Chapter 6 Perturbation Theory - uni-muenchen.de€¦ · Time dependent perturbation theory is relevant for the time-evolution of ... radiation ﬁeld (2.40) in Coulomb gauge (2.32)

Chapter 6

Perturbation Theory

6.1 Time independent perturbation theory

In this chapter we will assume that the HamiltonianH of a quantum mechan-ical system can be written as a small perturbation of a simple HamiltonianH0, i.e.

H = H0 + �H1 + �2H2 + · · · ⌘ H(�) = H0 + V (�) (6.1)

We will also assume thatH0 has an eigenvalue E0 which is n0-fold degenerate.Under rather general assumptions (eg. Messiah) the perturbation V (�) willresolve E0 analytically in maximally n0 di↵erent Eigenvalues Ek(�), withcorresponding eigenvectors | k(�) > which are also analytic in �,

H(�)| k(�) > = Ek(�)| k(�) >

Ek(�) = E0 + �Ek1 + �2Ek

2 + · · · (6.2)

| k(�) > = | k0 > +�| k

1 > +�2| k2 > + · · ·

with normalization condition

< k0 | k(�) >= 1 () < k

0 | kl >= 0, l > 0 (6.3)

In physical applications one usually stops at the second order in �, i.e.

(H0 � E0)| k0 > = 0 (6.4)

(H0 � E0)| k1 > +(H1 � Ek

1 )| k0 > = 0 (6.5)

(H0 � E0)| k2 > +(H1 � Ek

1 )| k1 > +(H2 � Ek

2 )| k0 > = 0 (6.6)

43

44 CHAPTER 6. PERTURBATION THEORY

Let M0 be the eigenspace to H0 with eigenvalue E0 and P0 the projectionoperator onto M0. In addition let Q0 = (1� P0) be the projection operatoronto the orthogonal complement Mc

0. Since P0H0 = H0P0 = E0P0, thesubspace M0 and its complement are invariant under H0 and H0 � E0 isinvertible on Mc

0. In other words the resolvent

R0 = �Q0(H0 � E0)�1Q0 (6.7)

is well defined on H.If E0 is non-degenerate, n0 = 1, then | 0 > is uniquely fixed by (6.4)

up to normalization. Taking the scalar product of (6.5) with < 0| we get

E1 =< 0|H1| 0 > (6.8)

On the other hand acting on (6.5) by Q0 we obtain

(H0 � E0)Q0| 1 > +Q0H1| 0 >= 0 (6.9)

Taking

| 1 >= Q0| 1 >= R0H1| 0 > (6.10)

we have satisfied all components of equation (6.5). Finally we act on (6.6)by P0 to obtain

P0H1| 1 > +P0(H2 � E2)| 0 >= 0 (6.11)

Upon substitution of | 1 > from (6.19) and taking the scalar product with< 0| we get

E2 = < 0|H1R0H1| 0 > + < 0|H2| 0 >

=X

✏n 6=E0

| < �n|H1| 0 > |2E0 � En

+ < 0|H2| 0 > (6.12)

where

H0|�n >= ✏n|�n > (6.13)

on Mc0. Similarly acting on (6.6) by Q0 we obtain

(H0 � E0)Q0| 2 > +Q0(H1 � E1)| 1 > +Q0H2| 0 >= 0 (6.14)

6.2. TIME DEPENDENT PERTURBATION THEORY 45

and thus by taking

| 2 >= Q0| 2 >= R0H2| 0 > +R0(H1 � E1)R0H1| 0 > (6.15)

we have satisfied all components of equation (6.6).If E0 is degenerate, then (6.5) together P0(H0 � E0) = 0 leads to the

n0-dimensional eigenvalue problem

(P0H1P0 � E1)| k0 >= 0 (6.16)

which determines the first order correction to the energy Ek1 and the cor-

responding eigenvectors | k0 > up to degeneracy. We then consider for a

given solution E1 of (6.16) with degeneracy n1, 1 n1 n0 the eigenspaceM1 ⇢ M0 of P0H1P0 with corresponding projector P1 = P1P0 = P0P1

and P1(H0 � E0) = P1(P0H1P0 � E1) = 0. The family of vectors {| k1 >},

k = 1, · · ·n1, is then found from (6.5)

