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Physics 452
Quantum mechanics II
Winter 2012
Karine Chesnel
Phys 452
Homework
Friday Mar 30: assignment # 20
11.1, 11.2, 11.4
Tuesday Apr 3: assignment #21
11.5, 11.6, 11.7
Sign up for the QM & Research presentations
On Friday April 6 or Monday April 9
Phys 452
Homework
Fri April 6 & Mon April 9
assignment # 24
Research &QM presentations
• Briefly describe your research project and how Quantum Mechanics
can help you or can be connected to your research field
• If no direct connection between your research and QM, mention one
topic of QM that could potentially be useful or that you particularly liked
• 2-3 minutes / student (suggested 2-3 transparencies)
Phys 452
Scattering
A classical geometrical view
Scattering
solid angle
d
Scattering angle
d
b
Incident
cross-section
Differential cross-section
d
D   
d
Impact parameter
Phys 452
Scattering
Scattering
solid angle
A classical geometrical view
Scattering angle
d
b
Incident
cross-section
Impact parameter
To be determined in specific situations
b db
D   
sin  d
 tot   d   D  d 
Example: Hard-sphere scattering
b  R cos  / 2
 tot   R 2
Phys 452
Scattering
Pb 11.1 Rutherford scattering
1. Conservation of energy
2. Conservation of angular momentum
3. Change of variable to express r(f)
4. Integration: to find fmax in terms of b etc
5 Relationship between fmax and 
6 Final relationship
b
b
q1q2
cot  / 2 
8 0 E
q1

r
f
q2
Phys 452
Scattering
Quantum treatment
Plane wave
 plane  Aeikz
Spherical wave
 sph
eikr
 Af  
r
Scattering
amplitude
Phys 452
Quiz 33
Which one of these statements describes best
the quantum treatment of scattering?
A. The incident wave is described by the initial state of the wave function
B. The scattered wave is described by the final state of the wave function
C. Only the scattered wave is the solution to the Schrödinger equation
D. The global solution for the wave function includes the sum
of the incident and scattered waves
E. The quantum theory of scattering has no physical analogy
with the classical theory of scattering
Phys 452
Scattering
Quantum treatment
Elastic
process
Spherical wave
Plane wave
In 3D
 ikz
eikr 
  A e  f   
r 

2
 sph dV  cst
2
Pb 11.2
In 2D
 sph dS  cst
2
In 1D
 sph dz  cst
Phys 452
Scattering
Quantum treatment
Plane wave
Spherical wave
 ikz
eikr 
  A e  f   
r 

Differential
cross-section
2
d
D   
 f  
d
Phys 452
X-ray Resonant Magnetic Scattering
q=k’-k
Interaction
photon spin / magnetic moment
’ e’
’

e

Sphoton
M

I (q ) 

iq rn
 fe
2
n
f  f non.res  f res
Electric dipolar transition E1 ( L2,3 : 2p 3d) edges)
f E1  (e' e) F (0)  i(e'e)  MF (1)  (e' M )(e  M ) F ( 2)
charge
magnetism
Phys 452
Scattering
Partial wave analysis
Develop the solution in
terms of spherical harmonics,
solution to a
spherically symmetrical potential
  r, ,f   R  r  Yl m  ,f 
kr
1
2
d 2u 
l (l  1) 


V
r




 u  Eu
2
2
2m dr
2m r 

2
V 0
Radiation zone
V 0
intermediate zone
Scattering zone
Phys 452
Scattering
Partial wave analysis
kr
V 0
V 0
V 0
1
Radiation zone
Intermediate
zone
Physical Solution
General Solution
Scattering zone
kr
1
d 2u
2


k
u
2
dr
R  r   eikr / r
d 2u l (l  1)
2

u


k
u
2
2
dr
r
R(r )
hl1  kr 
Hankel functions
 ikz

1
m
  r , , f   A e   cl ,m hl  kr  Yl  , f 
l .m


Geometrical considerations
Solve the Schrödinger
equation with potential V


  r , , f   A eikz  k  i l 1  2l  1 al hl1  kr  Pl  cos  
l


Partial wave amplitude
Phys 452
Scattering
Partial wave analysis
kr
1
V 0
V 0
Connecting intermediate and radiation zone
 ikz
eikr 
  r , , f   A e  f   
r 

with
when
kr
1
f      2l  1 al Pl  cos  
l
Differential
cross-section
Total
cross-section
D    f    
2
l'
*
2
l

1
2
l
'

1
a




l ' al Pl '  cos  Pl  cos  
l
   D   d  4   2l  1 al
l
2
Orthogonality of
Legendre polynomials
Phys 452
Scattering
Partial wave analysis
kr
1
Connecting all three regions and expressing the
Global wave function in spherical coordinates
 ikz
eikr 
  r ,   A e  f   
r 

V 0
V 0
Rayleigh’s formula

e   i l  2l  1 jl  kr  Pl  cos  
ikz
l 0

Jl Bessel functions

  r ,   A i l  2l  1 jl  kr   ikal hl1  kr  Pl  cos 
l
Total
cross-section
   D   d  4   2l  1 al
l
2
To be determined
by solving the
Schrödinger equation
in the scattering region
+ boundary conditions
Phys 452
Scattering
Partial wave analysis


  r ,   A i l  2l  1 jl  kr   ikal hl1  kr  Pl  cos 
l
Bessel function
Hankel function
h1(1) ( x)  jl ( x)  inl ( x)
Legendre polynomial
Phys 452
Scattering
Partial wave analysis
V 0
Boundary conditions
V 
  a,   0


A il  2l  1 jl  kr   ikal hl1  kr  Pl  cos    0
l
Exploiting Pn Pl   nl
Total
cross-section
(Pb 11.3)
Example: Hard-sphere scattering
  4   2l  1 al
al  i
jl  ka 
khl1  ka 
2
l
ka
1
  4 a 2
Phys 452
Scattering- Partial wave analysis
Pb 11.4
Spherical delta function shell
Assumption ka
V 0
Outside:
(low energy scattering)
f      2l  1 al Pl  cos    a0 P0  cos    a0
l
 sin  kr 
eikr 
  A
 a0

kr
r 

V 0
Inside:
 r   B
Continuity of
Boundary conditions
1
sin  kr 
kr

Discontinuity of 
Find a relationship
between a0 and (a,...
'
f  
 ' 
2m
2
D  
 a
