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

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

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

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

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

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)

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.

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

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

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

Ground state energy First excited state is triply degenerate having energy

Consider perturbations First order correction to ground state energy

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

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

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

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

Good unperturbed state

Find Eigen vector corresponding to different eigenvalue