1. Time-Independent Perturbation Theory 1
Source: D. Griffiths, Introduction to Quantum Mechanics (Prentice Hall, 2004)
R. Scherrer, Quantum Mechanics An Accessible Introduction (Pearson Int’l Ed., 2006)
R. Eisberg & R. Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles (Wiley, 1974)

2. Perturbation Theory
Perturbation Theory: A systematic procedure for obtaining approximate solutions to the unperturbed problem, by building on the known exact solutions to the unperturbed case.

3. Time-Independent Perturbation Theory
Schroedinger Equation for 1-D Infinite Square Well
Obtain a complete set of orthonormal eigenfunctions
If potential is perturbed slightly
Find the new eigenfunctions and eigenvalues of H.

4. Derivation of Corrections
New Hamiltonian:
H’ = perturbation H0 = unperturbed quantity
Write Ψn and En as power series of λ :
Insert
into
Ist order correction to the nth value
2nd order correction to the nth value

5. Derivation of Corrections
After insertion:
Collecting like powers of λ,

6. First Order Correction to Energy
Taking the inner product of:
This means: Multiplying by and integrating.
Replace
But H0 is hermitean, so
and
Therefore:
First order correction to energy: Expectation value of perturbation, in the unperturbed state.

7. First Order Correction to Wavefunction
Rewrite
Known function
Becomes inhomogeneous DE
Therefore:
satisfies

8. First Order Correction to Wavefunction
If l = n, m = n
Equals Zero
!st order energy correction
First order correction to wavefunction
If n = m, degenerate perturbation theory need to be used.

9. Example:
V(x)
V(x)
-d/3
d/3
Unperturbed State
Perturbed State
Unperturbed Wave function of Infinitely Deep Square Well

10. Perturbed Energy Levels are obtained from:
Energy is increased by 0.61 times the amount of additional potential energy at

11. To find the perturbed wave function:
and
Unperturbed levels are degenerate. Perturbation remove degeneracy.

12. Example
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.

13. Example: Continue

14. Problem 1

15. Problem 2