by using, e.g., 〈νjδνi|U |νiνj〉 = 〈δνiνj|U |νjνi〉 and changing the sum-mation indices i,j such that

δ

δΨ

1

2

∑

ij

[〈νjνi|U |νiνj〉 − 〈νiνj |U |νiνj〉] (7.43)

=∑

ij

[

〈δνjνi|U |νiνj〉+ 〈νjνi|U |νiδνj〉 − 〈δνiνj|U |νiνj〉 − 〈νiνj |U |νiδνj〉]

=∑

ij

[

〈δνjνi|U |νiνj〉 − 〈δνjνi|U |νjνi〉+ (H.c.)]

,

where again in the ‘−’ term we have swapped indices, and H.c means that there are twoterms which are the hermitian conjugates of the two others.

7.3.2.2 ‘Direct’ and ‘Exchange’ Operators

We defining these one-particle operators by their matrix elements (excessive use of Diracnotation, hurrah!)

〈µ|Ji|ν〉 ≡ 〈µνi|U |νiν〉〈δνj |Ji|νj〉 = 〈δνjνi|U |νiνj〉 (7.44)

〈µ|Ki|ν〉 ≡ 〈µνi|U |ννi〉〈δνj |Ki|νj〉 = 〈δνjνi|U |νjνi〉. (7.45)

Note that both these operators depend on the still to be determined single particle states|νi〉!.

We can now write the functional derivate in a very elegant manner,

δ

δΨ

1

2

∑

ij

[〈νjνi|U |νiνj〉 − 〈νiνj|U |νiνj〉] =N∑

j=1

〈δνj |J − K|νj〉+ (H.c.)

J ≡∑

i

Ji, K ≡∑

i

Ki, (7.46)

and the total functional derivative becomes

δF [Ψ]

δΨ=

N∑

j=1

〈δνj |H0 + λj + J − K|νj〉+ (H.c.). (7.47)

As we set the functional derivative to zero

δF [Ψ]

δΨ= 0

(

H0 + λj + J − K)

|νj〉 = 0, (7.48)

as all the deviations δνj are independent.


7.3.3 Hartree-Fock Equations

We write out Eq. (7.48) in detail, setting λj = −εj ,(

H0 + J − K)

|νj〉 = εj |νj〉 (7.49)

〈µ|J |ν〉 ≡∑

i

〈µνi|U |νiν〉, 〈µ|K |ν〉 ≡∑

i

〈µνi|U |ννi〉,

where we again stated the definition of the two operators J and K. How do theseequations look in the coordinate representation? Let’s write

(

H0 + J − K)

|νj〉 = εj |νj〉

〈r|(

H0 + J − K)

|νj〉 = εj〈r|νj〉

〈r|H0|νj〉+∑

i

〈rνi|U |νiνj〉 −∑

i

〈rνi|U |νjνi〉 = εj〈r|νj〉

H0ψνj(r) +

∑

i

∫

dr′ψ∗νi

(r′)U(|r − r′|)ψνi(r′)ψνj

(r)

−∑

i

∫

dr′ψ∗νi

(r′)U(|r − r′|)ψνj(r′)ψνi

(r)δσiσj= εjψνj

(r). (7.50)

These are the Hartree-Fock equations in the position representation; we write them outagain,

[

H0 +∑

i

∫

dr′|ψνi(r′)|2U(|r − r′|)

]

ψνj(r)

−∑

i

∫

dr′ψ∗νi


(r)δσiσj= εjψνj

(r). (7.51)

This looks like a set of j = 1, ..., N stationary Schrodinger equations, but things areactually more complicated as the equations are non-linear.

7.3.3.1 Direct Term

The direct term,

direct term

[∑

i

∫

dr′|ψνi(r′)|2U(|r − r′|)

]

ψνj(r) (7.52)

acts like a local one-particle potential on particle j: it depends on all the wave functionsψi(r

′) that have still to be determined. The direct term has a simple physical interpreta-tion: it is the potential at position r generated by the total density

∑

i |ψi(r′)|2 of all the

individual electrons in their states |νi〉 at position r′. The direct term can be interpretedas a ‘direct’ re-normalisation of the one-particle Hamiltonian H0.


7.3.3.2 Exchange Term

The exchange term,

exchange term∑

i

∫

dr′ψ∗νi


(r)δσiσj(7.53)

is more complicated and cannot be written as a simple re-normalisation of the one-particle Hamiltonian H0. Its spin-dependence indicates that it originates from the ex-change interaction between indistinguishable Fermions.

What we have achieved, though, is a self-consistent description of the interacting N -Fermion systems in terms of a single Slater determinant built from the states ψi(r, σ).Actually, for spin-independent H0 and U only the orbital parts ψi(r enter the Hartree-Fock equations, although the spin-indices do play a role in the exchange term.

7.3.3.3 Example: N = 2, ‘closed shell’

In the case N = 2, we are back to the Helium atom (N = 2 electrons). We assume

ψν1(r, σ) = ψ(r)| ↑〉, ψν2(r, σ) = ψ(r)| ↓〉, (7.54)

i.e. we only have two spin orbitals with opposite spin. From the Hartree-Fock equationsEq. (7.51), we obtain

[

H0 +2∑

i=1

∫

dr′|ψ(r′)|2U(|r − r′|)]

ψ(r)

−2∑

i=1

∫

dr′ψ∗(r′)U(|r − r′|)ψ(r′)ψ(r)δσiσj= εjψ(r). (7.55)

Formally, these are still two equations due to the label j (= 1, 2), but the two equationsare the same and we may set ε1 = ε2 = ε. The sum in the exchange part has only oneterm which is half the direct part, and therefore (we re-insert the explicit expression forH0)

[

− ~2

2m∆ + V (r) +

∫

dr′|ψ(r′)|2U(|r − r′|)]

ψ(r) = εψ(r). (7.56)

Since we have only one orbital wave function, we only have one equation.


7.3.3.4 Ground State Energy

The ground state energy in Hartree-Fock can be expressed using our equations Eq. (7.19)and Eq. (7.15),

〈Ψ|H|Ψ〉 = 〈Ψ|H0|Ψ〉+ 〈Ψ|U |Ψ〉 (7.57)

=

N∑

i=1

〈νi|H0|νi〉+1

2

∑

i6=j

〈νjνi|U |νiνj〉 − 〈νiνj|U |νiνj〉

=

N∑

i=1

〈νi|H0 +1

2

(

J − K)

|νi〉, (7.58)

where again we used the direct and exchange operators J and K, Eq. (7.49). Since the|νi〉 are the solutions of the HF equations Eq. (7.49),

(

H0 + J − K)

|νj〉 = εj |νj〉, (7.59)

we obtain

EΨ ≡ 〈Ψ|H |Ψ〉 =

N∑

i=1

[

εi −1

2〈νi|

(

J − K)

|νi〉]

(7.60)

=1

2

N∑

i=1

[

εi + 〈νi|H0|νi〉]

. (7.61)

8. MOLECULES

8.1 Introduction

Molecules are system consisting of electrons and nuclei. This definition covers the fullrange from rather simple molecules like H2 up to extremely complex situations withbillions of nuclei, or in principle even solids or fluids although one usually thinks ofsomething like a microscopic object. The question, of course, is what microscopic reallymeans. In principle, one could have molecules with macroscopic large numbers (like 1023)of electrons and nuclei. Would these behave as quantum or as classical objects?

Even for small molecules, there are in fact some fundamental, conceptual issues inthe field of molecular structure, cf. for example the article by B. T. Sutcliffe in ch. 35 ofVol. 1 of the ‘Handbook of Molecular Physics and Quantum Chemistry’, Wiley (2003).These are related to the question of whether or not molecular structure and properties ofmolecules can be strictly derived from a microscopic Schrodinger equation of an isolatedmolecule, including all the Coulomb interaction among the constituents. For example, thetotal Hamiltonian commutes with the parity operator which means that its eigenstatesare parity eigenstates and therefore can not must have zero expectation value of thestatic dipole moment. This would mean that there exist no molecules with static dipolemoments, which apparently is in contradiction to what we learn from chemistry. Anothersuch ‘paradoxon’ seems to be isomers of polyatomic molecules, and the concept of thechemical bond (‘deconstructing the bond’) is not an easy one, either. These seem to beopen questions.

8.1.1 Model Hamiltonian

We start from a Hamiltonian describing a system composed of two sub-systems, electrons(e) and nuclei (n)

H = He +Hn +Hen, (8.1)

where Hen is the interaction between the two systems. Note that the splitting of the Ha-miltonian H is not unique: for example, Hn could just be the kinetic energy of the nucleiwith their mutual interaction potential included into Hen (as in the BO approximation).

The set-up H = He + Hn + Hen is quite general and typical for so-called ‘system-bath’ theories where one would say the electrons are the ‘system’ and the nuclei are the‘bath’ (or vice versa!). In the theory of molecules, however, things are a little bit morecomplicated as there is a back-action from the electrons on the nuclei. This back-actionis due to the electronic charge density acting as a potential for the nuclei.

