1

Scattering theory Masatsugu Sei Suzuki Department of Physics SUNY at Binghamton (Date: April 16, 2015)

One of the best experimental methods to probe a microscopic structure of the target (system) is to examine the distribution of scattered particles (or wave) from the collision of particles with the target. The experimental results thus obtained are compared with the theory of scattering based on the quantum mechanics. Here we discuss the scattering theory in the quantum mechanics: the Born approximation and (ii) the Lippmann-Schwinger equation. Our understanding on the scattering has been greatly enhanced, thank to these two theories. Max Born (11 December 1882 – 5 January 1970) was a German born physicist and mathematician who was instrumental in the development of quantum mechanics. He also made contributions to solid-state physics and optics and supervised the work of a number of notable physicists in the 1920s and 30s. Born won the 1954 Nobel Prize in Physics (shared with Walther Bothe).

http://en.wikipedia.org/wiki/Max_Born ________________________________________________________________________

Julian Seymour Schwinger (February 12, 1918 – July 16, 1994) was an American theoretical physicist. He is best known for his work on the theory of quantum electrodynamics, in particular for developing a relativistically invariant perturbation theory, and for renormalizing QED to one loop order. Schwinger is recognized as one of the greatest physicists of the twentieth century, responsible for much of modern quantum field theory, including a variational approach, and the equations of motion for quantum fields. He developed the first electroweak model, and the first

2

example of confinement in 1+1 dimensions. He is responsible for the theory of multiple neutrinos, Schwinger terms, and the theory of the spin 3/2 field.

http://en.wikipedia.org/wiki/Julian_Schwinger ________________________________________________________________________ 1 Green's function in scattering theory

We start with the original Schrödinger equation.

)()()()(2

22

rrrr kEV

,

or

)()(2

)()2

(22

2 rrr VEk

.

under the potential energy )(rV . We assume that

22

2kEE k

,

We put

)()(2

)(2

rrr Vf

,

Using the operator

22 kL r .

3

we have the differential equation

)()()()( 22 rrrr fkL . Suppose that there exists a Green's function )(rG such that

)'()',()( )(22 rrrrr Gk , with

'4

)'exp()',()(

rr

rrrr

ik

G (Green function)

We will discuss about the derivation of this Green function later. Then )(r is formally given by

)'()'('4

)'exp('

2)()'()',(')()( )(

2)()( rr

rr

rrrrrrrrrr

V

ikdfGd

,

where )(r is a solution of the homogeneous equation satisfying

0)()( 22 rk , or

)exp()2(

1)( 2/3 rkkrr i

, (plane wave)

with

kk

Note that

)(

)'()'('

)'()',()(')()()()( )(222222

r

rrrr

rrrrrr

f

fd

fGkdkk

2 Born approximation

We start with

4

)'()'('4

)'exp('

2)()( )(

2)( rr

rr

rrrrr

V

ikd

,

)exp()2(

1)( 2/3 rkkr ir

, (plane wave).

Fig. Vectors r and r’ in calculation of scattering amplitude in the first Born

approximation

5

Fig. rr eu

Here we consider the case of )()( r

rr errr ''

rkek '

'')'(' rkerrr iikrrikik eeee r for large r.

r

1

'

1

rr

Then we have

O r'

r

r'ÿur

r-r'

6

)'()'('4

12)exp(

)2(

1)( )(''

22/3)( rrrrk rk

Ved

r

eir i

ikr

or

)],'([)2(

1)(

2/3)( kkrk f

r

eer

ikri

The first term denotes the original plane wave in propagation direction k. The second denotes the term: outgoing spherical wave with amplitude, ),'( kkf ,

)'()'('2

)2(4

1),'( )(''

22/3 rrrkk rk

Vedf i

.

The first Born approximation:

')'(2

'''2

)'('2

)'('2

4

1),'(

rkk

rkrk

rr

rrkk

i

ii

eVd

eVedf

when )()( r is approximated by

rk ier2/3

)(

)2(

1)(

.

