physics 452
DESCRIPTION
Physics 452. Quantum mechanics II Winter 2012. Karine Chesnel. Phys 452. Homework. Thu Apr 5 : assignment #22 11.8, 11.10, 11.11, 11.13 Tuesday April 10 : assignment #23 11.14, 11.18, 11.20. Sign up for the QM & Research presentations Fri April 6 or Mon April 9. Homework #24 20 pts. - PowerPoint PPT PresentationTRANSCRIPT
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
Phys 452Born formalism
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
Phys 452Born formalism
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
Phys 452Born formalism
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?
Phys 452Born approximation
'k
. 32
,2
iq rmf e V r d r
k
Scattering vectorq
42 sin / 2 sin / 2q k
Phys 452Born approximation
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
Phys 452Born approximation
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
Phys 452Born approximation
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
Phys 452Born approximation
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
Phys 452Born approximation
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 452Compton scattering
Phys rev. 21, 483 (1923)
Phys 452Compton scattering
Electromagnetic wave
Particle: photon
Classical treatment:Collision between particles
• Conservation of energy
• Conservation of momentum
Phys 452Compton scattering
Homework Compton problem (a): Derive this formula from the conservation laws
Compton experiments
Final wavelength vs. angle
Phys 452Compton scattering
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
Phys 452Compton scattering
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
Phys 452Compton scattering
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
Phys 452Compton scattering
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
Phys 452Compton scattering
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)