8. Molecules 96

There is no a priori reason why the nuclei and the electronic system should not betreated on equal footing. However, the theory has a small parameter

κ =(m

M

) 14

(8.2)

given by the ratio of electron mass m and a typical nuclear mass M ≫ m, and theexponent 1/4 is introduced for convenience in the perturbation theory used by Born andOppenheimer in their original paper. The smallness of this parameter makes it possibleto use an approximation which is called the Born-Oppenheimer approximation.

We assume there is a position representation, where q ≡ x1, ...,xN represents thepositons of all electrons, X ≡ X1, ...,XN the positions of all nuclei, and correspondin-gly for the momenta p and P ,

H = H(q, p;X,P ) = He(q, p) +Hn(X,P ) +Hen(q,X). (8.3)

Spin is not considered here. Also note that the interaction only depends on (q,X) andnot on the momenta.

8.2 The Born-Oppenheimer Approximation

This is the central aproximation used in many calculations.

8.2.1 Derivation

We now try to attack the Schrodinger equation HΨ = EΨ for the total system (electronplus nuclei).

8.2.1.1 Unsuccessful Attempt

A first guess to solve the stationary Schrodinger equation HΨ = EΨ for the total systemwould be a separation ansatz

H(q, p;X,P )Ψ(q,X) = EΨ(q,X)

Ψ(q,X) = ψe(q)φn(X) unsuccessful, (8.4)

which does not work because the interaction Hen(q,X) depends on both q and X.

8.2.1.2 More Successful Attempt

As Hen(q,X) depends on the positions of the nuclei X, let us try an ansatz

Ψ(q,X) = ψe(q,X)φn(X) successful (8.5)

where now the electronic part depends on the nuclear coordinates X as well. This looksunsymmetric: why shouldn’t one have Ψ(q,X) = ψe(q,X)φn(q,X)? First, there is an

8. Molecules 97

asymmetry in the problem in the form of M ≫ m, and Ψ(q,X) = ψe(q,X)φn(q,X) isno more better than Ψ(q,X) in the first place.

The idea with writing Ψ(q,X) = ψe(q,X)φn(X) is that the electronic part ψe(q,X)already solves part of the problem, i.e.

[He(q, p) +Hen(q,X)]ψe(q,X) = E(X)ψe(q,X), (8.6)

an equation in which X, of course, appears as an external classical parameter that com-mutes with all other variables. Consequently, the eigenvalue E(X) has to depend on X.We thus obtain

Hψeφn ≡ [He +Hn +Hen]ψeφn

= [Hn + E(X)]ψeφn (?) = Eψeφn (8.7)

where the last question mark indicated what we would like to have! Since Hn and E(X)depend on the nuclear coordinates only, one would like to use an equation like

[Hn + E(X)]φn(X) = Eφn(X), (8.8)

because then we would have achieved our goal. However, the operator Hn contains thenuclear momenta P which operate on the X in ψe(q,X), i.e.

Hψeφn = ψe [Hn + E(X)] φn + [Hnψeφn − ψeHnφn]

= Eψeφn + [Hnψeφn − ψeHnφn]. (8.9)

This shows that we are almost there if it wasn’t for the underlined term. One now triesto find arguments why this term can be neglected. If it can be neglected, then we haveachieved the full solution of the Schrodinger equation by the two separate equations

[He(q, p) +Hen(q,X)]ψe(q,X) = E(X)ψe(q,X) electronic part

[Hn + E(X)] φn(X) = Eφn(X) nuclear part. (8.10)

These two equations Eq. (8.10) are the central equations of the Born-Oppenheimerapproximation. Even without solving them, some quite interesting observations canalready be made:

• The electronic part is calculated as if the nuclei were at fixed positions X (‘clampednuclei’).

• The eigenvalue of the energy of the electronic part serves as a potential energy forthe nuclei in the nuclear part of the equations.

8.2.2 Discussion of the Born-Oppenheimer Approximation

We now have to justify the neglect of the underlined term in

Hψeφn = Eψeφn + [Hnψeφn − ψeHnφn]. (8.11)

8. Molecules 98

Up to here, everything was still fairly general. Now we make our choice for Hn as justthe kinetic energy of the nuclei,

Hn =N∑

i=1

P 2i

2Mi. (8.12)

We simplify the following discussion by writing

Hn =P 2

2M= − ~

2

2M∇2

X , (8.13)

which refers to a) a single relative motion of two nuclei of effective mass M , or alterna-

tively b) represents an ‘abstract notation’ for Hn =∑N

i=1P 2

i

2Mi(to which the following

transformations can easily be generalised).We write

Hnψeφn − ψeHnφn = − ~2

2M

[∇2

Xψe(q,X)φn(X)− ψe(q,X)∇2Xφn(X)

]

= − ~2

2M

[

∇X φn∇Xψe + ψe∇Xφn − ψe∇2Xφn

]

= − ~2

2M

[

2∇Xφn∇Xψe + φn∇2Xψe

]

. (8.14)

This term is therefore determined by the derivative of the electronic part with respect tothe nuclear positions X, and it has the factor 1/M in front. The ‘handwaving’ argumentnow is to say that the derivatives ∇Xψe and ∇2

Xψe are small.

8.2.3 Adiabaticity and Geometric Phases

The electronic part equation

[He(q, p) +Hen(q,X)]ψe(q,X) = E(X)ψe(q,X) (8.15)

usually should give not only one but a whole set of eigenstates,

[He +Hen] |α(X)〉 = Eα(X)|α(X)〉. (8.16)

Assume that for a fixed X we have an orthogonal basis of the electronic Hilbert spacewith states |α(X)〉, no degeneracies and a discrete spectrum Eα(X),

〈α(X)|β(X)〉 = δαβ . (8.17)

Adiabaticity means that when X is changed slowly from X → X ′, the correspondingstate slowly changes from |α(X)〉 → |α(X ′)〉 and does not jump to another α′ 6= α like|α(X)〉 → |α′(X ′)〉. In that case, we can use the |α(X)〉 as a basis for all X and write

Ψ(q,X) =∑

α

φα(X)ψα(q,X). (8.18)

8. Molecules 99

Now

H∑

α

|φα〉n ⊗ |ψα〉e =∑

α

[Hn + Eα(X)] |φα〉n ⊗ |ψα〉e, (8.19)

and taking the scalar product with a 〈ψα| of the Schrodinger equationHΨ = EΨ thereforegives

[〈ψα|Hn|ψα〉e + Eα(X)] |φα〉n = E|φα〉n (8.20)

This is the Schrodinger equation for the nuclei within the adiabatic approximation. Nowusing again

Hn = − ~2

2M∇2

X Hnψα(q,X)φα(X)

= − ~2

2M

[

ψα(q,X)∇2Xφα(X) + φα(X)∇2

Xψα(q,X)

+ 2∇Xφα(X)∇Xψα(q,X)]

(8.21)

and therefore the nuclear Schrodinger equation becomes

[〈ψα|Hn|ψα〉e + Eα(X)] |φα〉n = E|φα〉n [

− ~2

2M∇2

X + Eα(X)− 〈ψα|~

2∇2X

2M|ψα〉 − 〈ψα|

~2∇X

M|ψα〉∇X

]

|φα〉n

= E|φα〉n (8.22)

which can be re-written as[

− ~2

2M∇2

X + Eα(X)− ~2

2MG(X) − ~

2

MF (X)∇X

]

|φα〉n = E|φα〉n

G(X) ≡ 〈ψα|∇2Xψα〉, F (X) ≡ 〈ψα|∇Xψα〉 , (8.23)

where we followed the notation by Mead and Truhlar in their paper J. Chem. Phys. 70,2284 (1979). Eq. (8.23) is an important result as it shows that the adiabatic assumptionleads to extra terms F (X) and G(X) in the nuclear Schrodinger equation in BO appro-ximation on top of just the potential created by the electrons. In particular, the termF (X) is important as it leads to a non-trivial geometrical phase in cases where thecurl of F (X) is non-zero. This has consequences for molecular spectra, too. Geometricphases such as the Abelian Berry phase and the non-Abelian Wilczek-Zee holono-mies play an important role in other areas of modern physics, too, one example being‘geometrical quantum computing’. For more info on the geometric phase in molecularsystems, cf. the Review by C. A. Mead, Prev. Mod. Phys. 64, 51 (1992).

8. Molecules 100

8.2.4 Breakdown of the Born-Oppenheimer Approximation

This is a non-trivial, much discussed issue and in actual fact still the topic of presentresearch. From our discussion in the previus section we understand that adiabaticity islost if transitions between electronic states |α(X)〉 → |α′(X ′)〉 occur while change X.One example for is the so-called Landau-Zener tunneling between nearby energy levelsEα(X) and Eα′(X). Also discussed in this context are the Renner-Teller and the Jahn-Teller effects, cf. the short summary by B. T. Sutcliffe in ch. 36 of Vol. 1 of the ‘Handbookof Molecular Physics and Quantum Chemistry’, Wiley (2003).

