New Energy Levels for Degenerate Perturbation Theory

[ma18]

Homework Statement

The e-states of H^0 are

phi_1 = (1, 0, 0) , phi_2 = (0,1,0), phi_3 = (0,0,1) *all columns
with e-values E_1, E_2 and E_3 respectively.

Each are subject to the perturbation

H' = beta (0 1 0
1 0 1
0 1 0)

where beta is a positive constant

a) If E_1 =/ E_2 =/ E_3

What are the new energy levels according to first and second-order perturbation theory

b) If E_1 = E_2 = E_3

What are the new energy levels according to first degenerate perturbation theory

c) If E_1 =/ E_2 = E_3

What are the new energy levels according to first perturbation theory

Homework Equations

For first order non degenerate perturbation:

E_n ^1 = <phi_n ^ 0 | H' | phi_n ^ 0>

For second order perturbation

E_n ^2 = Σ (m=/n) of (|phi_m ^0 | H' | phi_n ^ 0>|^2)/(E_n ^ 0 - E_m ^0)

The Attempt at a Solution

a)

E_1 ^1 = < (1 | H' | (1 >
0 0
0) 0)

I am not sure how to deal with this as I just get zeroAny help pushing me in the right direction would be appreciated

 

[Orodruin]

Getting a zero only means that the first order correction to the energy vanishes. You should comtinue with the eigenstate perturbation and the second order energy correction.

 

[ma18]

Ok, thanks I have figured out a) but am still have trouble with the degenerate case

 

[Orodruin]

Can you show what you have attempted for the degenerate case?