Transcript Born approximation - BYU Physics and Astronomy

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
Phys 452
Class- schedule
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
Phys 452
Research and QM presentation
Template
As an experimentalist
In the lab …
…or doing simulations
or theory
Phys 452
Research and QM presentation
Template
Focus on
one physical
principle or
phenomenon
involved
in your research
Make a connection
with a topic covered in
Quantum Mechanics:
A principle
An equation
An application
Phys 452
Scattering
Quantum treatment
q
Plane wave
Spherical wave
 ikz
eikr 
  A e  f  ,   
r 

Easy formula to calculate f(,)?
or f(q)?
Phys 452
Born formalism
Worked together with
Max Born (1882-1970)
German physicist
Albert Einstein
(Nobel Prize 1921
Photoelectric effect)
Nobel prize in 1954
For interpretation of probability of density 
Werner Heisenberg
(Nobel Prize 1932
Creation 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 452
Born formalism
Born approximation:
The main impact of the interaction
is that an incoming wave of
direction k is just deflected in a
direction k ' but keeps same
amplitude and same wavelength.
Max Born (1882-1970)
One can express the scattering
factor
f  ,  
German physicist
In terms of wave vectors
Nobel prize in 1954
For interpretation of probability of density 
k,k '
Phys 452
Born formalism
Schrödinger equation

Solution
2
k

k  Q 
2
2m
2
V
2mE
Helmholtz equation
  r    G  r  r0 Q  r0  d 3r0

2

 k G r   
2
Green’s
function
3
r 
George Green
British Mathematician
1793 - 1841
Helmholtz
1821 - 1894
Phys 452
Born formalism
Using Fourier Transform of Helmholtz equation
and contour integral with Cauchy’s formula, one gets:
eikr
G r   
4 r
Green’s function
 r   0 r  
ik r  r
m
2
2
Pb 11.8
e 0
3
V
r

r
d




 r  r0 0 0 r0
Integral form of the Schrödinger equation
Phys 452
Born approximation
• First Born approximation
r
eikr ikr0
G  r  r0   
e
4 r
r0
  r    0  r   Aeik '.r
  r   Aeik .r
f  ,    
 0  r   Aeik '.r
m
2
2
e


i k 'k r
V  r  d 3r
Phys 452
Quiz 35b
When expressing the scattering factor as following
f  ,    
m
2
2
iq .r
3
e
V
r
d
  r

What approximation is done?
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
Phys 452
Born approximation
k'
Scattering vector
q
q  2k sin  / 2  
k
f  ,    
m
2
2
e
iq .r
V r  d r
3
4

sin  / 2 
Phys 452
Born approximation
• Low energy approximation
q.r
1
Examples:
f  ,    
m
2
V r  d r
3
2
• Soft-sphere
• Case of spherical potential

2m
f  ,     2  rV  r  sin  qr  dr
q0
• Yukawa potential
• Rutherford
scattering
Phys 452
Born approximation
Pb 11.10
Soft sphere potential
• Scattering amplitude
Case of spherical potential

f  ,     rV0 sin  qr  dr
V0
0
• Approximation at low E
qa
Develop sin  qa  and
1
cos  qa  to third order
Phys 452
Scattering – Phase shift
Pb 11.11 Yukawa potential
e  r
V r   
r

f  ,     e  r sin  qr  dr
0
Expand

1 iqr  iqr
sin  qr  
e e
2i

f 
1
 2  q2
Phys 452
Scattering- phase shifts
Pb 11.13
Spherical delta function shell (Pb 11.4)
V (r )    r  a 
V 0
• Low energy case
f  ,       r  a d 3r
V 0
• For any energy
f  ,     r  r  a  sin  qr dr
•
Compare results with pb 11.4
f  2
m a 2
2
Phys 452
Scattering – Born approximation
Pb 11.20 Gaussian potential
V  r   Ae

f  ,     re
 r2
 r2
sin  qr  dr
0
Integration by parts
f e
Differential cross- section
d
 f
d
   f d
2
Total cross- section
 q2 /4 
f has also a Gaussian
shape in respect to q
2
don’t forget that
q  2k sin  / 2
Phys 452
Born approximation
Impulse approximation
I   F dt
impulse
momentum
p
I
Deflection

Pb 11.14:
tan  
I
p
Rutherford scattering
q1
Step 1. Evaluate the transverse force F

Step 2. Evaluate the impulse I
Step 3. Evaluate the deflection 
b
Step 4. deduct relationship between b and 
r

q2
Phys 452
Born approximation
Impulse and Born series
  r    0  r    G  r  r 0 V  r0   r0  d 3r0
Unperturbed wave
(zero order)
Deflected wave
(first order)
Extending at higher orders
propagator
  r    0  r    GV    GVGV     GVGVGV  ...
Zero
order
First
order
Second order
Third order
See pb 11.15
Phys 452
Born approximation
Pb 11.16
Pb 11.17
Pb 11.18: build a reflection coefficient
Back scattering
(in 1D)
R
 m 
 2 
 k
• Delta function well:
V  x     x 
2 

R

2 ikx
e
 V  x  dx

e 2ikx  x dx

• Finite square well
a
-a
a
R
2ikx
e
 V0dx
a
2
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 452
Compton scattering
Arthur Compton
(1892-1962, Berkeley)
American physicist
Nobel prize in 1927
For demonstrating the “particle”
concept of an electromagnetic
radiation
January 13, 1936
Phys 452
Compton scattering
Phys rev. 21, 483 (1923)
Phys 452
Compton scattering
Classical treatment:
Collision between particles
• Conservation of energy
• Conservation of momentum
Electromagnetic wave
Particle: photon
Phys 452
Compton scattering
Compton experiments
Final wavelength vs. angle
Homework Compton problem (a): Derive this formula from the conservation laws
Phys 452
Compton 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…
We need to evaluate the Hamiltonian for this interaction
and solve the Schrodinger equation
Constraint 2: the energy of the photon and recoiled electron are high
So we need a relativistic quantum theory
Phys 452
Compton scattering
Quantum theory
• Klein – Gordon equation: relativistic electrons in an electromagnetic field

2
2
2
2
2 4

c

i


qA


m
c
2
t

momentum

Vector potential
A  A0  As
• Vector potential
• Interaction Hamiltonian (perturbation theory)


2
1
H
i   qA  mc 2
m
q2
H '  2 As . A0†
m
Energy at rest
Phys 452
Compton scattering
Quantum theory
q2
H '  2 As . A0†
m

i k '.r  ' t
As  As e

i k .r t
A0  A0e

'


q 2 As A0 i k 'k r  ' t 
H'2
e
 '.
m
Phys 452
Compton scattering
Quantum theory
Electron in a scattering state
  r , t    cp pd p
3
with
 p r ,t  
mc3
 2 
3
ei p.r  Et  /
E
First order perturbation theory to evaluate the coefficients:
1
cp'
mc 2
 0
3
 i
dt
d
p

H
'

c
p
p'
p
2  
Homework Compton problem (b): Show that
1
cp'


  E  E '    '  p  p ' k  k '  0
i 2
4
3
  q  mc A0 As   . '  d p
cp
2
E.E '
Phys 452
Compton scattering
Quantum theory
We retrieve the conservation laws:
p ' k '  p  k
E '  '  E  
Furthermore we can evaluate the cross-section:
2
 q
  k'
d
2


.

'
  
d  k '  4 0 mc 2   k 
2
2
Homework Compton problem (c): Evaluate  in case of
k'  k
(d): Compare to Rutherford scattering cross-section
(Thomson
scattering)