Introduction to

Time dependentTime-independent methods:

Kap. 7-lect2

),(ˆ),( trHtrt

i

),(),(ˆ trEtrH

Methods to obtain an approximate eigen energy, E and wave function:

perturbation methods

Perturbation theory Variational method Scattering theory

),()(),( trtctr nn

n

Ground/Bound states Continuum states

Non degenerate states

Methods to obtain an approximate expression for the expansion amplitudes.

Approximation methods in Quantum Mechanics

Degenerate states

GoldenRule

Scattering Theory:

• Classical Scattering:– Differential and total cross section– Examples: Hard sphere and Coulomb

scattering

• Quantal Scattering:– Formulated as a stationary problem– Integral Equation– Born Approximation– Examples: Hard sphere and Coulomb scattering

ddd

dd

d

d)sin(

0

2

0

scin

dNj

dtd

Number of scattered particles into :

insc jdj

per unit time:

dj

j

d

d

in

scDifferential Cross Section:

Total Cross Section:

Dimension: AreaInterpretation: Effective area for scattering.

Dimension: NoneInterpretation: ”Probability” for scattering into d

innj

ddscj

The Scattering Cross Section (To be corrected, see Endre Slide)

ddd

dd

d

d)sin(

0

2

0

Number of scattered particles into :

Differential Cross Section:

Total Cross Section:

innj

dd

The Scattering Cross Section

Nout

Nout

Quantal Scattering - No Trajectory! (A plane wave hits some object and a spherical wave emerges)

innj

dd

r

eCf

ikr

scattered ),(

scatteredinn

rkiinn Ce

• Solve the time independent Schrödinger equation• Approximate the solution to one which is valid far away from the scattering center• Write the solution as a sum of an incoming plane wave and an outgoing spherical wave.• Must find a relation between the wavefunction and the current densities that defines the

cross section.

Procedure:

scj

Current Density:

imm

ij *** Re)(

2

Incomming current density:

2C

m

kjin

Outgoing spherical current density:

2

22),(

mr

kfCjsc

r

eCf

ikr

sc ),(

ikzrkiin CeCe

zyx ez

fe

y

fe

x

frf

)(

eeer

fr

.......

2),(),(

rOr

eikCf

r

eCf

r

ikrikr

Example - Classical scattering:

d

b

bdbdd

ddd

dd )sin(

d

dbb

d

d

sin

b R

2

2cos

R

b

2sin

2

R

d

db

4

2R

d

d

2R

Hard Sphere scattering:

Independent of angles!= Geometrical Cross sectional area of sphere!

Example from 1D:

scatteredinn

0V

0

22

2V

m

kE

ikxikxscattered BeAeC

ikxBe

B

Af ),(

Forward scattering

Reflection

ikxinn Ce

ikxAe

In this case (since potential is discontinuous) we can find f excactly by ”gluing”

The Schrödinger equation - scattering form:

)( )( )(2

22

rErrVm

)( )()( 22 rrUrk

:get we)(2

)( and 2

with 2

22

rVm

rUm

kE

Now we must define the current densities from the wave function…

The final expression:

2

2

22

2

22

),(),(

f

mk

Cd

drfmrk

C

jd

drj

d

d

in

sc

2),(

fd

d

Summary

Then we have:

2),(

fd

d

…. Now we can start to work

)( )()( 22 rrUrk

Write the Schrödinger equation as:

)),(( r

efeC

ikrrki

Asymptotics:

Integral equation

)( )()( 22 rrUrk

)()(G ''22 rrrrk

With the rewritten Schrödinger equation we can introducea Greens function, which (almost) solves the problem for a delta-function potential:

Then a solution of:

can be written:

rdrrUrrGrr 30 )()()()()(

where we require: 0)( 022 rk

because….

rdrrUrrGkrkrk 3220

2222 )()()()()()()()(

This term is 0 This equals )( rr

Integration over the delta function gives result:

)()()()( 22 rrUrk

rdrrUrrGrr 30 )()()()()(

Formal solution:

Useless so far!

Must find G(r) in )()(G ''22 rrrrk

Note: sder rsi 33)2(

1)(

rsirsi ese 22and:

sdekssdek rsirsi 322322 )()(

sdks

erG

rsi3

222)2(

1)(

)()2(

1)()( 3

322 rrderGk rsi

r

erG

rki

4)(

Then:

The function: solves the problem!

”Proof”:

The integral can be evaluated, and the result is:

rdrrUrr

eer

rrikrki

3

01 )()(4

1)(

Inserting G(r), we obtain:

0)( 022 rk

implies that:

rkier )(0

rdrrUrr

eer

rrikrki

3)()(

4

1)(

At large r this can be recast to an outgoing spherical wave…..

The Born series:

rkier )(0

rdrrUrr

eer

rrikrki

3

12 )()(4

1)(

And so on…. Not necesarily convergent!

rdrrUrr

eer

rrikrki

3

01 )()(4

1)(

We obtains:

rdrrUrr

eer

rrikrki

3)()(

4

1)(

At large r this can be recast to an outgoing spherical wave…..

The Born series:

rkier )(0

rdrrUrr

eer

rrikrki

3

12 )()(4

1)(

And so on…. Not necesarily convergent!

)( )()( 22 rrUrk

Write the Schrödinger equation as:

)),(( r

efeC

ikrrki

Asymptotics:

SUMMARY

)1(

)21(

2

2

2/12

2

2

22

r

rrr

r

r

r

rrr

rrrrrr

The potential is assumed to have short range, i.e. Active only for small r’ :

rr

rikikrrrik eee

1)

rrr

11

ff k

p

r

rk

Asymptotics - Detector is at near infinite r

2)

),(

3)()(4

1)(

f

rkiikr

rki rdrrUer

eer f

Asymptotic excact result:

Still Useless!

The Born approximation:

rdrVem

r

eer rkki

ikrrki f

3)(

2)(

4

2)(

rdrVem

f rkki f 3)(

2)(

2),(

The scattering amplitude is then:

:) The momentum change Fourier transform of the potential!

Use incomming wave instead of )'(r Under integration sign:

Valid when:

1)( rkier

1)()(4

1 3

rdrrUrr

e rrik

Weak potentialsand/or large energies!

rdrVem

f rkkiB

q

f 3)(

2)(

4

2),(

fk

k

q

2sin2

kq

2

00

cos'

0

22

sin)(4

2)( ddedrrrV

mf iqrB

2cos

2sin2sin

dkdq

2cos

0

2sin)(

2drqrrV

q

m

Spheric Symmetric potentials:

Total Cross Section:

ddd

dd )sin(

qdqqfk

dfk

BB

22

02

0

2)(

2)sin()(2

Summary - 1’st. Born Approximation:

rdrVem

f rkkiB

q

f 3)(

2)(

4

2),(

qdqqfk

kB

22

02

)(2

2),( Bf

d

d

fkkq

Best at large energies!

Example - Hard sphere 1. Born scattering:

b R

2

4

2R

d

d

2R

Classical Hard Sphere scattering:

Quantal Hard Sphere potential:

qRRq

mrdrqrRr

q

mf B sin

2sin

22

02

RrrV )(

qRRq

m

d

d 22

2

2sin

2

Depends on angles - but roughly independent when qR << 1 Thats it!