Note that ),'( kkf is the Fourier transform of the potential energy with the wave vector Q; the scattering vector;

kkQ ' . Formally ),'( kkf can be rewritten as

kk

kkkk

V

Vf

ˆ'4

ˆ')2(2

),'(

2

2

32

where

7

)()2(

1

)('ˆ'

)'(33

3

rr

krrrkrkk

rkk Ved

VdV

i

with

)exp()2(

12/3 rkkr i

((Forward scattering))

Suppose that k' = k (Q = 0) Then we have

)'('2

)'('2

4

1)0(

2

''2

rr

rr rkrk

Vd

eVedf ii

Suppose that the attractive potential is a type of square-well

0

)( 0VrV

Rr

Rr

Then we have

32

032

0

3

2)0( R

VR

Vf

The total cross section (which is isotropic) is obtained as

23

202

4)0(4

R

Vf

.

The units of )0(f is cm, and the units of is cm2

.

3 Differential cross section

8

We define the differential cross section d

d as the number of particles per unit time

scattered into an element of solid angle d divided by the incident flux of particles. The probability flux associated with a wave function

ikzik ee 2/32/3 )2(

1

)2(

1)(

rkkrr ,

is obtained as

33

**

)2()2(

1)]()()()([

2

vk

zziJN kkkkzz

rrrr

Fig.

volume = 1k

z

Detectord

0

V(r)

incidentplane wave

sphericalwave

eikz fk(,)

reikz

z

unit area

k

vrel

relativevelocity

9

12ikze means that there is one particle per unit volume. Jz is the probability flow

(probability per unit area per unit time) of the incident beam crossing a unit surface perpendicular to OZ The probability flux associated with the scattered wave function

)()2(

12/3

f

r

eikr

r (spherical wave)

is

2

2

3

** )(

)2(

1)(

2 r

fk

rriJ rrrrr

drdA 2

dfv

drr

fvdAJN r

2

32

2

2

3)(

)2(

)(

)2(

where Jr is the probability flow (probability per unit area per unit time)

The differential cross section

dfN

Nd

d

d

z

2)(

or

2)(

f

.

First-order Born amplitude:

r

d

dS

Detector

dS r 2d

10

)'('2

)'('2

'2

)2(4

1),'(

32

)'(32

23

rr

rr

kkkk

rQ

rkk

Ved

Ved

Vf

i

i

,

which is the Fourier transform of the potential with respect to Q, where

kkQ ' : scattering wave vector.

2sin2

kQ Q for the elastic scattering.

The Ewald sphere is given by this figure. Note that the scattering angle is here. In the case of x-ray and neutron diffraction, we use the scattering angle 2, instead of .

Fig. Ewald sphere for the present system (elastic scattering). ki = k. kf = k’. Q =q = k –

k’ (scattering wave vector). 2

sin2

kQ .

((Ewald sphere)) x-ray and neutron scattering

11

Fig. Ewald sphere used for the x-ray and neutron scattering experiments. ki = k. kf = k’.

Q =q = k – k’ (scattering wave vector). Note that in the conventional x-ray and neutron scattering experiments, we use the angle 2, instead of for both the x-ray and neutron scattering,

4. Spherical symmetric potential

When the potential energy V(r) is dependent only on r, it has a spherical symmetry. For simplicity we assume that ’ is an angle between Q and r’.

'cos'' Qr rQ .

We can perform the angular integration over ’.

0 0

2'cos'2

0 0

2'cos'2

2)1(

)'('sin'''

)'('sin'2''2

4

1

)'('2

4

1)(

rVreddr

rVreddr

Vedf

iQr

iQr

i

rr rQ

Note that

)'sin('

2'sin'

0

'cos' QrQr

ed iQr

ff2

2q

OO1

Q

ki ki

k f

Ewald sphere

12

Then

02

0

22

)1(

)'sin()'(''21

)'sin('

2)'('2'

2

4

1)(

QrrVrdrQ

QrQr

rVrdrf

(spherical symmetry)

Then the differential cross section is given by

2

0

2

22

2)1( )'sin()'(''21

)(

QrrVrdr

Qf

d

d

Then we find that )()1( f is a function of Q.

)2

sin(2

kQ ,

x

y

z

Q

r '

'

13

where is an angle between k’ and k (Ewald’s sphere). 4. Low energy soft-sphere scattering

Suppose that

0

)( 0VrV

Rr

Rr

Then we get

3303

2

202

0

02)1(

)cos()sin(2

)cos()sin(21

)'sin(''21

)(

RQ

QRQRQRVR

Q

QRQRQRV

Q

QrrdrVQ

fR

When QR<<1,

303

1)cos()sin( 2

33

x

RQ

QRQRQR

.