Q0| k1 >= R0H1| k

0 > (6.17)

with | k0 >2 M1. Taking Q0| k

1 >= | k1 > eqn (6.17) determines | k

1 >uniquely. Alternatively we may write

| k1 >= R0H1P1| k

0 > (6.18)

of which only n1 vectors can be non-vanishing and distinct (question: whatis the role of P1 on the r.h.s. of (6.19)?). Acting with P1 on (6.6) we get

P1H1| k1 > +(P1H2P1 � Ek

2 )| k0 >= 0 (6.19)

which upon substitution of (6.17) leads to a n1-dimensional eigenvalue prob-lem of E2, etc.

6.2 Time dependent perturbation theory

Time dependent perturbation theory is relevant for the time-evolution oftransition probabilities. We will assume that the Hamiltonian is of the formH(t) = H0 + �H1(t) with H1(t) = H⇤

1 (t) bounded in H and (piecewise)continuous in t. We work in the interaction picture:

U(t, s) = U0(t, 0)UW (t, s)U⇤0 (s, 0) , U0(t, 0) ⌘ U0(t) = e�

i~H0t (6.20)


with

i~ d

dtUW (t, s) = HW (t)UW (t, s), UW (s, s) = 1 (6.21)

HW (t) = U⇤0 (t, 0)H1(t)U0(t, 0)

The solution for the evolution operator UW (t, s) can then be expanded as

UW (t, s) = 1+1X

n=1

U (n)W (t, s) (6.22)

= 1+1X

n=1

✓�i

~

◆ntZ

s

dr1 · · ·rn�1Z

s

drn HW (r1) · · ·HW (rn)

=: P

2

4exp[

tZ

s

dr HW (r)]

3

5

We now consider the special case of a time-independent perturbation,H1 6= H1(t). Let | 0 > be the normalized eigenvector ofH0 for the non-degenerateeigenvalue E0. we shall also assume that H0 has the spectral representa-tion

H0 = E0| 0 >< 0|+X

n

EnPn +

Z

E dP (E) (6.23)

with projection operators Pn for the discrete eigenvalues En and dP (E) forthe continuous spectrum

6.2.1 Transition from | 0 > to the discrete spectrum

Using that [Pn, U0(t, 0)] = 0 we find for the transition probability

wn(t) = < 0|U(0, t)PnU(t, 0)| 0 > (6.24)

= < 0|Uw(0, t)PnUw(t, 0)| 0 >

1 (6.25)

Upon substitution of (6.22) we obtain (using Pn| 0 >= 0)

wn(t) = w(2)n (t) +O(H3

1 ) (6.26)

6.2. TIME DEPENDENT PERTURBATION THEORY 47

with

w(2)n (t) = < 0|Uw(0, t)PnUw(t, 0)| 0 >

= < 0|H1PnH1| 0 > |�i

~

tZ

0

dr ei(En�E0)r/~|2 (6.27)

= 4< 0|H1PnH1| 0 >

(En � E0)2sin2(

En � E0

2~ t) (6.28)

The transition probability w(2)n (t) thus shows oscillatory behavior and is small

for ||H1||/|En � E0| << 1.

6.2.2 Transition from | 0 > to the continuous spectrum

Let � = [a, b] be a closed interval in the continuous spectrum and

P� =

Z

�

dP (E) (6.29)

be the projector on the corresponding subspace of H. The transition proba-bility from | 0 >, at t = 0, to P�H at time t is then ( P�| 0 >= 0)

w�(t) = w(2)� (t) +O(H3

1 ) (6.30)

w(2)� (t) = < 0|Uw(0, t)P�Uw(t, 0)| 0 >

= 4

Z

�

sin2(E � E0

2~ t)< 0|H1dP (E)H1| 0 >

(En � E0)2(6.31)

If E0 /2 � then we have for 0 < ~t<< |�|

w(2)� (t) ' 2

Z

�

�(E)dE

(En � E0)2(6.32)

with

�(E)dE :=< 0|H1dP (E)H1| 0 > (6.33)

(what is the dimension of �(E)?) so that w(2)� (t) approaches a constant

(t-independent) value provided �(E) is integrable, ie.R |�(E)|dE < 1.