8.3 The Hydrogen Molecule Ion H+2

The hydrogen molecule is an example of an diatomic molecule. These contain two nucleiof charge Z1e and Z2e; they are called homonuclear for two identical nuclei (Z1 = Z2)and heteronuclear (Z1 6= Z2) otherwise. For diatomic molecules the Born-Oppenheimerapproximation can be reliably justified (CHECK).

8.3.1 Hamiltonian for H+2

(Cf. Weissbluth [2] ch. 26 for this section). The Hamiltonian for the electronic part atfixed positions xa and xb of the two protons is a Hamiltonian for a single electron atposition x,

H(0)e =

p2

2m− e2

4πε0

[1

|x− xa|+

1

|x− xb|− 1

R

]

, (8.24)

where R ≡ |xa − xb| and the (fixed) Coulomb repulsion energy ∝ 1/R between the twonuclei has been included for later convenience. The eigenstates of this Hamiltonian canbe determined from an exact solution in ellipsoidal coordinates. The corresponding wavefunctions are called molecular orbitals (MO) because these orbitals spread out overthe whole molecule.

Instead of discussing the exact solution, it is more instructive to discuss an appro-ximate method that can also be used for more complicated molecules. This method iscalled LCAO (linear combination of atomic orbitals) and has a central role in quantumchemistry.

8.3.2 The Rayleigh-Ritz Variational Method

For a given Hamiltonian H we minimise the expectation value of the energy over asub-set of states |Ψ〉 that are linear combinations of n given states |ψi〉,

E = min〈Ψ|H|Ψ〉〈Ψ|Ψ〉 , |Ψ〉 =

n∑

i=1

xi|ψi〉. (8.25)

The |ψi〉 are assumed to be normalised but not necessarily mutually orthogonal, i.e., onecan have 〈ψi|ψj〉 6= 0.

8. Molecules 101

The energy E = E(x1, ..., xn) is therefore minimized with respect to the n coefficientsxi, i = 1, ..., n. It can be written as

E = minx1,...,xn

∑ni,j=1 x

∗iHijxj

∑ni,j=1 x

∗iSijxj

≡ minx

x†Hx

x†Sx, (8.26)

where one has introduced the matrices H and S with matrix elements

Hij = 〈ψi|H|ψj〉, Sij = 〈ψi|ψj〉. (8.27)

We find the minimum of

f(x) ≡x†Hx

x†Sx(8.28)

by setting the gradient to zero. We treat x and its complex conjugate x∗ as independentvariables and calculate

∂

∂x∗k

(

x†Hx)

=∂

∂x∗k

∑

ij

x∗iHijxj =∑

j

Hkjxj = (Hx)k

∇∗(

x†Hx)

= Hx (8.29)

Correspondingly,

∇∗(

x†Sx)

= Sx. (8.30)

Thus,

∇∗f(x) =Hx

x†Sx−

x†HxSx

(x†Sx)2=

Hx

x†Sx−f(x)Sx

x†Sx=

(H − f(x)S)x

x†Sx

(H − ES)x = 0, (8.31)

since E = f(x) at the minimum! A necessary condition for a minimum therefore is theequation (H − ES)x = 0, which has a solution for x only if

det∣∣H − ES

∣∣ = 0. (8.32)

Exercise: Check which equations one obtains when taking the derivative ∇ instead of∇∗ !

We summarise:

ERayleigh−Ritz ≡ min〈Ψ|H|Ψ〉〈Ψ|Ψ〉 , |Ψ〉 =

n∑

i=1

xi|ψi〉 (8.33)

(H − ES)x = 0

Hij ≡ 〈ψi|H|ψj〉, Sij ≡ 〈ψi|ψj〉,x ≡ (x1, ..., xn)T .

The minimization problem thus led us to an eigenvalue problem.

8. Molecules 102

8.3.3 Bonding and Antibonding

(Cf. Atkins and Friedman [3], ch. 8.3, for this section). We now apply the Rayleigh-Ritzvariational method to the Hydrogen Molecule Ion H+

2 , restricting ourselves to just n = 2real wave functions (atomic orbitals) ψi (i = 1, 2), i.e.

MO = Ψ = LCAO = x1ψ1 + x2ψ2 (8.34)

ψ1(r) = ψn=1,l=0,m=0(r− ra), ψ2(r) = ψn=1,l=0,m=0(r− rb)

with two hydrogen groundstate s-orbitals for nuclei at ra and rb, respectively.

8.3.3.1 Rayleigh-Ritz Results

We require the matrices H and S,

H =

(α ββ α

)

, S =

(1 SS 1

)

(8.35)

α ≡ 〈ψ1|H|ψ1〉 = 〈ψ2|H|ψ2〉, β = 〈ψ1|H|ψ2〉, S = 〈ψ1|ψ2〉.

We have to solve

det∣∣H − ES

∣∣ = 0 det

∣∣∣∣

α− E β − ESβ − ES α− E

∣∣∣∣= 0

(α− E)2 − (β − ES)2 = 0 α− E = ±(β −ES)

E+ =α+ β

1 + S, E− =

α− β1− S . (8.36)

This give the eigenvalues of the energy, E±. We find the eigenvectors (x1, x2) from

(α− E±)x1 + (β − E±S)x2 = 0 (8.37)

E+ : ((1 + S)α− (α+ β))x1 + ((1 + S)β − (α+ β)S)x2 = 0

(Sα− β)x1 + (β − Sα)x2 = 0 x1 = x2 ≡ x+.

E− : ((1 − S)α− (α− β))x1 + ((1− S)β − (α− β)S)x2 = 0

(−Sα+ β)x1 + (β − Sα)x2 = 0 x1 = −x2 ≡ x−. (8.38)

The normalisation constant is determined from

1 = 〈Ψ|Ψ〉 = x21 + x2

2 + 2x1x2〈ψ1|ψ2〉 = x21 + x2

2 + 2x1x2S

1 = x2+ + x2

+ + 2x2+S x+ =

1√

2(1 + S)(8.39)

1 = x2− + x2

− − 2x2−S x− =

1√

2(1 − S). (8.40)

8. Molecules 103

Summarising, we therefore have obtained the two molecular orbitals (MOs) with energiesE±,

E+ : Ψ+ =1

√

2(1 + S)(ψ1 + ψ2) bonding (8.41)

E− : Ψ− =1

√

2(1− S)(ψ1 − ψ2) antibonding. (8.42)

Note that the normalisation factor is different for the two MOs, this is due to the factthat the original AOs (atomic orbitals) are not orthogonal.

8.3.3.2 Explicit Calculation of α, β, S

This is required in order to find the values for the two energies E±, and also in order tofind out which of the two states Ψ± has lower energy! The calculations are performedby introducing elliptical coordinates 1 ≤ µ ≤ ∞, −1 ≤ ν ≤ 1, 0 ≤ φ ≤ 2π,

µ =ra + rbR

, ν =ra − rbR

(8.43)

and noting that the volume element in these coordinates is

dV =1

8R3(µ2 − ν2)dµdνdφ. (8.44)

The result for α, β, and S is found as a function of the (fixed) distance R between thetwo protons. Using this together with Eq. (8.36), one finally obtains

E+ = E1s +e2

4πε0a0

[1

R− j + k

1 + S

]

E− = E1s +e2

4πε0a0

[1

R+j − k1− S

]

R ≡ |ra − rb|/a0, S ≡(

1 +R+1

3R2

)

e−R (8.45)

j ≡ a0

∫

dV|ψ1s(r− ra)|2|r− rb|)

=1

R

(1− (1 +R)e−2R

)

k ≡ a0

∫

dVψ1s(r− ra)ψ1s(r− rb)

|r− ra|)= (1 +R)e−R.

Be careful because I haven’t checked these explicit expressions, which are from At-kins/Friedman [3] ch. 8.3.

REMARKS:

• The energies j and k are here written as dimensionless quantities.

• The energy j is due to the electron charge density around nucleus a in the Coulombfield of nucleus b. The energy k is an interference term.

8. Molecules 104

Abbildung 8.1: Energy level splitting for Ψ+ (a) and Ψ− (b), from Weissbluth [2].

• One has j > k and therefore the energy E+ corresponding to the bonding state isthe lower of the two: Occupation of the bonding orbital Ψ+ lowers the energyof the molecule and ‘draws the two nuclei together’, as we will see from the curveE+(R) below which represents the potential in BO approximation for the twonuclei. The bonding orbital corresponds to a wave function with even parity withrespect to with respect to reflections at the plane that lies symmetrically betweenthe two nuclei.

• The antibonding orbital Ψ− has a larger energy. It corresponds to an odd wavefunction.

• For the Hydrogen ion, sometimes one uses the notation (cf. Weissbluth [2] ch. 26.2)

Ψ+ ≡ 1σ+g bonding MO (8.46)

Ψ− ≡ 1σ+u antibonding MO, (8.47)

where the indices mean even for g (German ‘gerade’) and odd for u (German‘ungerade’) and σ+ referring to a symmetry (see below).

The charge distribution in Ψ+ and Ψ− is shown in Fig.(8.3.3.2).