Then we have

3

0203

2)1(

3

4

23

12)( RVVRf

and

23

20

2)1(

3

2)(

R

Vf

d

d

with

)2

sin(2

kQ ,

We make a plot of 2)1(2)1( )0(/)( QfQf as a function QR.

14

Fig. Plot of 2)1(2)1( )0(/)( QfQf as a function QR. The value of

2)1( )(Qf becomes

zero when QR = 4.49341 and 7.72525. 5 Yukawa potential Hideki Yukawa (23 January 1907 – 8 September 1981) was a Japanese theoretical physicis-t and the first Japanese Nobel laureate.

http://en.wikipedia.org/wiki/Hideki_Yukawa ________________________________________________________________________ The Yukawa potential is given by

rer

VrV

0)( ,

p 2p 3 pQR

-15

-10

-5

0

f Q 2

f Q = 0 2

15

where 0V is independent of r. 1/ corresponds to the range of the potential.

0

'020

0

'020)1( )'sin('

12)'sin(

'''

12)( Qredr

V

QQre

r

Vrdr

Qf rr

Here we use 0 for the reduced mass in order to avoid the confusion of the co-efficient

for the Yukawa potential with the reduced mass. Note that

220

' )'sin('

Q

QQredr r , (Laplace transformation)

220

20)1( 12

)(

Q

Vf

.

Since

)cos1(2)2

(sin4 2222 kkQ

so, in the first Born approximation,

222

2

200

2)1(

])cos1(2[

12)(

k

Vf

d

d

.

Note that as 0 , the Yukawa potential is reduced to the Coulomb potential, provided

the ratio /0V is fixed.

20 'eZZ

V

,

)2/(sin16

1)'()2()(

444

2220

2)1(

k

eZZf

d

d

.

Using 0

22

2k

Ek

, we have

)2/(sin

1'

16

1)(

4

222

kE

eZZf

d

d

16

which is the Rutherford scattering cross section (that can be obtained classically). 6 Validity of the first-order Born approximation

If the Born approximation is to be applicable, )(r should not be too different

from kr inside the range of potential. The distortion of the incident wave must be

small.

)(

'

2)( ')'(

'4'

2

rrrr

rkrrrr

Ve

dik

krr )( at the center of scattering potential at r = 0.

2/32/3

''

2 )2(

1

)2()'(

'4'

2

rk

rriikr e

Vr

ed

,

or

1)'('

'2

''

2 rkrr iikr

eVr

ed

.

For spherical potential, we have

'cos''

2''

)'('

''sin''2)'('

' ikrikr

iikr

erVr

edrdrerV

r

edI rkr

or

)'sin()'(''4

)'sin('

2)'(''2

''sin)'(''2

''sin)'(''2

'

'

0

'cos''

'cos''

krrVedrk

krkr

rVerdr

edrVerdr

edrVerdrI

ikr

ikr

ikrikr

ikrikr

where

)'sin('

2''sin

0

'cos' krkr

ed iQr

17

Then we get

1)'sin()'('2

0

'2

krrVedrk

ikr

.

At k , the integrand oscillates rapidly. It becomes zero for r’>1/k. Then we have

1'')'sin()'('2

220

/1

0

020

'2

k

VkrdrV

kkrrVedr

k

kikr

,

or

.10 E

V

____________________________________________________________________ 7 Lippmann-Schwinger equation

The Lippmann–Schwinger equation is equivalent to the Schrödinger equation plus the typical boundary conditions for scattering problems. In order to embed the boundary conditions, the Lippmann–Schwinger equation must be written as an integral equation. For scattering problems, the Lippmann–Schwinger equation is often more convenient than the original Schrödinger equation. (http://en.wikipedia.org/wiki/Lippmann%E2%80%93Schwinger_equation) ________________________________________________________________________

The Hamiltonian H of the system is given by

VHH ˆˆˆ0 ,

where H0 is the Hamiltonian of free particle. Let be the eigenket of H0 with the energy

eigenvalue E,

EH 0ˆ .

The basic Schrödinger equation is

EVH )ˆˆ( 0 . (1)

Both 0H and VH ˆˆ0 exhibit continuous energy spectra. We look for a solution to Eq.(1)

such that as 0V , , where is the solution to the free particle Schrödinger

equation with the same energy eigenvalue E.