If �(✏) = [E0 � ✏, E0 + ✏] for ✏ > 0, small and �(E) continuous in �(✏)then we have for late times 0 < ~

t<< ✏:

p(2)�(✏) ⌘w(2)

� (t)

t' 2�(E0)

~

1Z

�1

sin2 x

x2dx =

2⇡

~ �(E0) (6.34)

Thus, the transition rate p(2)�(✏) in second order perturbation theory ap-

proaches a constant p(2)E0indpendent of ✏.

Suppose now that H0 has an almost continuous spectrum around E0 withnormalized eigenvectors | n >, such that

< n|H1| 0 >'< n0 |H1| 0 > (6.35)

for E ' E0 and < n0 |H1| 0 > is a ”typical” matrix element. Then we have

w(2)� (t)

t= 4

X

{n 6=0 , |En�E0|<✏}

sin2(En�E02~ t)

(En � E0)2t| < n0 |H1| 0 > |2

' 4| < n0 |H1| 0 > |2

t

Z

�✏

sin2(En�E02~ t)

(E � E0)2⇢(E)dE

' 2⇡

~ | < n0 |H1| 0 > |2⇢(E0) (6.36)

where ⇢(E) is the density of states of H0 around E0. This is Fermi’s goldenrule.

6.3 Atom in the radiation field

We consider an atom with hamiltonian

HA0 =

X

i

~p2i2me

+ V (~qi) +X

i<j

W (~qi � ~qj) (6.37)

We will be interested in the transition rate between two bound states |�Ai >

with

HA0 |�A

i >= ✏i|�Ai >, ✏i 6= ✏j, < �A

i |�Aj >= �ij (6.38)

6.3. ATOM IN THE RADIATION FIELD 49

In the dipole approximation the minimal coupling to the quantizedradiation field (2.40) in Coulomb gauge (2.32) with hamiltonian (2.44) tofirst order in e is given by

H1 =X

i

e

mec~pi · ~A(~q0) (6.39)

where ~q0 is the position of the atom. The Hilbert space of the combinedsystem is given by HA ⌦ Hrad with Hrad as in (2.29). We take the initialstate of the combined system at t = 0 to be

| 1 >= |�A1 > ⌦|n1 >=: |�A

1 , n1 > (6.40)

where n1 = n~k1, · · · , n~kj

, · · · are the photon occupation numbers for the wave

vectors ~k1, · · · ,~kj, · · · .Let us now compute the transition probability at time t, in lowest order

in time-dependent perturbation theory

w21(t) =< 1(t)|P2| 1(t) >=X

n2

< 1(t)|�A2 , n2 >< n2,�

A2 | 1(t) > (6.41)

into the subspace HA2 ⌦Hrad where the atom is in the state |�A

2 >. We have

w21(t) =2

~2 | < n2,�A2 |H1| 1(t) > |21� cos(E2�E1

~ t)

(E2 � E1)2/~2(6.42)

with

H0|�Ai , ni >= (✏i +

X

~k,�

n(~k,�)~!(~k))|�Ai , ni > (6.43)

It is clear from (2.40) that H1 changes the photon number by ±1 correspond-ing to a photon emission/ absorption by the atom. We will treat thesetwo cases separately.

During an absorption of a (~k,�)-photon we have

n2(~k0,�0) = n1(~k

0,�0), (~k,�) 6= (~k,�) (6.44)

and

n2(~k,�) = n1(~k,�)� 1 (6.45)


and hence

1

~(E2 � E1) =1

~(✏2 � ✏1)� !(~k,�) =: !21 � !(~k,�) (6.46)

In Hrad eqn. (2.40) leads to the transition amplitude

< n2| ~A(~q)|n1 >=

n1(~k,�)~c22⇡!(~k)L3

!

