1. Perturbation Theory Lecture 1 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. Approximate Methods:
Approximate methods for stationary states corresponds
to time independent Hamiltonian.
Examples: perturbation theory, variational method,
and the WKB method.
Perturbation theory: Exactly solvable part + small
correction
In case we are not able to reduce the Hamiltonian
of the problem to exactly solvable part+ small
correction then we use variation method or
WKB method

3. Time independent perturbation theory
is splited into two parts, exactly solvable part
and small perturbative part
(1)
Small perturbative part can be written as
(2)
From (1)
-----(3)

4. Non-degenerate perturbation theory
We have
(4)
Perturbed Eigen value and Eigen state is expanded as
------(5)
------(6)
For λ = 0 we get unperturbed part.

5. Use (5) and (6) in (3), we get
----(7)
Now we compare powers of λ on both sides

6. Zero power of λ
----(8)
1st power of λ
----(9)
2nd power of λ
----(10)

7. Now we have to find eigen values and eigen functions
not much different from
----(11)
Using (6) in (11), we get
----(12)
Coefficients of various powers of λ vanishes
----(13)

8. First order correction to :
Multiplying both sides of (9) by
(14)
Where we used = 0
and
Thus, from (5), using (14)
(15)

9. Expand in terms of basis:
------(16)
From (9), multiplying by
(17)

10. Using (17) in (16), we get
----(18)
We have
(19)

11. Second order correction:
Multiply both sides by
(20)
Where we used
Using (18) in (20)
----(21)

12. Thus En upto 2nd order
Validity :