4. Quantum mechanics II Winter 2011
Physics 452
Quantum mechanics II
Winter 2011
Karine Chesnel

5. Homework Phys 452 Today Mar 30: assignment # 20 11.1, 11.2, 11.4
Friday Apr 1: assignment #21
11.5, 11.6, 11.7
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Next week, W April 6 or F April 8

6. Scattering Phys 452 Quantum treatment Spherical wave Plane wave
Relationship
with cross-section

7. Scattering Phys 452 Partial wave analysis Develop the solution in
terms of spherical harmonics,
Solution to Coulomb potential
Radiation zone
intermediate zone
Scattering zone

8. in the scattering region
Phys 452
Scattering
Partial wave analysis
General solution where V~0:
Wave function in spherical coordinates
Rayleigh’s formula
Scattered waves
To be determined
by solving the
Schrödinger equation
in the scattering region
+ boundary conditions
Total
cross-section
for large r

9. Scattering Phys 452 Partial wave analysis (Pb 11.3)
Example: Hard-sphere scattering
Boundary conditions
Exploiting
Total
cross-section

10. Scattering- Partial wave analysis
Phys 452
Scattering- Partial wave analysis
Pb 11.4
Spherical delta function shell
Assumption (low energy)
Outside:
Inside:
Boundary conditions
at r = a
Continuity of
Discontinuity of
In spherical
coordinates
Find a relationship
between a0 , a and a...

11. Scattering Phys 452 Phase - shifts Physical representation in 1D
wall
Physical representation in 3D

12. Scattering Phys 452 Phase – shifts and interference effects
Physical representation in 1D
wall
interference

13. Quiz 32 Phys 452 In scattering and interference processes,
the phase shift depends on the wavelength
True
False

14. Scattering Phys 452 Phase - shifts If If Asymptotic behavior at
Outgoing
spherical
wave
Incoming
spherical
wave If

15. Scattering Phys 452 Phase - shifts Partial wave amplitude Phase shift
Connecting the asymptotic behavior at
Partial wave amplitude
Phase shift

16. Scattering Phys 452 Phase - shifts Scattering amplitude Scattering
Cross-section

17. Scattering – phase shift
Phys 452
Scattering – phase shift
Pb 11.5
Reflection against a wall
1) Solve the Schrödinger equation
In region 1
In region 2
2) Continuity at boundary:
3) Identify the phase shift
wall
Region 1
Region 2

18. Scattering – Phase shift
Phys 452
Scattering – Phase shift
Pb Hard sphere scattering
(Pb 11.3)
Boundary conditions
We found
Express the phase shift: using
Express in terms of functions and

19. Scattering- phase shifts
Phys 452
Scattering- phase shifts
Pb 11.7
Spherical delta function shell (Pb 11.4)
Do NOT do the assumption
Outside:
Inside:
Boundary conditions
Continuity of
Discontinuity of
Express in terms of and