1. LECTURE 15

2. Example: Second Order Perturbation
Suppose we put a delta-function bump in the centre of the infinite square well.
where α is a constant. Find the second-order correction to the energies
for the above potential.

3. Example: Continue

4. Degenerate Perturbation Theory
If two or more Eigen-States share the same energy:
and ∞

5. Degenerate Perturbation Theory
Let be the “good” unperturbed state in generic form with a and b adjustable.
Matrix Element of H’
Perturbed energies of degenerate eigenstates:

6. Potential is zero in the box and infinite outside.
PARTICLE IN A BOX
Potential is zero in the box and infinite outside. x y z a c b
when x =0, x = a
y = 0, y = b
z = 0, z = c
…………..(1)
…………..(2)
(3)
(4)
Substitute Equation (1) into Equation (2)
(5)

7. Insert (3), (4) and (5) into (1) and divide by XYZ
…………..(6)
Separate Equation (6) by making each component a constant
Using Boundary conditions: At

8. Degeneracy – Number of states that have the same energy.
Lowest Energy:
No quantum number of a particle in a square box is zero because if so
Degeneracy – Number of states that have the same energy.
Example: Second level above ground level. ,
and
are degenerates if a = b = c, Degeneracy = 3

9. Example: Degeneracy Consider the 3-D infinite cubical well:
This was solved in Lecture 8
The stationary states are:
The allowed energies are:
Ground State
The first excited state is triply degenerate:

10. Perturbation on Degenerate Energy Levels: Example
Ist order correction of ground state:

11. Perturbation on Degenerate Energy Levels: Example
Construct matrix W :
The z integral is zero. It can be shown that:
Therefore:

12. Perturbation on Degenerate Energy Levels: Example
The eigenvalues are:
The first order in l:
The “good” unperturbed states are linear combinations of the form:

13. Perturbation on Degenerate Energy Levels: Example
The coefficients (a, b and g) form the eigenvectors of the matrix W:
The “good” states are:

14. Fine Structure of Hydrogen: Relativistic Correction
Relativistic Momentum and Kinetic Energy:
Expanding T in the powers of small number of (p/mc) since p << mc in the non-relativistic limit.
The lowest order relativistic correction to Hamiltonian:

15. Fine Structure of Hydrogen: Relativistic Correction
First order correction:
Schroedinger Equation in unperturbed state:
Substitute:
and
Relativistic Correction

16. Spin-Orbit Coupling
Electron in orbit around a nucleus. Electrons point of view: proton is circling around it. A magnetic field B is set up, in electron frame. The Hamiltonian
m :Dipole moment of e.
Magnetic field of proton:
Hydrogen atom from electron’s perspective.
B is in the same direction as L
Use to eliminate mo in favor of eo.

17. Spin-Orbit Coupling Magnetic dipole moment of the electron.
Magnetic dipole moment of ring of charge
Moment inertia, mr2 X angular velocity (2p/T)
Gyromagnetic Ratio:
Classical Value
Actual Value
A ring of charge rotating about its axis.
Kinematic Correction: Thomas Precession:

18. Spin –Orbit Coupling Total Angular Momentum:
As a result of Spin-Orbit coupling, H’so no longer commutes with L and S. H’so commute with L2, S2 and J.
Eigenvalues of L.S
In terms of En:
Fine-Structure=Realativistic Correction + Spin-Orbit Coupling
Combining with Bohr Formula: Pg. 149Eq Griffith.

19. Spin-Orbit Coupling
Energy Levels of Hydrogen, including Fine Structure