1. Perturbation Theory Lecture 2 Books Recommended:
Quantum Mechanics, concept and applications by
Nouredine Zetili
Introduction to Quantum Mechanics by D.J. Griffiths
Cohen Tanudouji, Quantum Mechanics II
Introductory Quantum Mechanics, Rechard L. Liboff

2. Degenerate Perturbation Theory
Two fold degeneracy:
Consider
------(1)
Linear combination
(2)
is eigen state of H0
-----(3)

3. Perturbation will lift the degeneracy.
When perturbation is switched off, the upper state
will reduce to some linear combination of
unperturbed states and lower will reduce to
some other linear combination.
But we don't know what these combinations are and
hence, we cannot wind corrections to even upto first
order using Perturbation theory
we discussed earlier.
We need to find α and β
in eq. (2).

4. We need to solve SE
(4)
Where
-----(5)
Also
---(6)
Using (5) and (6) in (4)
-(7)

5. From (7)
----(8)
Consider inner product with
Using (1) and (2) in above, we get
---(9)

6. Writing (9) in form
---(10)
Where
---(11)
Consider inner product with in Eq. (8),
---(12)

7. Multiply (12) by Wab and use (10) to eleminate
βWab, we get
----(13)
For non-zero α, we get
----(14)
Which has roots, (using )
----(15)

8. If α = 0, then β = 1 and from (10)
------(16)
And thus, form (12)
(17)
Check this from (15). It comes from + sign.
For α = 1 and β = 0, we have root corresponding to
–Ve sign.

9. Thus as special case we have
Which can be obtained from non-degenerate theory

11. Higher Order Perturbation Theory
For 2-fold degeneracy, we can write (10) and (12) as
For n-fold degeneracy, we will have n by n matrix
Whose elements will be

12. Example: Consider 3-Dim infinite potential well
Stationary states
Energies

13. Ground state energy
First excited state is triply degenerate
having energy

14. Consider perturbations
First order correction to ground state energy

15. To find the correction to the energy of first
excited state which is triply degenerate
We need to construct the matrix W
(recall result of derivation)
Diagonal elements will be same as derived
above

16. For off diagonal elements
Last integral over z will be zero and hence
Wab = 0
Similarly

17. Also
Thus
Now we find Eigen values of above matrix and
Corresponding Eigen vectors

18. Characteristic equation (without V0/4)
Eigenvalues
Thus, upto first order we have

19. Good unperturbed state

20. Find Eigen vector corresponding to different
eigenvalue