Quantum Mechanics – Applications and Atomic Structures

Jeong Won KangDepartment of Chemical Engineering

Korea University

Applied Statistical Mechanics Lecture Note - 3

Subjects

Three Basic Types of Motions (single particle) Translational Motions Vibrational Motions Rotational Motions

Atomic Structures Electronic Structures of Atoms One-electron atom / Many-electron atom

Free Translational Motion

ψψ Edxd

m=− 2

22

2

ψψ EH =

All values of energies are possible (all values of k) Momentum Position See next page

2

22

2 dxd

mH −=

ikxikxk BeAe −+=ψ

OR

Solutions

mkEk 2

22=

kpe xikx +=→ kpe x

ikx −=→−

Probability Density

B = 0

A = 0

A = B

ikxAe=ψ

ikxBe−=ψ

kxAcos2=ψ

22 A=ψ

22 B=ψ

kxA 22 cos4=ψnodes

A particle in a box

m

Consider a particle of mass m is confined between two walls

LxxVLxV

==∞=<<=

and 0for 0for 0

ψψψ ExVdxd

m=+− )(

2 2

22

kxDkxCk cossin +=ψ mkEk 2

22=

All the same solution as the free particlebut with different boundary condition

A particle in a box

Applying boundary condition

Normalization

0)(0)0(

====

Lxx

ψψ

kxDkxCk cossin +=ψ

0=D0sin =kLC πnkL =

1,2,3,...n )/sin()( == LxnCxn πψ

12

)/(sin2

0

22

0

2 === LCdxLxnCdx

LL

n πψ

2/12

=

LC 1,2,3,...n )/sin(2)(

2/1

=

= Lxn

Lxn πψ mL

hnEn 8

22

=

n cannot be zero

A particle in a box- Properties of the solutions

n : Quantum number (allowable state)

Momentum

1,2,3,...n )/sin(2)(2/1

=

= Lxn

Lxn πψ

Lnk)-e(e

LiLxn

Lx -ikxikx

nππψ =

=

= 2

21)/sin(2)(

2/12/1

kpe xikx +=→ kpe x

ikx −=→−

Probability : 1/2 Probability : 1/2

A particle in a box- Properties of the solutions

n cannot be zero (n = 1,2,3,…) Lowest energy of a particle (zero-point energy) If a particle is confined in a finite region (particle’s

location is not indefinite), momentum cannot be zero

If the wave function is to be zero at walls, but smooth, continuous and not zero everywhere, the it must be curved possession of kinetic energy

mLhE

8

2

1 =

Wave function and probability density

)/(sin2)( 22 LxnL

x πψ

=

With n→∞, more uniform distribution: corresponds to classical prediction

“correspondence principle”

Orthogonality and bracket notation

0'* = τψψ dnn

1 0' == nnnn

)'( 0''* nnnndnn ≠== τψψ

'' nnnn δ=

Orthogonality

Dirac Bracket Notation

<n| “BRA”|n’> “KET”

Kronecker delta

Wave functions corresponding to differentenergies are orthogonal

Orthonormal=Orthogonal + Normalized

Motion in two dimension

ψψψ Eyxm

=

∂∂+

∂∂− 2

2

2

22

2 )()(),( yYxXyx =ψ

XEdx

Xdm X=− 2

22

2

YEdx

Ydm Y=− 2

22

2

YX EEE +=

1

1

2/1

1

sin2)(1 L

xnL

xX nπ

=

2

2

2/1

2

sin2)(2 L

ynL

yYnπ

=

2

2

1

1

2/1

2

2/1

1, sinsin22),(

21 Lyn

Lxn

LLyxnn

ππψ

= m

hLn

LnE nn 2

2

22

22

21

21

, 21

+=

Separation of Variables

Solution

The wave functions for a particle confined to a rectangular surface

2,2 21 == nn1,1 21 == nn 1,2 21 == nn2,1 21 == nn

Degeneracy

Degenerate : Two or more wave functions correspond to the same energy If L1=L2 then