12

~e(~k,�)ei~k·~q0 =:

q

n1(~k,�) ~A~k,� (6.47)

In HA we have with ~D =P

i

e~qi (see worksheet 9)

< �A2 |X

i

e~pime

|�A1 >=

i

~ < �A2 |[HA

0 , ~D]|�A1 >=: i!21

~D21 (6.48)

Altogether we get for the probability to absorb a (~k,�)-photon

wa21((~k,�), t) =

2

~2n1(~k,�)!2

21

c2| ~D21 · ~A~k,�|2

1� cos(E2�E1~ t)

(E2 � E1)2/~2(6.49)

and hence for the total absorption rate

dwa21

dt=

X

~k,�

dwa21((~k,�), t)

dt=

=2!2

21

~2c2X

~k,�

n1(~k,�)| ~D21 · ~A~k,�|2sin((!21 � !(~k))t)

!21 � !(~k)(6.50)

We note in passing that one can obtain this last result for the ”absorption”rate by treating the electro-magnetic field as a classical external field ~Aclass

~k,�

provided we set

~Aclass~k,�

:= ~A~k,�

q

n1(~k,�) (6.51)

If we consider an ensemble of atoms with ~q0 arbitrary in V and arbitraryorientation of ~D21 we take the average

| ~D21 · ~A~k,�|2 ! 1

3| ~D21|2| ~A~k,�|2 !

1

3| ~D21|2 1

V

Z

d3q| ~A~k,�(~q)|2

=1

3| ~D21|2 ~c

22⇡

!(~k)V(6.52)


With this the total absorption rate per atom becomes

dwa21

dt=

4⇡!221

3~2 | ~D21|2 1V

X

~k,�

n1(~k,�)~!(~k)1

!(~k)2sin((!21 � !(~k))t)

!21 � !(~k)

(6.53)

For large volume V ! 1 we may assume a quasi continuous occupationnumber of photons and then

1

V

X

~k,�

n1(~k,�)~!(~k)f(!(~k)) '1Z

0

d! u(!)f(!) (6.54)

where u(!) is the spectral energy density of the radiation field. Supposenow that t >> 2⇡

!for ! in the support of u(!). Then we have

1Z

0

d! u(!)f(!)sin((!21 � !)t)

!21 � !'(

⇡ u(!21)f(!21) !21 > 0

0 !21 < 0(6.55)

Thus for ✏2�✏1 = ~!21 > 0 we find a constant transition rate for stimulatedabsorption

dwa21

dt=

4⇡2

3~2 u(!21)| ~D21|2 (6.56)

in agreement with Fermi’s golden rule with ⇢(E) = u(!)/~! (why?).For ✏2�✏1 = ~!21 < 0 there is no absorption but instead we have emission

of a (~k,�)-photon, ie.

n2(~k0,�0) = n1(~k

0,�0), (~k,�) 6= (~k,�) (6.57)

and

n2(~k,�) = n1(~k,�) + 1 (6.58)

Then using

< n2| ~A(~q)|n1 >=q

n1(~k,�) + 1 ~A⇤~k,�

(6.59)


and

1

V

X

~k,�

f(!(~k)) '1Z

0

d!!2

⇡2c3f(!) (6.60)

we and up with

dwe21

dt=

4⇡2

3~2

u(!21) +~|!21|3⇡2c3

�

| ~D21|2 (6.61)

for the sum of stimulated and spontaneous emission. In particular, anexcited atomic state can be relaxed even in the absence of a radiation field.

Historical Notes

In a 1905 paper Einstein postulated that light itself consists of localized par-ticles (quanta). Einsteins light quanta were nearly universally rejected byall physicists, including Max Planck and Niels Bohr. This idea only becameuniversally accepted in 1919, with Robert Millikans detailed experiments onthe photoelectric e↵ect, and with the measurement of Compton scattering.Einsteins paper on the light particles was almost entirely motivated by ther-modynamic considerations. He was not at all motivated by the detailedexperiments on the photoelectric e↵ect, which did not confirm his theory un-til fifteen years later. Einstein considers the entropy of light at temperatureT, and decomposes it into a low-frequency part and a high-frequency part.The high-frequency part, where the light is described by Wiens law, has anentropy which looks exactly the same as the entropy of a gas of classical par-ticles. Since the entropy is the logarithm of the number of possible states,Einstein concludes that the number of states of short wavelength light wavesin a box with volume V is equal to the number of states of a group of localiz-able particles in the same box. Since (unlike others) he was comfortable withthe statistical interpretation, he confidently postulates that the light itself ismade up of localized particles, as this is the only reasonable interpretationof the entropy. This leads him to conclude that each wave of frequency fis associated with a collection of photons with energy hf each, where h isPlancks constant. He does not say much more, because he is not sure howthe particles are related to the wave. But he does suggest that this ideawould explain certain experimental results, notably the photoelectric e↵ect.