8. Molecules 105

Abbildung 8.2: Charge distribution in Ψ+ (a) and Ψ− (b), from Weissbluth [2].

8.3.3.3 Symmetries of MOs in LCAO

(cf. Weissbluth [2] ch. 26.2). This leads to a classification of MOs according to theirsymmetry under symmetry transformations. The most important classes are the σ mo-lecular orbitals which form σ-bonds, and the π molecular orbitals which formπ-bonds, cf. Fig.

8.3.3.4 Molecular Potential Energy

Within BO approximation, the energies E±(R) enter the nuclear Hamiltonian (cf. Eq. (??)with εα = E±) for the wave functions χ

∑

i=a,b

P2i

2M+ E±(R)

χ±(xa,xb) = Eχ±(xa,xb) (8.48)

of the nuclear system with R = |xa−xb|, cf. Eq. (8.24). Clearly, a separation in center-ofmass and relative motion is easily done here. The potential energy for the nuclei is givenby the function E±(R), cf. Eq. (8.45),

E±(R) = E1s +e2

4πε0a0

[1

R∓ j(R)± k(R)

1± S(R)

]

, (8.49)

with the explicit expression for j(R), k(R), S(R), in Eq. (8.45). The parametric ei-genenergies of the electronic system become the potential for the nuclei, which is thecharacteristic feature of the BO approximation. The corresponding potential curves areshown in Fig.(8.3.3.4).

• The potential energy E+(R) of the bonding molecular orbital has a minimum atR = R0. This determines the equilibrium position of the two nuclei. Occupation ofthe bonding MO helps to bond the nuclei together and thereby form the molecule.

8. Molecules 106

Abbildung 8.3: σ-type and π-type LCAO-MOs, from Weissbluth [2]

• The potential energy E−(R) of the antibonding molecular orbital has no local mi-nimum. Therefore, the antibonding state is an excited state in which the moleculedissociates.

8. Molecules 107

Abbildung 8.4: E±(R), Eq. (8.49), for the H+

2 -ion in Born-Oppenheimer approximation andusing the MO-LCAO Rayleigh-Ritz method, from Weissbluth [2].

8.4 Hartree-Fock for Molecules

We now discuss a method to calculate molecular orbitals within the Hartree-Fock me-thod. Let us start from Eq. (7.49),

(

H0 + J − K)

|νj〉 = εj|νj〉 (8.50)

〈µ|J |νj〉 ≡∑

i

〈µνi|U |νiνj〉, 〈µ|K|νj〉 ≡∑

i

〈µνi|U |νjνi〉,

and assume a closed shell situation and a Hamiltonian H0 + U which is diagonal inspin-space, i.e. does not flip the spin. The counter j runs from 1 to 2N , there are Norbitals with spin up and N orbitals with spin down. The index j thus runs like

j = 1 ↑, 1 ↓, 2 ↑, 2 ↓, ...,N ↑,N ↓ . (8.51)

We write

ψνj=2k−1≡ ψk| ↑〉, ψνj=2k

≡ ψk| ↓〉, (8.52)

8. Molecules 108

because j = 2k, k = 1, ..., N corresponds to spin-orbitals with spin ↓ and j = 2k − 1,k = 1, ..., N corresponds to spin-orbitals with spin ↑. Use the Fock operator

F ≡ H0 + J − K (8.53)

and let us, for example, set j = 2k − 1 to obtain

F|ψk〉 ⊗ | ↑〉 = εk↑|ψk〉 ⊗ | ↑〉 (8.54)

(8.55)

and expand the orbital wave function as

(MO) ψk =M∑

l=1

clkφl, (LCAO) (8.56)

with l = 1, ...,M given atomic orbitals. Inserting yields

〈↑ | ⊗ 〈φl′ |F|M∑

l=1

clk|φl〉 ⊗ | ↑〉 = εk↑

M∑

l=1

clk〈φl′ |φl〉

M∑

l=1

F↑↑l′l clk = εk↑

M∑

l=1

Sl′lclk (8.57)

Sl′l ≡ 〈φl′ |φl〉, F↑↑l′l ≡ 〈↑ | ⊗ 〈φl′ |F|φl〉 ⊗ | ↑〉.

8.4.1 Roothan Equations

The equations Eq. (8.57) are called Roothan equations (they are usually written forspin-independent Fock-operator F . We summarize the situation so far:

• We have l = 1, ...,M atomic orbitals (AOs) φl for k = 1, ...,N molecular orbitals(MOs) expressed as linear combinations (LCAO) of the AOs.

• We define the matrix C as the matrix of the coefficients, Clk

= clk, S as the matrixof the overlaps, S

lk= Slk, and F as the Fock matrix.

• As H0 + U is diagonal in spin-space so is the Fock-operator F whence there areno mixed terms F↓↑ or F↑↓.

We can then write the Roothan equations as

F↑↑C = S C ε↑ (8.58)

F↓↓C = S C ε↓, (8.59)

where ε are diagonal matrices for the energies εk. Now these look like simultaneouslinear equations but of course they are not, because the Fock-operator depends on the

8. Molecules 109

coefficients ckl that we try to determine: recall F ≡ H0 + J − K with

J↑↑l′l ≡

2N∑

i=1

〈l′νi|U |νil〉 (8.60)

K↑↑l′l ≡

2N∑

i=1

〈l′νi|U |lνi〉〈σi| ↑〉,

where we first considered spin up. The i-sum runs over spin-orbitals, i.e. AOs includingthe spin. We now assume U to be spin-independent.

Jl′l ≡∑

σ=0,1

N∑

k=1

〈l′ν2k−σ|U |ν2k−σl〉 = 2N∑

k=1

〈l′ψk|U |ψkl〉

Kl′l ≡∑

σ=0,1

N∑

k=1

〈l′ν2k−σ|U |lν2k−σ〉〈σ| ↑〉 =

N∑

k=1

〈l′ψk|U |lψk〉,

where σ = 0 corresponds to spin down and σ = 1 corresponds to spin up, and the orbitalpart of the spin-orbital ν2k−σ is ψk by definition. Now everything is expressed in termsof orbitals only and the spin has just led to the factor of two in front of the direct term!

We now use the LCAO expansion ψk =∑

m cmkφm and thus obtain

Jl′l = 2

N∑

k=1

M∑

m,m′=1

c∗m′kcmk〈l′m′|U |ml〉 =

M∑

m,m′=1

Pm′m〈l′m′|U |ml〉

Kl′l =

N∑

k=1

M∑

m,m′=1

c∗m′kcmk〈l′m′|U |lm〉 =1

2

M∑

m,m′=1

Pm′m〈l′m′|U |lm〉

Pm′m ≡ 2

N∑

k=1

c∗m′kcmk, populations (8.61)

where the populations depend on the c’s: it is them who are responsible for the non-linearity (self-consistent character) of the Roothan equations. Summarizing, the matrixelements of the Fock operator are given by

F↑↑l′l =

(

H0)↑↑

l′l(8.62)

+

M∑

m,m′=1

Pm′m[cij](

〈l′m′|U |ml〉 − 1

2〈l′m′|U |lm〉

)

.

9. TIME-DEPENDENT FIELDS

9.1 Time-Dependence in Quantum Mechanics

The basis equation is the Schrodinger equation. For a given (time-dependent) Hamilto-nian H(t), the time evolution of a Dirac ket is

i~∂t|Ψ(t)〉 = H(t)|Ψ(t)〉. (9.1)

There are usually two steps in solving a given physical problem

1. find H(t) for the problem at hand.

2. solve the corresponding Schrodinger equation.

Depending on the problem, one often has one of the following cases:

H(t) = H time-independent Hamiltonian (9.2)

H(t) = H(t+ T ) periodic time-dependence (period T ) (9.3)

H(t) arbitrary time-dependence in Hamiltonian (9.4)

The second case occurs, e.g., in the interaction of atoms with monochromatic electricfields like

E(r, t) = E0 cos (kr− ωt) . (9.5)

An explicit time-dependence in the Hamiltonian usually represents classical fields orparameters that can be controlled from the outside and which are not quantum variables.

With respect to the interaction between atoms or molecules and light, there are twogroups of problems one has to sort out:

1. find the correct Hamiltonian H(t) (in fact not so easy).

2. find appropriate techniques to solve the Schrodinger equation (at least in principleone knows how to do that).