18

)ˆ(ˆ0HEV .

Since EH 0ˆ or )ˆ( 0HE = 0, this can be rewritten as

)ˆ()ˆ(ˆ00 HEHEV ,

which leads to

VHE ˆ))(ˆ( 0 ,

or

VHE ˆ)ˆ( 10

.

The presence of is reasonable because must reduce to as V vanishes.

Lippmann-Schwinger equation:

)(10

)( ˆ)ˆ( ViHEkk

by making Ek (= )2/22 k slightly complex number (>0, ≈0). This can be rewritten as

)(10

)( ˆ'')ˆ(' ViHEdr k rrrkrr

where

rkkr ie2/3)2(

1

,

and

''ˆ'0 kk kEH ,

with

22

' '2

kEk

.

19

The Green's function is defined by

'10

)(0 '4

1')ˆ(

2)',(

2

rr

rrrrrr

ik

k eiHEG

((Proof))

'""')'2

('"'2

')ˆ(2

122

10

2

2

rkkkkkrkk

rr

iEdd

iHEI

k

k

or

iE

ed

iEd

iEddI

k

i

k

k

22

)'('

3

122

122

'2

)2(

'

2

'')'2

(''2

'")"'()'2

('"'2

2

2

2

k

k

rkkkrk

rkkkkkrkk

rrk

where

22

2k

kE .

Then we have

)(')2(

'

)'(2

)2(

'

2

22

)'('

3

222

)'('

3

2

ikk

ed

i

edI

i

i

rrk

rrk

k

kk

k

So we have

)(''

)2(

1)'(

22

)'('

3)(

0 ikk

edG

i rrk

krr

20

where >0 is infinitesimally small value. In summary, we get

)()(02

)( ˆ')',('2 VGd rrrrkrr

or

)()(02

)( ')'()',('2 rrrrrkrr VGd

.

More conveniently the Lippmann-Schwinger equation can be rewritten as

)(10

)( ˆ)ˆ( ViHEkk

with

10

2)(

0 )ˆ(2

ˆ

iHEG k

,

and

ViHEVG kˆ)ˆ(ˆˆ2 1

0)(

02

When two operators A and B are not commutable, we have very useful formula as follows,

AAB

BBAB

ABA ˆ1

)ˆˆ(ˆ1

ˆ1

)ˆˆ(ˆ1

ˆ1

ˆ1

,

We assume that

)ˆ(ˆ0 iHEA k , )ˆ(ˆ iHEB k

VHHBA ˆˆˆˆˆ0

Then

10

1110 )ˆ(ˆ)ˆ()ˆ()ˆ( iHEViHEiHEiHE kkkk ,

or

21

110

10

1 )ˆ(ˆ)ˆ()ˆ()ˆ( iHEViHEiHEiHE kkkk .

For simplicity, we newly define the two operators by

100 )ˆ()(ˆ iHEiEG kk

1)ˆ()(ˆ iHEiEG kk

where )(ˆ0 iEG k denotes an outgoing spherical wave and )(ˆ

0 iEG k denotes an

incoming spherical wave. We note that

)(ˆ2

)ˆ(2

ˆ0

21

0

2)(

0

iEGiHEG kk .

Then we have

)](ˆˆ1)[(ˆ

)(ˆˆ)(ˆ)(ˆ)(ˆ

0

00

iEGViEG

iEGViEGiEGiEG

kk

kkkk

)](ˆˆ1)[(ˆ

)(ˆˆ)(ˆ)(ˆ)(ˆ

0

00

iEGViEG

iEGViEGiEGiEG

kk

kkkk

Then )( can be rewritten as

k

kk

k

k

k

]ˆ)(ˆ1[

ˆ)(ˆ

ˆ)](ˆ(ˆ)(ˆ

ˆ)](ˆˆ1)[(ˆ

ˆ)(ˆ

)(0

)(

)(0

)(0

)(

ViEG

ViEG

ViEGViEG

ViEGViEG

ViEG

k

k

kk

kk

k

or

kk ViHEk

ˆˆ

1)(

.

