1. Quantum mechanics II Winter 2012
Physics 452
Quantum mechanics II
Winter 2012
Karine Chesnel

2. Phys 452
Homework
Friday Jan 20: Homework # 4
Pb 6.6, 6.8, 6.9, 6.10

3. Example of non-degenerate system
Phys 452
Perturbation theory
Example of non-degenerate system
Spring constant expansion in the harmonic oscillator V x

4. Degenerate perturbation theory
Phys 452
Degenerate perturbation theory
Example of
degenerate system 1s 2s 2p 3d
Atomic
energy levels
perturbation
“lift the degeneracy”

5. Quiz 5 Phys 452 Taking the electron spin into account,
in how many energy levels, the (3d) band of Fe atom
is susceptible to split, under the application of a magnetic field?
A. 2
B. 3
C. 5
D. 6
E. 10

6. Degenerate perturbation theory
Phys 452
Degenerate perturbation theory
Two-fold degeneracy
are the eigenvalues of
In fact, we are solving for

7. Degenerate perturbation theory
Phys 452
Degenerate perturbation theory
Two-fold degeneracy E
d=2 E0

8. Degenerate perturbation theory
Phys 452
Degenerate perturbation theory
Two-fold degeneracy
In fact, we are solving for If
Then, using
Choose a “good” clever basis
(to make W diagonal)
Pb 6.6

9. Degenerate perturbation theory
Phys 452
Degenerate perturbation theory
Higher –order degeneracy
Find eigenvalues of
d = 3
3 energies
Find eigenvalues of
d = N
N energies
Pb 6.10
Try to find a “good” basis
to make W diagonal

10. Degenerate perturbation theory
Phys 452
Degenerate perturbation theory
General method
Start with an ortho-normal basis of the unperturbed states
If the state is non-degenerate:
If the state is degenerate: build l
Diagonalize W : the eigenvalues are
Pb 6.8 & 6.9

11. Quiz 5b Phys 452 Here is a list of energy levels for a given system
(V,5V,3V,3V,2V, 3V,5V,5V,V,7V)
How many levels are degenerate and non-degenerate?
A. (3,1)
B. (2,3)
C. (3,2)
D. (1,4)
E. (4,1)

12. Degenerate perturbation theory
Phys 452
Degenerate perturbation theory
Example: cubic infinite square well z
Perturbation:
in a sub-cell y l x
Build W
for the first excited state
Diagonalize W

13. Degenerate perturbation theory
Phys 452
Degenerate perturbation theory
Example: problem 6.8 z
Perturbation: y l ? x
Build W for the first excited state
Diagonalize W

14. Degenerate perturbation theory
Phys 452
Degenerate perturbation theory
Problem 6.9
e is the pertubation
Identify the unperturbed states
determine which one are degenerate or non degenerate
Find the exact solution of the perturbed Hamiltonian
Compare with the first order correction
Use W for degenerate state
Compare with the second order correction for non- degenerate states