9.1.1 Time-evolution with time-independent H

(Set ~ = 1 in the following). In this case, the initial value problem

i∂t|Ψ(t)〉 = H|Ψ(t)〉, |Ψ(t0)〉 = |Ψ〉0 (9.6)

9. Time-Dependent Fields 111

is formally solved as

|Ψ(t)〉 = U(t, t0)|Ψ〉0, U(t, t0) ≡ e−iH(t−t0), t ≥ t0, (9.7)

where we introduced the time-evolution operator U(t, t0) as the exponential of theoperator −iH(t− t0) by the power series

ex =∞∑

n=0

xn

n!. (9.8)

Things are simple, however, when we use the solutions of the stationary Schrodingerequation

H|n〉 = εn|n〉, (9.9)

where the eigenstates |n〉 form a complete basis and one has

〈n|Ψ(t)〉 =∑

m

〈n|e−iH(t−t0)|m〉〈m|Ψ〉0 (9.10)

=∑

m

〈n|m〉e−iεm(t−t0)〈m|Ψ〉0

=∑

m

δnme−iεm(t−t0)〈m|Ψ〉0

= e−iεn(t−t0)〈n|Ψ〉0 |Ψ(t)〉 =

∑

n

|n〉〈n|Ψ(t)〉 =∑

n

|n〉e−iεn(t−t0)〈n|Ψ〉0,

where the underlined terms are the expansion coefficients of |Ψ(t)〉 in the basis |n〉.

9.1.2 Example: Two-Level System

Consider

H =

(0 Tc

Tc 0

)

. (9.11)

We calculate the time-evolution operator U(t, t0) ≡ e−iH(t−t0) by two methods:


9.1.2.1 Power Series

Use

H = Tcσx, σx ≡(

0 11 0

)

σ0x = 1, σ1

x = σx, σ2x = 1

eασx = 1 +α

1!σx +

α2

2!1 +

α3

3!σx +

α4

4!1 + ...

= cosh(α)1 + sinh(α)σx

U(t, t0) ≡ e−iH(t−t0) = cosh(−i(t− t0)Tc)1 + sinh(−i(t− t0)Tc)σx

= cos[(t− t0)Tc]1− i sin[(t− t0)Tc]σx. (9.12)

9.1.2.2 Eigenvectors

We diagonalise H according to

H = SDS−1, (9.13)

where D is the diagonal matrix of the eigenvalues and S the orthogonal matrix of theeigenvectors. This yields

eαH = eαSDS−1= SeαDS−1 (9.14)

which follows from the definition of the power series (Exercise: CHECK)! For H = Tcσx,we already calculated the EVs in an earlier chapter,

(0 Tc

Tc 0

)

=1√2

(1 11 −1

)(ε+ 00 ε−

)1√2

(1 11 −1

)

(9.15)

with ε± = ±Tc. Thus,

U(t, t0) ≡ e−iH(t−t0) = Se−i(t−t0)DS−1

=1

2

(1 11 −1

)(e−i(t−t0)Tc 0

0 e+i(t−t0)Tc

)(1 11 −1

)

=1

2

(1 11 −1

)(e−i(t−t0)Tc e−i(t−t0)Tc

e+i(t−t0)Tc −e+i(t−t0)Tc

)

=

(cos[(t− t0)Tc] −i sin[(t− t0)Tc]−i sin[(t− t0)Tc] cos[(t− t0)Tc]

)

= cos[(t− t0)Tc]1− i sin[(t− t0)Tc]σx. (9.16)


9.1.2.3 Quantum Oscillations in Two-Level Systems

We can now easily calculate these: use an initial condition

|Ψ〉0 = αL|L〉+ αR|R〉 =(αL

αR

)

|Ψ(t)〉 = U(t, t0)|Ψ〉0 =cos[(t− t0)Tc]1− i sin[(t− t0)Tc]σx

|Ψ〉0

= αL cos[(t− t0)Tc]− iαR sin[(t− t0)Tc] |L〉+ αR cos[(t− t0)Tc]− iαL sin[(t− t0)Tc] |R〉. (9.17)

Check out a few examples:αL = 1, αR = 0 (particle initially in left well): in this case, the probabilities for theparticle to be in the left (right) well at time t ≥ t0 are

|〈L|Ψ(t)〉|2 = cos2[(t− t0)Tc] (9.18)

|〈R|Ψ(t)〉|2 = sin2[(t− t0)Tc] quantum-mechanical oscillations.

9.2 Time-dependent Hamiltonians

There are almost no exact analytical solutions when the Hamiltonian, H(t), is time-dependent. A few exceptions do exist, however.

9.2.1 Spin 12 in Magnetic Field

This case is, for example, extremely important for NMR (nuclear magnetic resonance).Even here the Hamiltonian H(t) is in general not exactly soluble, its form is

H(t) ≡ B(t)σ ≡ Bx(t)σx +By(t)σy +Bz(t)σz

≡(Bz(t) B∗

‖(t)

B‖(t) −Bz(t)

)

, B‖(t) ≡ Bx(t) + iBy(t), (9.19)

where the Pauli-matrices are defined as

σx ≡(

0 11 0

)

, σy ≡(

0 −ii 0

)

, σz ≡(

1 00 −1

)

. (9.20)

Why is that so difficult? Let us write the Schrodinger equation

i∂t|Ψ(t)〉 = H(t)|Ψ(t)〉, |Ψ(t)〉 ≡(ψ1(t)ψ2(t)

)

id

dtψ1(t) = Bz(t)ψ1(t) +B∗

‖(t)ψ2(t)

id

dtψ2(t) = B‖(t)ψ1(t)−Bz(t)ψ2(t). (9.21)


We assume B‖ 6= 0 and write (omit the t-dependence for a moment)

ψ1 =iψ2 +Bzψ2

B‖(9.22)

iψ2 = B‖ψ1 +B‖ψ1 − Bzψ2 −Bzψ2

=B‖B‖

[iψ2 +Bzψ2]− iB‖[Bzψ1 +B∗‖ψ2]− Bzψ2 −Bzψ2

=B‖B‖

[iψ2 +Bzψ2]− iBz[iψ2 +Bzψ2]− iB‖B∗‖ψ2 − Bzψ2 −Bzψ2

= iB‖B‖

ψ2 +

[

B‖B‖Bz − iB2

z − i|B‖|2 − Bz

]

ψ2. (9.23)

This is a second order ODE with time-dependent coefficients, which in general is notsolvable in terms of known functions (it can of course be solved numerically quite easily).

9.2.1.1 Constant B

In this case we must of course recover our usual two-level system:

iψ2 = −i[B2z + |B‖|2]ψ2 = −i|B|2ψ2 (9.24)

ψ2 + |B|2ψ2 = 0 (9.25)

ψ2(t) = ψ2(0) cos |B|t+ψ2(0)

|B| sin |B|t (9.26)

For constant B, the eigenvalues of the Hamiltonian

H ≡(Bz B∗

‖B‖ −Bz

)

(9.27)

are given by (Bz − ε)(−Bz − ε) − |B‖|2 = 0 or ε± = ±√

B2z + |B‖|2 = ±|B|. Therefore,

Eq. (9.24) describes quantum mechanical oscillations with angular frequency of half thelevel splitting 2|B| between ground and excited state, in agreement with our specificexample Bz = 0, B‖ = Tc from section 9.1.2.3.

9.2.1.2 Rotating Field

This is defined as for constant field in z direction and an oscillating field in the x-y plane,

Bz(t) = B0 = const, B‖(t) = B1eiωt. (9.28)


Our equation for ψ2 thus becomes

iψ2 = iB‖B‖

ψ2 +

[

B‖B‖

Bz − iB2z − i|B‖|2 − Bz

]

ψ2

= −ωψ2 + i[ωB0 −B20 − |B1|2]ψ2

ψ2 − iωψ2 + [B20 + |B1|2 − ωB0]ψ2 = 0. (9.29)

This can be solved using the exponential ansatz method ψ2(t) = ce−izt which yields aquadratic equation for z,

z2 − ωz + [ωB0 −B20 − |B1|2] = 0

z± =ω

2± 1

2

√

ω2 + 4B20 − 4ωB0 + 4|B1|2 =

ω

2± 1

2ΩR

ΩR ≡√

(ω − 2B0)2 + 4|B1|2 Rabi-frequency. (9.30)

Note that the term 2B0 in the Rabi-frequency is determined by the level-splitting ∆ ≡2B0 in absence of the time-dependent field B‖(t). The solution for ψ2(t) (from whichψ1(t) follows immediately) therefore is

ψ2(t) = c1ei“

ω2+

ΩR2

”

t+ c2e

i“

ω2−ΩR

2

”

t

= eiω2

t

[

c′1 cosΩR

2t+ c′2 sin

ΩR

2t

]

. (9.31)

We can choose, e.g., the initial condition ψ2(0) = 1 from which follows

ψ2(t) = eiω2

t

[

cosΩR

2t+ c′2 sin

ΩR

2t

]

0 = ψ1(0) =iψ2 +Bzψ2

B‖

∥∥∥∥∥

t=0

=−ω

2 + iΩR

2 c′2 +B0

B1

c′2 = −iω − 2B0

ΩR(9.32)

This leads to

|ψ2(t)|2 = cos2ΩR

2t+

(ω − 2B0)2

Ω2R

sin2 ΩR

2t (9.33)

=(ω − 2B0)

2

Ω2R

+4|B1|2Ω2

R

cos2ΩR

2t Rabi-Oscillations.

Note that the quantum-mechanical oscillations at constant B (e.g., the case B =(Tc, 0, 0) in Eq. (9.18)) occur for a time-independent Hamiltonian. The Rabi-oscillationsoccur in a time-dependent Hamiltonian containing a time-dependent term (‘time-dependentfield’). These two often get mixed up in the literature.