((Note)) Significance of Lippmann Schwinger equation

Translation from S. Sunagawa, Quantum theory of scattering (in Japanese)

22

The method of Lippmann-Schwinger (LS) was developed after the World War II. Thank to this method, the theoretical study on the scattering has a significant progress. One of the reason is that LS equation can be used for a complicated case such that the scattering does not occur due to the potential. That is to say, when the target a

complicated structure consisting of many particles. In this case, the Hamiltonian 0H in

the propagator )(0

ˆ G cannot be expressed in simple way, leading to the difficulties in

expressing the concrete form, )(0

ˆ G . In the LS equation, 0H in )(0

ˆ G can be replaced by

the corresponding Hamiltonian without knowing any knowledge of 0H . So the

theoretical treatment can be done smoothly. The second reason is that we can also consider the interaction between fields as well as the potential form. In other words, the LP equation can be applied to the quantum field theory. 7 The higher order Born Approximation

From the iteration, )( can be expressed as

...ˆ)(ˆˆ)(ˆˆ)(ˆ

)ˆ)(ˆ(ˆ)(ˆ

ˆ)(ˆ

000

)(00

)(0

)(

kkk

kk

k

ViEGViEGViEG

ViEGViEG

ViEG

kkk

kk

k

The Lippmann-Schwinger equation is given by

kkk TiEGViEG kkˆ)(ˆˆ)(ˆ

0)(

0)(

,

where the transition operator T is defined as

kTV ˆˆ )(

or

kkk TiEGVVVT kˆ)(ˆˆˆˆˆ

0)(

This is supposed to hold for any k taken to be any plane-wave state.

TiEGVVT kˆ)(ˆˆˆˆ

0 .

The scattering amplitude ),'( kkf can now be written as

23

kkkkk TVf ˆ')2(4

12ˆ')2(4

12),'( 3

2)(3

2

.

Using the iteration, we have

...ˆ)(ˆˆ)(ˆˆˆ)(ˆˆˆˆ)(ˆˆˆˆ0000 ViEGViEGVViEGVVTiEGVVT kkkk

Correspondingly we can expand ),'( kkf as follows:

.......),'(),'(),'(),'( )3()2()1( kkkkkkkk ffff with

kkkk Vf ˆ')2(4

12),'( 3

2)1(

,

kkkk ViEGVf kˆ)(ˆˆ')2(

4

12),'( 0

32

)2(

,

kkkk ViEGViEGVf kkˆ)(ˆˆ)(ˆˆ')2(

4

12),'( 00

32

)3(

.

fk

V

fk '

fk

V

V

G0

fk '

fk

V

V

G0

G0

V

fk '

24

Fig. Feynman diagram. First order, 2nd order, and 3rd order Born approximations.

kk is the initial state of the incoming particle and '' kk is the final state

of the incoming particle. V is the interaction. 8 Optical Theorem

The scattering amplitude and the total cross section are related by the identity

tot

kf

4)]0(Im[ ,

where

),()0( kkff : scattering in the forward direction.

dd

dtot

.

This formula is known as the optical theorem, and holds for collisions in general.

Ok

ktot

k ktot

25

Fig. Optical theorem. The intensity of the incident wave is /k . The intensity of the forward wave is )]0(Im[)/4()/( fk . The waves with the total intensity

totkf )/()]0(Im[)/4( is scattered for all the directions, as the scattering

spherical waves. ((Proof))

]ˆIm[2

)2(4

1

]ˆIm[2

)2(4

1)],(Im[

)(2

3

23

V

Tf

k

kkkk

)(

0)( ˆ)(ˆ ViEG kk

,

or

)(0

)( ˆ)(ˆ ViEG kk,

or

)(ˆˆ0

)()( iEGV k k.

Then

)(0

)()()( )(ˆˆIm[(ˆIm[ iEGVV kk.

Now we use the well-known relation

)]ˆ(ˆ

1[

ˆ1

)(ˆ

0

0

0

0

HEiHE

P

iHEiEG

k

k

k

k

Then

]ˆ)ˆ(ˆIm

ˆˆ

1ˆ[ImIm[(ˆIm[

)(0

)(

)(

0

)()()()(

VHEiV

VHE

PVVk

26

The first two terms of this equation vanish because of the Hermitian oparators of V and

VHE

PV ˆˆ

1ˆ0

.

