Let H=H_{0}+H'=H_{0}+λW. Let {φ^{i}_{p}>} denote an orthonormal eigenbasis of H_{0}, H_{0}φ^{i}_{p}>=E_{0}^{p}φ^{i}_{p}>. Here i denotes the degeneracy. Consider a particular degenerate eigenvalue E_{0}^{p} of H_{0}. Assume that this eigenvalue is gfold degenerate, i = 1, 2, ...,g. To find E_{1}^{p} we use
Multiplying from the left by <φ^{i}_{p} we obtain
This is an eigenvalue equation for the operator W in the subspace E(0,p) of vectors with the eigenvalue of H_{0} equal to E_{0}^{p}. We find g eigenvalues E_{1}^{p,i} which may or may not be degenerate. We then find the corresponding g eigenvectors. To solve for the eigenvalues we set det(WE_{1}^{p})=0 in the subspace E(0,p). If we find a nondegenerate eigenvalue E_{1}^{p,i}, then the corresponding eigenvector is uniquely defined. If the eigenvalue E_{1}^{p,i} is degenerate then the corresponding eigenvector is still not uniquely defined. The degeneracy may or may not be removed in higher order.
Assume that χ^{j}_{p}> is an eigenvector with eigenvalue E_{0}^{p} of H_{0}, but with a nondegenerate eigenvalue E_{1}^{p,j}. χ^{j}_{p}> is uniquely defined. Firstorder perturbation theory has removed the degeneracy. Let y^{0}_{p}>=χ^{j}_{p}>.
(H_{0}E_{0}^{p})y_{p}^{1}> =( E_{1}^{p,j}W)y_{p}^{0}> .
y_{p}^{0}> is an eigenstate of the unperturbed Hamiltonian. We may expand
in terms of the basis vectors.
<y_{p}^{0}y_{p}^{1}> = 0.
Multiply from the left by <φ^{i}_{p''}.
We can solve for all b^{i}_{p'} but not for the b^{i}_{p}, they remain undetermined. However, this does not preclude us from finding E_{2}^{p,j} , the secondorder energy correction.
.
All the matrix elements multiplying the b^{i}_{p} are zero, since we have diagonalized the matrix of W. Therefore, if firstorder perturbation theory removes the degeneracy of a set of degenerate eigenvalues, we can proceed to find higherorder corrections to the now uniquely defined eigenvalues just like in the non degenerate case.
θ is the angular coordinate. We have chosen units with
(a)
Find the complete set of eigenvalues and eigenfunctions of H_{0}.
(b)
Use perturbation theory to find the first and secondorder corrections
to the ground state energy E^{0} of H_{0}
due to the perturbation V for 0<α<½
.
(c)
For α=½
the ground state energy of H_{0} is degenerate. Find
the firstorder correction to E^{0} for this case.
Solution:
(a) Try
(b) Let 0 < α < ½. Then all states are non degenerate and the ground state has n=0.
(c) If α=½ then all states are degenerate. The ground state is any linear combination of n=0 and n=1. We have to diagonalize the matrix of V in the subspace spanned by

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