Further Reading

For a more detailed treatment of time-independent perturbation theory seee.g. Messiah, Quantum mechanics I+II. Absorption and emission of photonsby atoms can be found eg. in Sakurai, ”Advanced Quantum Mechanics”.Einstein’s original derivation of Planks radiation law can be found in C.Kittel ”Elementary Statistical Physics” (1958).


Chapter 7

Scattering Theory

In many experiment, especially in particle physics interactions between twoinitially widely separated system are investigated. Generically the projectileand the target have a complicated structure and inelastic processes arecommon, including particle production. In other cases such as structureanalysis via neutron scattering processes with repeated scattering are ofrelevance. Here we will consider the simple case of elastic scattering o↵ acentral potential spinless particles.

7.1 Central Potential

For potentials that vanish su�ciently fast (V (r) / 1r1+✏ , ✏ > 0) one looks for

stationary solutions with the asymptotic form

(~k, ~q) ' ei~k·~q + f(k, ✓)

eikr

r(7.1)

The di↵erential cross section is then given by (see QMI or my Germannotes)

d�

d⌦= |f(k, ✓)|2 (7.2)

In analogy to the free particle we should be able to write (~k, ~q) as a partialwave decomposition

(~k, ~q) =1X

l=0

blfl(r)Pl(cos ✓) (7.3)

55

56 CHAPTER 7. SCATTERING THEORY

where Pl(x) are the Legendre polynomials and fl(r) = gl(r)/r satisfies theradial equation

g00l (r) +✓

~k2 � l(l + 1)

r2� 2mV (r)

~2

◆

gl(r) = 0 (7.4)

A classical result by Weyl and Kodaira shows that for V (r) / 1r1+✏ , ✏ > 0

at large r and V (r) = O(r�2+✏) for small r the scattering solutions have theasymptotic behaviour

fl(r) 'r

2

⇡

sin(kr � l⇡2 + �l)

r(7.5)

In particular, the plane wave ei~k·~q has the following expansion (see eg. Saku-

rai, p.398)

ei~k·~q =

1X

l=0

il(2l + 1)jl(kr)Pl(cos ✓) (7.6)

kr!1'1X

l=0

il(2l + 1)

2ikr

⇣

ei(kr�l⇡2 ) � e�i(kr� l⇡

2 )⌘

Pl(cos ✓)

and therefore

(~k, ~q) ' ei~k·~q + f(✓)

eikr

r(7.7)

!=

1X

l=0

bl

r

2

⇡

1

2ir

⇣

ei(kr�l⇡2 +�l) � e�i(kr� l⇡

2 +�l)⌘

Pl(cos ✓)

implies

bl

r

2

⇡e�i�l =

il(2l + 1)

k(7.8)

Upon substitution into (7.7) and using (7.6) we then find

f(k, ✓) =1

k

1X

l=0

(2l + 1)ei�l(k) sin(�l(k))Pl(cos ✓) (7.9)

7.2. SHORT RANGED POTENTIALS 57

so that f(k, ✓) and therefore the cross section is uniquely determined interms of the scattering phases �l(k). Furthermore due to the orthogonalitycondition

1Z

�1

Pl(x)Pl0(x) =2�ll0

2l + 1(7.10)

the total cross section satisfies the optical theorem

�T =4⇡

k2

1X

l=0

(2l + 1) sin2 �l(k) =4⇡

kImf(k, 0) (7.11)

In words: The total cross section is the sum of the partial wave crosssections �l =

4⇡k2(2l+ 1) sin2 �l(k) and the sum equals the imaginary part of

the forward scattering amplitude just like in classical optics (e.g. J. Jackson,class. ED, page 453). For each l we have �l 4⇡

k2(2l + 1) with maximum

for resonances, �l(k) = (n + 12)⇡. The partial wave expansion is useful if

either most of the �l(k) are negligible or if one partial wave dominates. Thefirst case arises at low energies or short ranged potentials whereas the lattersituation arises for example in pion-nucleon scattering at E ' 189Mev inthe laboratory frame due to the N⇤ resonance.