2

2

2,1 85mLhE =

Ly

Lx

Lyx ππψ 2sinsin2),(2,1

=

Ly

Lx

Lyx ππψ sin2sin2),(1,2

= 2

2

1,2 85mLhE =

the same energywith differentwave functions

Degeneracy

Ly

Lx

Lyx ππψ 2sinsin2),(2,1

=

Ly

Lx

Lyx ππψ sin2sin2),(1,2

=

degeneratenot are functions wave the, if 21 LL ≠The degeneracy can be traced to the symmetry of the system

Motion in three dimension

ψψψψ Ezyxm

=

∂∂+

∂∂+

∂∂− 2

2

2

2

2

22

2

3

3

2

2

1

1

2/1

3

2/1

2

2/1

1,, sinsinsin222),,(

321 Lyn

Lyn

Lxn

LLLzyxnnn

πππψ

=

mh

Ln

Ln

LnE nnn 2

2

23

23

22

22

21

21

,, 321

++=

Solution

Tunneling

If the potential energy of a particle does not rise to infinity when it is in the wall of the container and E < V , the wave function does not decay to zero

The particle might be found outside the container (leakage by penetration through forbidden zone)

Use of tunneling

STM (Scanning Tunneling Microscopy)

5 nm

fu

Si(111)-7x7

Vibrational Motion

Harmonic Motion

Potential Energy

Schrödinger equation

kxF −=

2

21 kxV =

ψψ Ekxdxd

m=+− 2

2

22

21

2

m1 m2

k : force constant

21

21

mmmm

+=μ

reduced mass

Solutions

αxy =

Wave Functions

Energy Levels

4/12

=

mkα

,...2,1,0 )21(

2/1

=

=+= v

mkvEv ωω

2/2

)( yvvv eyHN −=ψ

Energy Levels,...2,1,0 )

21(

2/1

=

=+= v

mkvEv ωω

1 ω=−+ vv EE

21

0 ω=E

Reason for zero-point energy the particle is confined : position is not

uncertain, momentum cannot be zero

2

21 kxV =

Potential energy

Quantization of energy : comes from B.C

±∞== xv at 0ψ

Wave functions

Quantum tunneling

The properties of oscillators

Expectation values

Mean displacement and mean square displacement

Mean potential energy and mean kinetic energy

Virial Theorem If the potential energy of a paticle has the form V=axb, then its mean

potential and kinetic energies are related by

dxvv vv∞

∞−

Ω=Ω=Ω ψψ ˆˆ *

2/12

)()

21(

mkvx +=0=x

2/12

)()

21(

21

21

mkvkxV +==

VbEK =2

vEV21= vK EE

21=

Rotation in Two Dimensions : Particles on a ring

A Rotational motion can be described by its angular momentum J J : vector Rate at which a particle circulates Direction : the axis of rotation

A particle of mass m constrained to move in a circular path of radius r in xy-plane V = 0 everywhere Total Energy = Kinetic Energy

locityangular ve : inertial ofmoment : 2

ω

ωmrII

IJ=

=

IJE

prJmpE

z

z

2

2/

2

2

=

±==

Rotation in Two Dimension

Wave Functions

Energy Levels

,...2,1,0 ±±=lm

Im

IJE lz

ml 22

222 ==

2/1)2()(

πφψ

φl

l

im

me=

Schrödinger eqn.

)()2( φψπφψll mm =+ψ

φψ

22

2 2

IEdd =

B.C

The wave function have to be

single-valued

Wave functions

States are doubly degenerate except for ml = 0

The real parts of wave functions

Rotation in Three dimension

A particle of mass m free to move anywhere on the 3D surface or a sphere

Colatitude (여위도) : θ Azimuth (방위각) : φ

Rotation in Three dimension

llllml −−−= ,...,2,1,

harmonics spherical : ),(, φθlmlY

Wave Functions

Energy Levels

,...,2,1,0=l

IllE

2)1( +=

Schrödinger eqn.