9.2.2 Landau-Zener-Rosen problem

This is another exactly solvable case for a two-level system. To be discussed later in thecontext of adiabatic and non-adiabatic transitions between energy levels.

9.3 Time-Dependent Perturbation Theory

9.3.1 Model Hamiltonian

This is written in the form

H(t) = H0 + V (t) (9.34)

with the time-dependence in the perturbation V (t). The case H(t) = H0(t) + V witha constant perturbation operator but a time-dependent ‘free part’ also exists, but isslightly less used.

The term ‘time-dependent’ perturbation theory, however, primarily refers to per-turbation theory for the time-dependence of the wave function and is also used fortime-independent Hamiltonians H(t) = H.

9.3.2 The Interaction Picture

This is introduced in order to facilitate the solution of the Schrodinger equation

d

dt|Ψ(t)〉 = −iH(t)|Ψ(t)〉 (9.35)

We define

|Ψ(t)〉I ≡ eiH0t|Ψ(t)〉 (9.36)

and derive the new Schrodinger equation for |Ψ(t)〉I ,

d

dt|ΨI(t)〉 = iH0|ΨI(t)〉+ eiH0t d

dt|Ψ(t)〉

= iH0|ΨI(t)〉 − ieiH0t [H0 + V (t)] e−iH0t|ΨI(t)〉= −ieiH0tV (t)e−iH0t|ΨI(t)〉 ≡ −iVI(t)|ΨI(t)〉. (9.37)

The Schrodinger equation therefore is transformed into the interaction picture

d

dt|Ψ(t)〉 = −iH(t)|Ψ(t)〉 ↔ d

dt|ΨI(t)〉 = −iVI(t)|ΨI(t)〉

|Ψ(t)〉I ≡ eiH0t|Ψ(t)〉, VI(t) ≡ eiH0tV (t)e−iH0t. (9.38)


9.3.3 First Order Perturbation Theory

This is achieved by doing the first iteration in

d

dt|ΨI(t)〉 = −iVI(t)|ΨI(t)〉

|ΨI(t)〉 = ΨI(t0)〉 − i∫ t

t0

dt′VI(t′)|ΨI(t

′)〉

= |ΨI(t0)〉 − i∫ t

t0

dt′VI(t′)|ΨI(t0)〉+O(V 2

I ). (9.39)

We take t0 = 0 for simplicity and the initial state therefore is |ΨI(0)〉 = |Ψ(0)〉,

|ΨI(t)〉 = |Ψ(0)〉 − i∫ t

0dt′VI(t

′)|Ψ(0)〉 +O(V 2I ). (9.40)

This can be worked out in some more detail by assuming a basis |n〉 of eigenstates ofH0,

H0|n〉 = εn|n〉, 1 =∑

n′

|n′〉〈n′| (9.41)

〈n|ΨI(t)〉 = 〈n|Ψ(0)〉 − i∑

n′

∫ t

0dt′〈n|VI(t

′)|n′〉〈n′|Ψ(0)〉 +O(V 2I ).

Let us assume the initial state |Ψ(0)〉 = |m〉 is an eigenstate of H0, then

〈n|ΨI(t)〉 = δnm − i∫ t

0dt′〈n|VI(t

′)|m〉+O(V 2I ). (9.42)

The probability to find the system in state |n〉 after time t is then a transition proba-bility. Use |〈n|Ψ(t)〉|2 = |〈n|ΨI(t)〉|2 (EXERCISE:CHECK!) to find within first orderperturbation theory

Pm→n(t) = |〈n|Ψ(t)〉|2 =

∣∣∣∣

∫ t

0dt′〈n|VI(t

′)|m〉∣∣∣∣

2

first order

|Ψ(0)〉 ≡ |m〉 6= |n〉. (9.43)

9.3.3.1 ‘Sudden switching’ Hamiltonian V (t) = V θ(t)

In this case

〈n|VI(t′)|m〉 = 〈n|eiH0t′V e−iH0t′ |m〉 = e−i(εn−εm)t′〈n|V |m〉

Pm→n(t) = |〈n|V |m〉|2∣∣∣∣

∫ t

0dt′e−i(εn−εm)t′

∣∣∣∣

2

= |〈n|V |m〉|2∣∣∣∣∣

e−i(εn−εm)t − 1

εn − εm

∣∣∣∣∣

2

= |〈n|V |m〉|2 4sin2 εn−εm

2 t

(εn − εm)2(9.44)


As for the sin2 function, we now use the representation of the Dirac Delta-function,

Theorem:For any integrable, normalised function f(x) with

∫∞−∞ dxf(x) = 1,

limε→0

1

εf(x

ε

)

= δ(x). (9.45)

Here, we apply it with f(x) = 1π

sin2(x)x2

limε→0

1

ε

1

π

sin2(x/ε)

(x/ε)2= δ(x), lim

t→∞t

2π

sin2(∆Et/2)

(∆Et/2)2= δ(∆E)

limt→∞

1

tPm→n(t) = lim

t→∞|〈n|V |m〉|2 2π

t

2π

sin2 εn−εm

2 t

[(εn − εm)t/2]2

= 2π |〈n|V |m〉|2 δ(εn − εm). (9.46)

This is an extremely important result, and we therefore highlight it here again, introdu-cing the transition rate Γm→n,

Γm→n ≡ limt→∞

1

tPm→n(t) =

2π

~2|〈n|V |m〉|2 δ(εn − εm) (9.47)

The total transition rate into any final state |n〉 is, within first order perturbation-theory in V , given by the sum over all n,

Γm ≡ 2π

~2

∑

n

|〈n|V |m〉|2 δ(εn − εm)

Fermi’s Golden Rule. (9.48)

9.3.4 Higher Order Perturbation Theory

(This is also discussed in Merzbacher [4] though with a slightly different notation.We start from the time-dependent Schrodinger equation

i∂t|Ψ(t)〉 = H(t)|Ψ(t)〉. (9.49)

The state |Ψ(t)〉 at time t is obtained from the state |Ψ(t0)〉 at time t0 by application ofthe time evolution operator U(t, t0) via

|Ψ(t)〉 = U(t, t0)|Ψ(t0)〉. (9.50)

If H(t) = H is time-independent, we have

U(t, t0) = e−iH(t−t0), time-independent Hamiltonian. (9.51)

For arbitrary H(t), we have

i∂tU(t, t0) = H(t)U(t, t0), U(t0, t0) = 1. (9.52)


We now assume a form

H(t) = H0 + V (t). (9.53)

We solve this differential equation by introducing the interaction picture time-evolutionoperator U(t, t0),

U(t, t0) = eiH0tU(t, t0)e−iH0t0 (9.54)

i∂tU(t, t0) = −H0U(t, t0) + i(−i)eiH0tH(t)U(t, t0)e−iH0t0

= −H0U(t, t0) + eiH0t[H0 + V (t)]e−iH0teiH0tU(t, t0)e−iH0t0

= −H0U(t, t0) + eiH0t[H0 + V (t)]e−iH0tU(t, t0)

= V (t)U (t, t0) (9.55)

V (t) = eiH0tV (t)e−iH0t. (9.56)

9.3.5 States

Using Eq. (9.36), in the interaction picture

|Ψ(t)〉I ≡ eiH0t|Ψ(t)〉 = eiH0tU(t, t0)|Ψ(t0)

= eiH0te−iH0tU(t, t0)eiH0t0 |Ψ(t0)〉

= U(t, t0)|Ψ(t0)〉I . (9.57)

9.3.5.1 Successive Interation

We can obtain the time-evolution operator U(t, t0) in the interaction picture by succes-sive iteration:

i∂tU(t, t0) = V (t)U(t, t0) (9.58)

U(t, t0) = 1− i∫ t

t0

dt1V (t1)U(t1, t0)

= 1− i∫ t

t0

dt1V (t1) + (−i)2∫ t

t0

dt1

∫ t1

t0

dt2V (t1)V (t2) + ...

= 1 +

∞∑

n=1

(−i)n∫ t

t0

dt1

∫ t1

t0

dt2...

∫ tn−1

t0

dtnV (t1)...V (tn).

There is a compact notation that slightly simplifies things here: time-ordered productsof operators are defined with the time-ordering operator T which orders a productof operators V (t1)...V (tn) with arbitrary times t1,...,tn such that the ‘earliest’ operatoris left and the ‘latest’ operator is right. For example,

T [V (t1)V (t2)] = θ(t1 − t2)V (t1)V (t2) + θ(t2 − t1)V (t2)V (t1), (9.59)

where

θ(t ≥ 0) = 1, θ(t < 0) = 0. (9.60)


Using the time-ordering operator, one can then show

U(t, t0) = 1 +

∞∑

n=1

(−i)nn!

∫ t

t0

dt1

∫ t

t0

dt2...

∫ t

t0

dtnT [V (t1)...V (tn)]

≡ T exp

[

−i∫ t

t0

dt′V (t′)

]

. (9.61)

Note that now the upper limit of all integrals is the same t and that there is the additional1/n! in front of each term.

