physics 452

Physics 452

Quantum mechanics II

Winter 2012

Karine Chesnel

HomeworkPhys 452

Thu Apr 5: assignment #2211.8, 11.10, 11.11, 11.13

Tuesday April 10: assignment #2311.14, 11.18, 11.20

Sign up for the QM & Research presentationsFri April 6 or Mon April 9

Homework #2420 pts

Class- schedule

Phys 452

Today April 4: Born approximation, Compton effect

Friday April 6 : research & QM presentations I

Mon. April 9 : research & QM presentations II

Wed. April 11: FINAL REVIEW

Treats and vote for best presentation In each session

Research and QM presentationPhys 452

Template

As an experimentalistIn the lab …

…or doing simulationsor theory

Research and QM presentationPhys 452

Template

Focus onone physical principle or

phenomenoninvolved

in your research

Make a connection with a topic covered in Quantum Mechanics:

A principleAn equation

An application

Phys 452Scattering

Quantum treatment

Plane wave Spherical wave

,ikr

ikz eA e f

r

Easy formula to calculate f()?

q

or f(q)?

Phys 452Born formalism

Max Born (1882-1970)

German physicist

Nobel prize in 1954For interpretation of probability of density

Worked together with

Albert Einstein(Nobel Prize 1921Photoelectric effect) Werner Heisenberg

(Nobel Prize 1932Creation of QM)

Phys 452

Quiz 35a

What is the main idea of the Born approximation?

A. To develop a formalism where we express the wave function in terms of Green’s functions

B. To use Helmholtz equation instead of Schrödinger equation

C. To find an approximate expression for when far away from the scattering center for a given potential V

D. To express the scattering factor in terms of scattering vector

E. To find the scattering factor in case of low energy


Max Born (1882-1970)

German physicist

Nobel prize in 1954For interpretation of probability of density

Born approximation:

The main impact of the interactionis that an incoming wave of direction is just deflected in a direction but keeps same amplitude and same wavelength.

One can express the scattering factor

In terms of wave vectors

,f

, 'k k

'kk


2 2 3k G r r

2 22

2mk Q V

30 0 0r G r r Q r d r

Solution

Schrödinger equation 2mEk

Helmholtz equation

Helmholtz1821 - 1894

Green’sfunction

George GreenBritish Mathematician1793 - 1841


4

ikreG r

rGreen’s function

0

30 0 0 02

02

ik r rm er r V r r d r

r r

Integral form of the Schrödinger equation

Using Fourier Transform of Helmholtz equationand contour integral with Cauchy’s formula, one gets:

Pb 11.8

Phys 452Born approximation

00 4

ikrikre

G r r er

• First Born approximation

0r r

.ik rr Ae

'.0

ik rr Ae

'.0

ik rr r Ae

' 32

,2

i k k rmf e V r d r

Phys 452

Quiz 35b

When expressing the scattering factor as following

A. The potential is spherically symmetrical

B. The wavelength of the light is very small

C. This scattering factor is evaluated at a location relatively close to the scattering center

D. The incoming wave plane is not strongly altered by the scattering

E. The scattering process is elastic

. 32

,2

iq rmf e V r d r

What approximation is done?


'k

. 32

,2

iq rmf e V r d r

k

Scattering vectorq

42 sin / 2 sin / 2q k


20

2, sin

mf rV r qr dr

q

• Case of spherical potential

32

,2

mf V r d r

• Low energy approximation . 1q r

Examples:

• Soft-sphere

• Yukawa potential

• Rutherford scattering


Soft sphere potentialPb 11.10

• Scattering amplitude

• Approximation at low E

0V

Case of spherical potential

0

0

, sinf rV qr dr

1qa

Develop and to third order sin qa cos qa

Phys 452Scattering – Phase shift

re

V rr

Pb 11.11 Yukawa potential

1sin

2iqr iqrqr e e

i Expand

0

, sinrf e qr dr

2 2

1f

q

Phys 452Scattering- phase shifts

Spherical delta function shell (Pb 11.4)

