H ^ ^ = H 0 ^ ^ + V ^ ^ The perturbed eigenstate, for no degeneracy, is given by- | ψ ψ 0 ⟩ ⟩ = | n 0 ⟩ ⟩ + ∑ ∑ k ≠ ≠ 0 V k 0 / ( E 0 − − E k ) | n k ⟩ ⟩ The perturbed energy eigenket as well as higher order energy shifts diverge when E 0 = E k , i.e., in the presence of degeneracy in energy levels. Assuming H 0 ^ ^ possesses N degenerate eigenstates | m ⟩ ⟩ with the same energy eigenvalue E, and also in general some non-degenerate eigenstates. A perturbed eigenstate | ψ ψ j ⟩ ⟩ can be written as a linear expansion in the unperturbed degenerate eigenstates as- | ψ ψ j ⟩ ⟩ = ∑ ∑ i | m i ⟩ ⟩ ⟨ ⟨ m i | ψ ψ j ⟩ ⟩ = ∑ ∑ i c j i | m i ⟩ ⟩ [ H 0 ^ ^ + V ^ ^ ] ψ ψ j ⟩ ⟩ = [ H 0 ^ ^ + V ^ ^ ] ∑ ∑ i c j i | m i ⟩ ⟩ = E j ∑ ∑ i c j i | m i ⟩ ⟩ where E j refer to the perturbed energy eigenvalues. Since E is a degenerate eigenvalue of H 0 ^ ^ , ∑ ∑ i c j i V ^ ^ | m i ⟩ ⟩ = ( E j − − E ) ∑ ∑ i c j i | m i ⟩ ⟩ = Δ Δ E j ∑ ∑ i c j i | m i ⟩ ⟩ Premultiplying by another unperturbed degenerate eigenket ⟨ ⟨ m k | gives- ∑ ∑ i c j i [ ⟨ ⟨ m k | V ^ ^ | m i ⟩ ⟩ − − δ δ i k ( E j − − E ) ] = 0 This is an eigenvalue problem, and writing V i k = ⟨ ⟨ m i | V ^ ^ | m k ⟩ ⟩ , we have- | V 11 − − Δ Δ E j V 12 … … V 1 N V 21 V 22 − − Δ Δ E j … … V 2 N ⋮ ⋮ ⋮ ⋮ ⋱ ⋱ ⋮ ⋮ V N 1 V N 2 … … V N N − − Δ Δ E j | .