7.2 Short ranged potentials

Suppose that V (r) = 0 for r > R and let fl(k, r) be the regular solution ofthe radial equations (7.4). For r > R we then have

fl = f>l =

r

2

⇡(jl(kr) cos(�l)� nl(kr) sin(�l)) (7.12)

where jl(x) and nl(x) ⌘ yl(x) are the spherical Bessel functions of thefirst and second kind respectively( see eg. http://www.phys.ufl.edu/ lex/pdfs/sphericalbessel.pdf ). If we let

�l(k) ⌘ d(log f<l (r))

dr(7.13)

then the matching condition

�l(k) =d(log f>

l (r))

dr(7.14)


implies

tan(�l(k)) =kj0l(kR)� �l(k)jl(kR)

kn0l(kR)� �l(k)nl(kR)

(7.15)

and thus the logarithmic derivative of fl at r = R completely determinesthe scattering phases for all l. Measuring the �0l therefore provides littleinformation about the wave function.

7.2.1 Threshold behaviour

For small energies or small range R we can expand jl(x) and nl(x) leadingto

tan(�l(k)) ' kl(kR)l�1 � (kR)l�l

(2l + 1)!!(2l � 1)!!⇣

k(l+1)(kR)l+2 +

�l

(kR)l+1

⌘

=(kR)l+2[l �R�l]

(2l + 1)!!(2l � 1)!![l + 1 +R�l](7.16)

For l + 1 + R�l(k) 6= 0 we have sin(�l) / k2l+1 for k ! 0 and the scatteringamplitude f(k, ✓) is isotropic at k ' 0.

7.2.2 Application: Hard sphere scattering

Suppose the potential is of the form

V (r) =

(

1 r R

0 r > R(7.17)

and thus

tan(�l(k)) =jl(kR)

nl(kR)l=0= � tan(kR) (7.18)

In particular �0 ' �kR. For more general short range potential �0 ' �kasstill holds where as is the scattering length which can di↵er considerablyfrom the rangr of the potential. For small kR where the total cross sectionis dominated by the s-wave we have with (7.11)

�T ' 4⇡

k2sin2 �0(k) = 4⇡R2 (7.19)

which is four times the geometric cross section.

7.3. BORN APPROXIMATION FOR PHASE SHIFTS 59

7.3 Born approximation for phase shifts

Let gl(k, r) be the regular solution of (7.4) with gl(k, 0) = 0 (why?) and

asymptotic behaviour from (7.5). Furthermore let g0l (k, r) =q

2⇡krjl(kr) be

the regular solution for V ⌘ 0. Then we have

2m

~2

RZ

0

V (r)g0(r)g(r)dr =

RZ

0

(g0(r)g00(r)� g(r)g000(r))dr

=h

g0(R)g0(R)� g(R)g00(R)i

R!1=

2k

⇡sin(kR� l⇡

2) cos(kR� l⇡

2+ �l) � 2k

⇡cos(kR� l⇡

2) sin(kR� l⇡

2+ �l)

= �2k

⇡sin(�l(k)) (7.20)

This then leads the exact relation for R ! 1,

sin(�l(k)) = �p2⇡m

~2

1Z

0

V (r)gl(r, k)jl(kr)rdr (7.21)

In analogy with the Born approximation for the scattering amplitude (seeQMI or my German notes) we then obtain the first Born approximation forthe phase shifts by writing gl(r, k) = g0l (r, k) +O(V ) and therefore

sin(�l(k)) = sin(�(1)l (k)) +O(V 2) (7.22)

with

sin(�(1)l (k)) = �p2⇡m

~2k

1Z

0

V (r)[krjl(kr)]2dr (7.23)

For large k and integrable potential (R |V (r)|dr < 1) the boundedness of

|⇢jl(⇢)| al for 0 ⇢ 1 the implies

sin(�(1)l (k)) = O(1/k), k ! 1 (7.24)

and the successive Born approximations for the phase shifts converge forlarge k (Taylor, page 143).