10. ROTATIONS AND VIBRATIONS OF

MOLECULES

10.1 Vibrations and Rotations in Diatomic Molecules

Here, we follow Weissbluth [2] ch. 27, and Landau-Lifshitz III [5].

10.1.1 Hamiltonian

Before deriving the Hamiltonian, a short excursion to classical mechanics of two particles:

10.1.1.1 Angular Momentum of Two Particles

If two particles have positions r1 and r2 and momenta p1 and p2, the angular momentumof the total system of the two particles is

L = r1 × p1 + r2 × p2. (10.1)

We introduce center-of-mass and relative coordinates according to

R =m1r1 +m2r2

m1 +m2, r = r2 − r1, (10.2)

and furthermore momenta

P = p1 + p2 (10.3)

p =m1p2 −m2p1

m1 +m2. (10.4)

Note that the relative momentum p is not just the difference of the individual momenta.It is rather defined such that in terms of

µ ≡ m1m2

m1 +m2reduced mass, (10.5)

one has

µr = µ(r2 − r1) = µ

(p2

m2− p1

m1

)

= p. (10.6)

10. Rotations and Vibrations of Molecules 122

Using these definitions, one checks

L = r1 × p1 + r2 × p2 (10.7)

= R×P + r× p. (10.8)

This is the sum of a center-of-mass angular momentum, R ×P, and a relative angularmomentum, r× p.

10.1.1.2 Born-Oppenheimer Approximation

We recall the Born-Oppenheimer Approximation for the total wave function Ψ(q,X) ofa molecule, cf. Eq. (8.18),

Ψ(q,X) =∑

α

φα(X)ψα(q,X), (10.9)

where q stands for the electronic, X for the nuclear coordinates, and the sum is overa complete set of adiabatic electronic eigenstates with electronic quantum numbers α.This form leads to the Schrodinger equation in the adiabatic approximation Eq. (8.20),

[〈ψα|Hn|ψα〉e + Eα(X)] |φα〉n = E|φα〉n. (10.10)

Here, Eα(X) is the potential acting on the nuclei. We now specify the kinetic energy ofthe nuclear part for a diatomic molecule,

Hn =P2

2M+

p2

2µ. (10.11)

Exercise Check that Hn =p2

12m1

+p2

22m2

.

The effective nuclear Hamiltonian corresponding to an electronic eigenstate α thusis

Hn,α = 〈ψα|P2

2M|ψα〉+ 〈ψα|

p2

2µ|ψα〉+ Eα(r), (10.12)

which is a sum of a center-of-mass Hamiltonian and a Hamiltonian for the relative motionof the two nuclei. Only the latter is interesting because it contains the potential Eα(r).Note that both center-of-mass and relative Hamiltonian still contain the geometricalphase terms, cf. Eq. (8.23), which however we will neglect in the following.

10.1.2 Angular Momentum

Neglecting the geometric phase terms, Eq. (8.23), we have in three spatial dimensions

Hreln,α =

p2

2µ+ Eα(r) = − ~

2

2µ∆r + Eα(r)

= − ~2

2µ

(∂2

∂r2+

2

r

∂

∂r

)

+J2

2µr2+ Eα(r), (10.13)


where J is the relative angular momentum operator of the nuclei. We have a three-dimensional problem which however due to the radial symmetry of Eα(r) is reduced toa one-dimensional radial eqaution, very much as for the case of the hydrogen atom! Wecould write the eigenfunctions of Hrel

n,α as

Ψ(r) = R(r)YJM(θ, φ) (10.14)

with the corresponding angular quantum numbers J andM of the nuclear relative motionseparated off in the spherical harmonics.

Instead of dealing with the angular momentum operator of the nuclei, one wouldrather descrive rotations of the whole molecule by the total angular momentum K of themolecule

K = J + L + S, (10.15)

where L is the total angular momentum of all electrons and S is the total spin.

10.1.2.1 Spin S = 0

This is the simplest case. The total angular momentum of the nuclei is then

J = K− L. (10.16)

Since we have neglected geometric phase terms, we can replace ∆r by its expectationvalue in the electronic state α under consideration,

∆r = 〈ψα|∆r|ψα〉 (10.17)

J2 = 〈ψα|J2|ψα〉 = 〈ψα|(K− L)2|ψα〉. (10.18)

This allows one to express everything in terms of total angular quantum numbers K asfollows: We first write

J2 = 〈ψα|(K− L)2|ψα〉 (10.19)

= 〈ψα|K2|ψα〉 − 2〈ψα|KL|ψα〉+ 〈ψα|L2|ψα〉.

First, K2 is conserved and can be replaced by its eigenvalue K(K + 1) whence

〈ψα|K2|ψα〉 = K(K + 1). (10.20)

Second, 〈ψα|L2|ψα〉 only depends on the electronic degrees of freedom and can thereforebe simply added to the potential Eα(r).

Finally, we assume that the electronic state α is an eigenstate of the z componentLz with eigenvalue Λ of the electronic angular momentum. Then,

〈ψα|KL|ψα〉 = K〈ψα|L|ψα〉 = KezΛ. (10.21)


On the other hand, we have Jez = 0 since the angular momentum of the two nuclei isperpendicular to the molecule axis ez ∝ r, thus

(Kz − Lz) = 0 Kz = Lz (10.22)

and

〈ψα|KL|ψα〉 = KezΛ = LzΛ

= 〈ψα|Lz|ψα〉Λ = Λ2. (10.23)

Summarizing, we now have for the radial part

− ~2

2µ

(∂2

∂r2+

2

r

∂

∂r

)

+K(K + 1)

2µr2− 2Λ2

2µr2+〈ψα|L2|ψα〉

2µr2+ Eα(r)

≡ − ~2

2µ

(∂2

∂r2+

2

r

∂

∂r

)

+K(K + 1)

2µr2+ Uα(r).

Thus we have finally arrived at the form for the effective potential energy,

K(K + 1)

2µr2+ Uα(r). (10.24)

The first term K(K+1)2µr2 is the centrifugal energy as in the hydrogen problem. Since Kz =

Lz with fixed eigenvalue Λ for the given state α, the eigenvalues of the total angularmomentum must fulfill

K ≥ Λ. (10.25)

10.1.3 Radial SE

Our SE for the radial motion of the two nuclei has the form[

− ~2

2µ

(∂2

∂r2+

2

r

∂

∂r

)

+K(K + 1)

2µr2+ Uα(r)

]

Rα;Kv(r) = εα;KvRα;Kv(r). (10.26)

We therefore have two sets of quantum numbers K and v that describe the rotationaland vibrational and state of the molecule for a given electronic state α. Setting

Rα;Kv(r) =1

rPα;Kv(r) (10.27)

leads to a standard one-dimensional SE with a ‘proper’ d2

dr2 kinetic energy term,

[

− 1

2µ

d2

dr2+ Uα(r) +

K(K + 1)

2µr2

]

Pα;Kv(r) = εα;KvPα;Kv(r), r ≥ 0. (10.28)


10.1.3.1 Harmonic Approximation

The rotation term K(K+1)2µr2 is assumed as small, and the potential Uα(r) is expanded

around a minimum rα,

Uα(r) = Uα(rα) +1

2

d2

dr2Uα(r = rα)(r − rα)2 + ... (10.29)

Here, rα can be considered as the equilibrium distance of the two nuclei which clearlystill depends on the electronic quantum number α. If the higher order terms in the Taylorexpansion are neglected, and K(K+1)

2µr2 replaced by K(K+1)2µr2

α, the approximate SE becomes

[

− 1

2µ

d2

dr2+K(K + 1)

2µr2α+ Uα(rα) +

1

2µω2

α(r − rα)2]

P harmα;Kv (r)

= εα;KvPharmα;Kv (r), ω2

α =1

µ

d2

dr2Uα(r = rα). (10.30)

This is the equation of a linear harmonic oscillator apart from the fact that r ≥ 0.However, |r− rα| has been assumed to be small anyway and within this approximation,

the energy levels are therefore those of a linear harmonic oscillator shifted by K(K+1)2µr2

α+

Uα(rα),

εharmα;Kv =

K(K + 1)

2µr2α+ Uα(rα) + ωα

(

v +1

2

)

. (10.31)

10.1.3.2 The Energy Spectrum

The structure of the energy spectrum is determined by the magnitude of the three termsK(K+1)

2µr2α

, Uα(rα), and ωα

(v + 1

2

). These differ strongly due to their dependence on the

relative nuclei mass µ. In terms of the small dimensionless parameter m/µ (where m isthe electron mass), we have

Uα = O(1), electronic part (10.32)

ωα

(

v +1

2

)

= O(m/µ)1/2, vibrational part (10.33)

K(K + 1)

2µr2α= O(m/µ), rotational part. (10.34)

In spectroscopic experiments, one determined energy differences δE which therefore arebroadly determined by

δEel ≫ δEvib ≫ δErot. (10.35)

10.1.4 Spin S > 0

Things get a little bit more complicated for S > 0 which leads to the so-called Hund’scases a, b, c and d. For more details cf. Landau-Lifshitz III [5], or Atkins-Friedman.