ψψ Em

=∇− 2

2

Wave functions

(2l + 1) fold degeneracy

Orbital momentum quantum number

Magnetic Quantum number

llllml −−−= ,...,2,1,

,...,2,1,0=l

Location of angular nodesWave functions

Angular Momentum

The energy of a rotating particle Classically, Quantum mechanical

Angular MomentumMagnitude of angular momentum Z-component of angular momentum

IJE2

2

=

,...2,1,0 2

)1(2

=+= lI

llE

{ } 2/1)1( hll +=

lm=

Space Quantization

The orientation of rotating body is quantized : rotating body may not take up an arbitrary orientation with respect to some specified axis

Space Quantization

The vector model

lx, ly, lz do not commute with each other. (lx, ly, lz are complementary observables)

uncertainty principle forbids the simultaneous, exact specification of more than one component (unless l=0).

-If lz is known, impossible to ascribe values to lx, ly.

lx = h/2πi {-sinφ(∂/∂θ) -cotθ cosφ(∂/∂ϕ)}

ly = h/2πi {cosφ(∂/∂θ) -cotθ sinφ(∂/∂ϕ)}

ly = h/2πi (∂/∂ϕ)

Angular momentum L (lx, ly, lz)

The Vector Model

The vector representing the state of angular momentum lies with its tip on any point on the mouth of the cone.

Experiment of Stern-Gerlach

Otto Stern and Walther Gerlach (1921) Shot a beam of silver atoms through an inhomogeneous magnetic field Evidence of space quantization

The classically expected result

The Observed outcomingusing silver atoms

Spin

Stern and Gerlach observed two bands using silver atoms (2) The result conflicts with prediction : (2l+1) orientation (l

must be an integer) Angular momentum due to the motion of the electron

about its own axis : spin Spin magnetic number : ms

ms =s, s-1,…,-s Spin angular momentum

Electron spin s = ½ Only two states

866.043 2/1

=

=

Fermions and Bosons

Electrons : s=1/2 Photons : s=1

Particles with half-integral spin (s=1/2)Elementary particles that constitute matters Electrons, nucleus

Fermions

Particles with integral spin (s=0,1,…)Responsible for the forces that binds fermions Photons

Bosons

Atomic Structureand Atomic Spectra

Electronic Structure of Atoms One Electron Atom : hydrogen atomMany-Electron Atom (polyelectronic atom)

Spectroscopy Experimental Technique to determine electronic structure of atoms. Spectrum

• Intensity vs. frequency (ν), wavelength (λ), wave number (ν/c)

Topics

Structure and Spectra of Hydrogen Atoms

Electric discharge is passed through gaseous hydrogen , H2 molecules and H atoms emit lights of discrete frequencies

Spectra of hydrogenic atoms

Balmer, Lyman and Paschen Series ( J. Rydberg) n1 = 1 (Lyman) n1 = 2 (Balmer) n1 = 3 (Paschen) n2 = n1 + 1, n1 +2 , … RH = 109667 cm -1 (Rydberg constant)

Ritz combination principle The wave number of any spectral line is the difference between two terms

−= 2

221

11~nn

RHν

21~ TT −=ν2n

RT Hn =

The wave lengths can be correlated by two integers Two different states

Spectra of hydrogenic atoms

Ritz combination principle Transition of one energy level to another level with emission of energy as photon

Bohr frequency condition

21 hcThcThvE −==Δ

원자에서 방출되거나 흡수된 electromagnetic radiation 은

주어진 특정 양자수로 제한된다.

따라서 원자들은 몇 가지 주어진 상태만을 가질 수 있음을

알 수 있다.

남은 문제는 양자역학을 이용하여 허용된 에너지 레벨을

구하는 것이다.