Therefore,

kkk THETVHEVV kˆ)ˆ(ˆˆ)ˆ(ˆˆIm[ 0

)(0

)()(

or

)2

'(ˆ'')

2

'(ˆ''ˆ'ˆIm[

22222)(

k

ETdk

ETTdV kk

kkkkkkkkk

)]'([))]')('(2

[)]'(2

[

)2

'

2()

2

'(

2222

2

222222

kkk

kkkkkk

kkkE

or

)'()2

'(

2

22

kkk

kE

2

2

22

2)(

ˆ''

)'(ˆ''''ˆIm[

kk

kkk

Tdk

kkTdkdkk

V

Thus

]ˆ''16

4

ˆ'')(2

)2(4

1

]ˆIm[2

)2(4

1)]0(Im[

2

4

24

2

223

)(2

3

kk

kk

k

Tdk

Tdk

Vf

Since

27

2

4

24

2

4

26

2

2

ˆ'16

ˆ'4

)2(16

1

),'('

kk

kk

kk

T

T

fd

d

,

2

4

24ˆ''

16

'' kk Tdd

ddtot

Then we obtain

tot

kf

4)]0(Im[ .

9. Summary

Comparison between the partial wave approximation (low-energy scattering) ans high-energy scattering (Born approximation). (i) Although the partial wave expansion is “straightforward”, when the energy of incident

particles is high (or the potential weak), many partial waves contribute. In this case, it is convenient to switch to a different formalism, the Born approximation.

(ii) At low energies, the partial wave expansion is dominated by small orbital angular

momentum. ________________________________________________________________________ REFERENCE J.J. Sakurai, Modern Quantum Mechanics, Revised Edition (Addison-Wesley, Reading

Massachusetts, 1994). A.G. Sitenko and P.J. Shepherd, Lectures in Scattering Theory (Pergamon Press, Oxford, 1971). E. Merzbacher, Quantum Mechanics 3rd edition (John Wiley & Sons, Inc., New York, 1998). David S. Saxon, Elementary Quantum Mechanics (Holden-Day, San Francisco, 1968). A. Das, Lectures on Quantum Mechanics, 2nd edition (World Scientific Publishing Co.Pte. Ltd,

2012). ___________________________________________________________________________ APPENDIX - I

Free particle wave function satisfies the Schrödinger equation

kE 22

2

,

where m is the mass of particle,

28

2

22kEk

,

is the energy of the particle, and k is the wave number. This equation can be rewritten as

0)( 22 k . This equation is solved in a formal way as

mkrrmk ),,(

),,(),,()(2

12

22

rEr

rp mkkmkr

L

(separation variables), where L is the angular momentum:

km(r,,) Rk(r )Ym(,) with

),()1(),( 22 mm YY L

Since rrri

pr

1

, we have

)]([1

)()1

(1

)(2

222 rrR

rrrRr

rrir

rrirRp kkkr

or

)()()1(1

)]([1 2

22

2

rRkrRr

rrRrr kkk

or

0)()]1(1

[)]([1

22

2

2

rRr

krrRrr kk .

In the limit of r →∞, we have

29

0)]([)]([ 22

2

rrRkrrRr kk .

Then we get

r

eR

ikr

kl

(outgoing and incoming spherical waves)

________________________________________________________________________ APPENDIX II For the potential

)()( 0 RrRVrV

Calculate in Born approximation the quantities )(f and d

d. Specify the limits of validity of

your calculation for both high- and low-energy scattering, respectively.. ((Schaum, Quantum Mechanics)) (a)

)sin(2

)'sin()'(''21

)'sin()'(''21

)(

2

20

0

02

02

)1(

QRQ

RV

QrRrRVrdrQ

QrrVrdrQ

f

.

22

22

2

30

2)1( )(sin2)(

RQ

QRRVf

d

d

where

2sin2

kQ

Validity of the Born approximation

1)'()1(')'sin()'('2

0

'22

0

'2

rVedrk

krrVedrk

ikrikr

30

or

k

rVedr ikr2

0

'2 )'()1('

for the potential with the spherical symmetry. In the present case

k

RredrRV ikr2

0

'20 )'()1('

or

1)sin(2

20 kRk

RV

(i) In the limit of kR<<1 (low energy limit),

12

)sin(2

22

020

RVkR

kRV .

(ii) For high energies,

1)sin(2

20 kRk

RV