10.1.5 Beyond the Harmonic Approximation

The harmonic approximation has to break down somewhere. A diatomic molecule withits two nuclei harmonically bound would never be able to dissociate into two individualatoms or ions.

One way is to introduce phenomenological potentials with fitting parameters, e.g.the Morse potential

Uα(r)→ UMorseα (r) ≡ Dα

[

1− e−βα(r−rα)]2, (10.36)

where Dα is the depth of the minimum below the asymptote and represents the disso-ciation energy of the molecule.

10.2 Selection Rules

10.2.1 Dipole Approximation

Assume system of charges qn localised around a spatial position r0 = 0. The coupling toan electric field E(r, t) within dipole approximation is then given by

Hdip(t) = −dE(t), d ≡∑

n

qαrα, (10.37)

where E(t) ≡ E(r0, t) is the electric field at r0 = 0. The dipole approximation is valid ifthe spatial variation of E(r, t) around r0 is important only on length scales l with l≫ a,where a is the size of the volume in which the charges are localised. For a plane waveelectric field with wave length λ one would have l ∼ λ.

10.2.2 Pure Rotation

Pure rotational transitions are between states where only rotational quantum numbersare changed,

|KmK , v, α〉 → |K ′m′K , v, α〉 (10.38)

leaving the vibrational quantum number(s) v and the electronic quantum number(s) αunchanged. Such transitions then depend on matrix elements of the dipole operator,

〈KmK |d|K ′m′K〉. (10.39)

The calculation of this matrix element, using spherical harmonics, yields the purelyrotational selection rules

∆K = ±1, ∆mK = 0,±1. (10.40)

Writing the rotational part of the energy as

εrot(K) = BK(K + 1) (10.41)

∆εrot(K) ≡ B(K + 1)(K + 2)−BK(K + 1) = 2B(K + 1).

The distance between the corresponding spectral lines is constant, ∆εvib(K + 1) −∆εvib(K) = 2B.


10.2.3 Pure Vibration

In this case, we have to deal with the harmonic oscillator.

10.2.3.1 Recap of the Harmonic Oscillator

The Hamiltonian of the harmonic oscillator

Hosc =p2

2m+

1

2mω2x2 (10.42)

can be re-written using the ladder operators

a ≡√mω

2~x+

i√2m~ω

p, a† ≡√mω

2~x− i√

2m~ωp (10.43)

x =

√

~

2mω

(

a+ a†)

, p = −i√

m~ω

2

(

a− a†)

, (10.44)

as

Hosc = ~ω

(

a†a+1

2

)

. (10.45)

The commutation relation is

[x, p] = i~, [a, a†] = 1. (10.46)

The eigenfunctions of the harmonic oscillator are n-phonon states,

Hosc|n〉 = εn|n〉, εn = ~ω

(

n+1

2

)

, n = 0, 1, 2, ...

|n〉 ↔ ψn(x) =(mω

π~

)1/4 1√n!2n

Hn

(√mω

~x

)

e−mω2~

x2, (10.47)

where Hn are the Hermite polynomials.The ladder operators are also called creation (a†) and annihiliation (a) operators.

They act on the states |n〉 as

a†|n〉 =√n+ 1|n+ 1〉, a|n〉 =

√n|n− 1〉, a|n〉 = 0. (10.48)

The state |0〉 is called ground state.

10.2.3.2 Pure Vibrational Dipole Transitions

Pure vibrational transitions are between states where only vibrational quantum numbersare changed,

|KmK , v, α〉 → |KmK , v′, α〉. (10.49)


Such transitions then depend on matrix elements of the dipole operator,

〈v|dα|v′〉, (10.50)

where |v〉 is an harmonic oscillator eigenstate (we write v instead of n now), and

dα = 〈α|d|α〉 (10.51)

is the diagonal matrix element of the dipole operator between the adiabatic electroniceigenstates |α〉.

Remember that the harmonic potential came from the Taylor expansion of the Born-Oppenheimer energy,

Uα(r) ≈ Uα(rα) +1

2

d2

dr2Uα(r = rα)(r − rα)2

Hosc =p2

2µ+

1

2mω2

αx2 = ~ωα

(

a†a+1

2

)

(10.52)

ω2α =

1

µ

d2

dr2Uα(r = rα) (10.53)

where the harmonic oscillator coordinate x = r − rα.The dipole moment operator dα depends on the electronic wave functions α and

thus parametrically on the coordinate x that describes the internuclear separation. WeTaylor-expand

dα(x) = dα(0) + d′α(0)x+O(x2). (10.54)

For transitions between v and v′ 6= v, one therefore has to linear approximation

〈v|dα|v′〉 = d′α(0)〈v|x|v′〉 = d′

α(0)

√

~

2µω〈v|a+ a†|v′〉

= d′α(0)

√

~

2µω

(δv+1,v′

√v + 1 + δv−1,v′

√v). (10.55)

The vibrational selection rule thus is

∆v = ±1. (10.56)

The corresponding energy differences determine the transition frequency,

∆εvib(v) = ~ωα, (10.57)

which means that a purely vibrational, harmonic spectrum just consists of a singlespectral line!


Abbildung 10.1: Franck-Condon-Principle. Left: Classical picture, right: quantum-mechanicalpicture. From Prof. Ed Castner’s lecture http://rutchem.rutgers.edu/.

10.2.4 Vibration-Rotation Spectra

Vibrational and Rotational transitions are coupled, and one now has to discuss thevarious transition possibilities. This leads to a description in terms of P-, Q-, andR-branch for the allowed transitions in diatomic molecules. For further reading, cf.Atkins/Friedman [3] ch. 10.11 or Weissbluth [2] ch. 27.2

10.3 Electronic Transitions

Transitions between two molecular states in general involve all quantum numbers: elec-tronic, vibrational, and rotational, i.e.

|KmK , v, α〉 → |K ′m′K , v

′, α′〉. (10.58)

10.3.1 The Franck-Condon Principle

Here, a good description is in Atkins/Friedman ch. 11.4.For simplicity, we leave out the rotations here and just discuss electronic and vibra-

tional transitions. In a classical picture (with respect to the large mass nuclear motion),one considers the two potential curves Uα(r) and Uα′(r) and argues that the electronictransition occurs so fast that the nuclear system has no time to react: before and after


the transition, the nuclear coordinate X is the same. This, however, means that thedistance |x′| ≡ |X − rα′ | from the equilibrium position rα′ after the transition and thedistance |x| ≡ |X − rα| from the equilibrium position rα before the transition are notthe same: when the nuclei are in equilibrium before the transition (X = rα, x = 0), theirnew coordinate x′ relative to the new equilibrium rα′ is x′ ≡ X− rα′ = rα− rα′ 6= 0 afterthe transition.

The total dipole moment operator is a sum of electronic and nuclear dipole moment,

d = −e∑

i

qi + e∑

s

ZsXs = de + dn. (10.59)

The transition matrix element in Born-Oppenheimer approximation is (α 6= α′)

〈α′v′|de + dn|αv〉 =

∫

dqdXψ∗α′(qX)φ∗α′,v′(X)(de + dn)φα,v(X)ψα(qX)

=

∫

dXφ∗α′,v′(X)

[∫

dqψ∗α′(qX)deψα(qX)

]

φα,v(X)

+

∫

dXφ∗α′,v′(X)φα,v(X)dn

∫

dqψ∗α′(qX)ψα(qX)

=

∫

dXφ∗α′,v′(X)

[∫

dqψ∗α′(qX)deψα(qX)

]

φα,v(X) + 0

≈ 〈α′|de|α〉S(v, v′), S(v, v′) ≡ 〈v′|v〉. (10.60)

Here it was assumed that the integral

∫

dqψ∗α′(qX)deψα(qX) ≈ 〈α′|de|α〉 (10.61)

does not depend on the nuclear coordinates X.The transition between two electronic levels α and α′ is therefore determined by

the dipole matrix element 〈α′|de|α〉 and the Franck-Condon factors S(v, v′), whichare the overlap integrals of the corresponding vibronic states. As these states belongto different electronic states α and α′, the overlaps are not zero, and there is also noselection rule for ∆v.

LITERATURVERZEICHNIS

[1] S. Gasiorowicz, Quantum Physics (Wiley, New York, 2003).

[2] M. Weissbluth, Atoms and Molecules (Academic Press, New York, 1978).

[3] P. W. Atkins and R. S. Friedman, Molecular Quantum Mechanics, 3 ed. (OxfordUniversity Press, Oxford, 1997).

[4] E. Merzbacher, Quantum Mechanics, 3 ed. (John Wiley, Weinheim, 1998).

[5] L. D. Landau and E. M. Lifshitz, Quantum Mechanics, Vol. 3 of Landau and Lifshitz,Course of Theoretical Physics (Pergamon Press, Oxford, 1965).

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the zero–vector because the norm of a|niis ka|nik= √ n>0. Therefore, 0