The Structure of Hydrogenic Atoms

Coulombic potential between an electron and hydrogen atom ( Z : atomic number , nucleus charge = Ze)

Hamiltonian of an electron + a nucleus

reZeV

0

2

4π−=

rZe

mmVEEH N

Ne

enucleusKelectronK

0

22

22

2

,, 422ˆˆ

πε−∇−∇−=++=

Electron coordinate

Nucleus coordinate

Separation of Internal motion

Full Schrödinger equation must be separated into two equations Atom as a whole through the space Motion of electron around the nucleus

Separation of relative motion of electron from the motion of atom as a whole (Justification 13.1)

rZeH

0

22

2

42 πεμ−∇−=

eNe mmm1111 ≈+=

μ

The Schrödinger Equation

ψψ EH = ψψπε

ψμ

Er

Ze =−∇−0

22

2

42

YllY )1(2 +−=Λ

2

2

0

2

2)1(

4 rll

rZeVeff μπε

++−=

ERRVdrdR

RdrRd

eff =−

+− 2

2 2

22

μ

),()(),,( φθφθψ YrRr =

Separation of Variable : Energy is centrosymmetricEnergy is not affeced by

angular component (Y)(Justification 13.2)

R : Radial Wave EquationNew solution required

Y : All the same equation as in Chap.12 (Section 12.7, the particle on a sphere)Spherical Harmonics

The Radial Solutions

0

2aZr=ρ

ERRVdrdR

RdrRd

eff =−

+− 2

2 2

22

μ

Solution

nln

l

lnln eLn

NrR 2/,,, )()( ρρρ −

=

r)in lexponentia (decaying r)in polynomial()( ×=rR

...3,2,1 with 32 222

02

42

=−= nne

eZEnπ

μAllowed Energies

Radial Solution

2

20

04

ema

e

πε=Reduced distance Bohr Radius

Associated Laguerre polynomials

Hydrogenic Radial Wavefunctions

The Radial Solutions

Atomic Orbital and Their Energies

Quantum numbers

n : Principal quantum number (n=1,2,3,…)• Determines the energies of the electron

an electron with quantum number l has angluar momentum

An electron with quantum number ml has z-component of angular momentum

lmln ,,

...3,2,1 with 32 222

02

42

=−= nne

eZEnπ

μ

{ } 1,..,1,0 with 1)l(l 1/2 −=+ nl

l±±±= ,...,2,1,0m with m ll

Shell

Sub Shell

Bound / Unbound State Bound State : negative energy Unbound State : positive energy (not quantized)

Ryberg const. for hydrogen atom

Ionization energy Mininum energy required to remove an electron

form the ground state

32 22

02

4

eehcR H

H πμ−=

32 22

02

4

1e

ehcRE HH π

μ=−=

Energy of hydrogen at ground state n=1

HhcRI =

Ionization Energy

The energy levels

Shells and Subshells Shell

n = 1 (K), 2 (L), 3 (M), 4(N) Subshell ( l=0,…. , n-1)

l = 0 (s), 1 (p), 2 (d), 3(f), 4(g), 5 (h), 6 (i) Examples

n = 1• l = 0 only 1s (1)

n = 2• l = 0, 1 2s (1) , 2p (3)

n = 3• l = 0, 1, 2 3s (1), 3p (3), 3d (5)

number of orbitals in nth shell : n2

n2 –f old degenerate

ml s are limited to the value -l,..,0, …+l

Ways to depicting probability density

Electron densities (Density Shading)

Boundary surface (within 90 % of electron probability)

Radial Nodes

Wave function becomes zero ( R(r) = 0 ) For 2s orbital :

For 3s orbital :

Zar /2 0=

Zar /90.1 0= Zar /10.7 0=

0

0

Radial Distribution Function

Wave Function probability of finding an electron in any region

Probability Density (1s)

Probability at any radius r = P (r ) dr

Radial Wave Function : R(r )

Radial Distribution Function : P(r) dr 을 곱하면 확률이 된다. 1s orbital 에 대하여 ,

0/22 aZre−∝ψ

22 )()( rRrrP =

2ψ

224)( ψπrrP =

0/2230

34)( aZreraZrP −=

Radial Distribution Function

Bohr radius

Wave function Radial Distribution Function

p-Orbital

Nonzero angular momentum l > 0

p-orbital l = 1

• ml = 0

• ml = + 1

• ml = -1

lr∝ψlr∝ψ

zpψ

ypψxpψ

D-orbital

n = 3l = 2ml = +2, +1, 0,1,2

Spectroscopic transitions and Selection rules

Transition (change of state)

All possible transitions are not permissible Photon has intrinsic spin angular momentum : s = 1 d orbital (l=2) s orbital (l=0) (X) forbidden