For integrable potential with compact support we can be more preciseabout the l-dependence. For this we first recall that in classical mechanicsthere is no scattering for impact parameterb b > R on a potential withsupport in [0, R]. Furthermore, if p is the momentum of the in-falling particle,

then bp it it angular momentum. If we then identify b ' ~p

l(l+1)

~k . This

suggests that the phase shift are small forp

l(l + 1) � kR. Indeed theradial component g0l (⇢), ⇢ = kr, of the free particle wave function changesqualitatively form a growing function to an oscillatory behaviour for ⇢ 'p

l(l + 1). For kR <<p

l(l + 1) the function rjl(kr) is small in the support

of the potential V (r) and therefore sin(�(1)l (k)) should be small unless gl(kr)is large. This happens typically for resonances (see next subsection).

In order to emphasize the importance of integrability we consider againthe hard sphere scattering but now at high energy. According to the abovediscussion we should have

�T ' 4⇡

k2

(l'kR)X

l=0

(2l + 1) sin2 �l(k) (7.25)

with

sin2 �l(k) =tan2 �l(k)

1 + tan2 �l(k)

=jl(kR)2

jl(kR)2 + nl(kR)2(7.26)

' sin2(kR� ⇡l

2), kR >> 1

In particular the scattering phase does not decay for large k. Noting thatsin2 �l+1(k) = cos2 �l(k) we can approximate sin2 �l(k) ' 1

2 . Then

�T ' 2⇡

k2

(l'kR)X

l=0

(2l + 1) ' 2⇡R2 (7.27)

which is twice the geometric cross section.

7.4. RESONANCES 61

7.4 Resonances

In the absence of resonance we have seen that partial wave cross sections �lshow the simple threshold behaviour

�l =4⇡

k2(2l + 1) sin2(�l) / k4l (7.28)

On the other hand for E ' Er with l+ 1+R�l(Er) = 0 we have to be morecareful and expand

l + 1 +R�(E) ' (E � Er)�0l(Er)

l �R�(E) ' 2l + 1 (7.29)

We hen have for E ' Er,

cot(�l) ' �2(E � Er)

�r

(7.30)

�r = � 2k2l+1r R2l

[(2l + 1)!!]2�0l(Er)

If we then write

sin2(�l) =1

1 + cot2(�l)(7.31)

obtain for the resonant partial wave cross section the Breit-Wigner For-mula

�l =4⇡

k2(2l + 1)

�2r

4(E � Er)2 + �2r

(7.32)

For E = Er the partial wave cross sections �l then saturates the unitaritylimit 4⇡

k2(2l + 1). The with at half maximum, �r decreases with grow-

ing angular momentum l. This can be interpreted as a suppression of thetunneling probability through the increasing potential barrier.

To better understand the resonant behaviour of �l we first show that �r

is always positive. In deed let gi(r) := gl(ki, r). The we have

(k21 � k2

2)

g1(R)g2(R)

RZ

0

g1(r)g2(r)dr =g02(R)

g2(R)� g01(R)

g1(R)

= �l(E2)� �l(E1) (7.33)


and thus

�0l(E) = lim

E2 6=E1!E

�l(E2)� �l(E1)

E2 � E1

= � 2m

~2g2l (k,R)

RZ

0

gl(k, r)2dr < 0 (7.34)

for R > 0 and thus �r � 0. Now using that cot(�l � ⇡2 ) = � tan(�l) we get

�l =⇡

2+ arctan

✓

E � Er

�r/2

◆

mod n⇡ (7.35)

and

d�ldE

|E=Er =2

�r

(7.36)

Historical Notes

Further Reading

The partial wave expansion described in this section can be found in Saku-rai’s Modern Quantum Mechanics, section 7.6. An in depth discussion ofscattering theory in quantum mechanics can be found in J. Taylor, scatter-ing theory. Resonant behaviour in low energy scattering in nuclear physicscan be found e.g. in Blatt and Weisskopf, theoretische Kernphysik.

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