0V

3,f r a d r

( )V r r a

0V

Pb 11.13

• Low energy case

2

22m a

f

• For any energy

, sinf r r a qr dr

• Compare results with pb 11.4

Phys 452Scattering – Born approximation

2rV r Ae

Pb 11.20 Gaussian potential

Integration by parts

2

0

, sinrf re qr dr

2 /4qf e f has also a Gaussianshape in respect to q

2f d Total cross- section

Differential cross- section2df

d

2 sin / 2q k don’t forget that


Impulse approximation

pmomentum I

impulse

Deflection tanI

p

I F dt

Step 2. Evaluate the impulse I

Step 3. Evaluate the deflection

Pb 11.14: Rutherford scattering

b

r

q1

q2

Step 1. Evaluate the transverse force F

Step 4. deduct relationship between b and


Impulse and Born series

30 0 0 00

r r G r r V r r d r

Unperturbed wave(zero order)

Deflected wave(first order)

Extending at higher orders

0 ...r r GV GVGV GVGVGV

Zeroorder

Firstorder

Second order Third order

propagator

See pb 11.15


Pb 11.18: build a reflection coefficient

• Delta function well:

V x x 2ikxR e x dx

20

aikx

a

R e V dx

• Finite square well

-a a

Pb 11.16

Pb 11.17

22

22

ikxmR e V x dx

k

Back scattering(in 1D)

See pb 11.17

Phys 452

Quiz 35

Compton scattering essentially describes:

A. The scattering of electrons by matter

B. The scattering of high energy photon by light atoms

C. The scattering of low energy photons by heavy atoms

D. The scattering of lo energy neutrons by electrons

E. The scattering of high energy electrons by matter

Phys 452Compton scattering

January 13, 1936

Arthur Compton (1892-1962, Berkeley)

American physicist

Nobel prize in 1927For demonstrating the “particle”concept of an electromagneticradiation


Phys rev. 21, 483 (1923)


Electromagnetic wave

Particle: photon

Classical treatment:Collision between particles

• Conservation of energy

• Conservation of momentum


Homework Compton problem (a): Derive this formula from the conservation laws

Compton experiments

Final wavelength vs. angle


Quantum theory

Photons and electrons treated as waves

Goal: Express the scattering cross-section

Constraint 1: we are not in an elastic scattering situation So the Born approximation does not apply…

Constraint 2: the energy of the photon and recoiled electron are high So we need a relativistic quantum theory

We need to evaluate the Hamiltonian for this interactionand solve the Schrodinger equation


Quantum theory

• Klein – Gordon equation: relativistic electrons in an electromagnetic field

2 2

2 2 2 42

c i qA m ct

0 sA A A

• Vector potential

• Interaction Hamiltonian (perturbation theory)

2 21H i qA mc

m

Vector potentialmomentum Energy at rest

2†0' 2 .s

qH A A

m


Quantum theory

2†0' 2 .s

qH A A

m

2

' '0' 2 '.i k k r tsq A AH e

m

.

0 0

i k r tA A e

'. ''

i k r t

s sA A e


Quantum theory

Electron in a scattering state

3, p pr t c d p

with

3

. /

3,2

i p r Etp

mcr t e

E

First order perturbation theory to evaluate the coefficients:

2

1 03' ''

2p p p p

mcc i dt d p H c

Homework Compton problem (b): Show that

1 02 4 3

' 0

' ' ' '. '

2 . 'p s p

E E p p k kic q mc A A d p c

E E


Quantum theory

' 'p k p k

' 'E E

We retrieve the conservation laws:

Furthermore we can evaluate the cross-section:

2 222

2' 0

'. '

4k

d q k

d mc k

(d): Compare to Rutherford scattering cross-section

'k kHomework Compton problem (c): Evaluate in case of (Thomson scattering)

physics 452

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