• Photon cannot carry away enough angular momentum

...3,2,1 with 32 222

02

42

=−= nne

eZEnπ

μn1, l1,ml1 n2, l2,ml2

PhotonhvE =Δ

Selection rule for hydrogenic atoms

Selection rule Allowed Forbidden

Grotrian diagram

1,0 ±=Δ lm1±=Δl

Structures of Many-Electron Atoms

The Schrödinger equation for many electron-atoms Highly complicated All electrons interact with one another Even for helium, approximations are required

Approaches Simple approach based on H atom

• Orbital Approximation Numerical computation technique

• Hartree Fock self consistent field (SCF) orbital

Pauli exclusion principle

Quantum numbersPrincipal quantum number : n Orbital quantum number : l Magnetic quantum number : ml

Spin quantum number : ms

Two electrons in atomic structure can never have all four quantum numbers in common

All four quantum number “Occupied”

Penetration and Shielding

Unlike hydrogenic atoms 2s and 2p orbitals are not degenerate in many-electron atoms electrons in s orbitals generally lie lower energy

than p orbital electrons in many-electron atoms experiences

repulsion form all other electrons shielding

Effective nuclear charge, shielding constant σ−= ZZeff

constant shielding : chargenuclear effective :

σeffZ

Penetration and Shielding

An s electrons has a greater “penetration” through inner shell than p electrons

Energies of electrons in the same shell s < p < d < f

Valence electrons electrons in the outer most shell of an atom in its

ground state largely responsible for chemical bonds

The building-up principle

The building-up principle (Aufbau principle ) The order of occupation of electrons 1s 2s 2p 3s 3p 4s 3d 4p 5s 4d 5p 6s There are complicating effects arising from electron-electron

repulsion (when the orbitals have very similar energies) Electrons occupy different orbitals of a given subshell

before doubly occupying any one of them Example : Carbon

• 1s2 2s2 2p2 1s2 2s2 2px12py

1 (O) 1s2 2s2 2px2 (X)

Hund’s maximum multiplicity principle An atom in its ground state adopts configuration with the greatest

number of unpaired electrons

favorable unfavorable

3d and 4s orbital

Sc atom (Z=21) [Ar]3d3

[Ar]3d24s1

[Ar]3d14s2

3d has lower energy than 4s orbital

3d repulsion is much higher than 4s (average distance from nucleus is smaller in 3d)

Ionization energy

Ionization energy I1 : The minimum energy required to remove an electron from many-

electron atom in the gas phase I2 : The minimum energy required to remove a second electron from

many-electron atom in the gas phase

Li :2s1

Be :2s2

B : 2s2p1

Self-consistent field orbital

Potential energy of electrons

Central difficulty presence of electron-electron interaction Analytical solution is hopeless Numerical techniques are available

• Hartree-Fock Self-consistent field (SCF) procedure (HF method)

+−=i ji ijoio re

ere

ZeV'

,

22

421

4 επεπ

Schrödinger equation for Neon Atom1s22s22p6 , 2p electrons

)(

)()()(

4

)()()(2

)(4

)(2

122

1212

22*

22

12212

2*

22

121

2

1221

2

rE

rdr

rre

rdr

rre

rr

Ze

rm

pp

ii

pi

o

pi

ii

o

po

pe

ψ

ψτψψ

πε

ψτψψπε

ψπε

ψ

=

−

+

−

∇−

Sum over orbitals (1s, 2s)

Kinetic energy of electron

Potential energy (electron – nucleus)

Columbic operator (electron – electron charge density)

Exchange operator (Spin correlation effect)

HF Procedure

There is no hope solving previous eqn. analytically Alternative procedure (numerical solution)

Assume 1s, 2s orbital

Solve 2p

Solve 2s

Solve 1s

Compare E andwave function

Converged Solution

Example

Orbitals of Na using SCF calculation

Quiz

What is a degeneracy ? Explain four quantum number.Why transition between two states are not always

allowed ? Explain difficulties of solving Schrödinger

equations for many-electron atoms.