special case we also consider the ground state of a hydrogen-like atom. This is
because that state is nondegenerate, and so the problem can be dealt with in parallel
to the cases of the one-dimensional physical systems. We assume that the quantum
sates ji i (i ¼ 0, 1, 2, Á Á Á) constitute the orthonormal eigenvectors such that
j
h j ii ¼ δ ji :
ð5:4Þ
Remember that the notation has already appeared in (2.53); see Chap. 13 as well.
5.1.1 Quantum State and Energy Level Shift Caused by
Perturbation
One of the most important applications of the perturbation method is to evaluate the
shift in quantum state and energy level caused by the applied external field. To this
end, we expand both the quantum sate and energy as a power series of λ. That is, in
(5.1) we expand jΨ i i and E i such that
j Ψ i i ¼j ii þ λ j ϕ
1
ð Þ
i i þ λ
2
j ϕ
2
ð Þ
i i þ λ
3
j ϕ
3
ð Þ
i i þ Á Á Á,
ð5:5Þ
E i ¼ E
0
ð Þ
i þ λE
1
ð Þ
i þ λ
2 E
2
ð Þ
i þ λ
3 E
3
ð Þ
i þ Á Á Á,
ð5:6Þ
where j ϕ
1
ð Þ
i i, j ϕ
2
ð Þ
i i, j ϕ
3
ð Þ
i i, etc. are chosen as correction terms for the state jii that
is associated with the unperturbed system. Once again, we assume that the state ji i
(i ¼ 0, 1, 2, Á Á Á) represents the nondegenerate normalized eigenvector that belongs to
the eigenenergy E
0
ð Þ
i of the unperturbed state. Unknown state vectors j ϕ
1
ð Þ
i i, j ϕ
2
ð Þ
i i,
j ϕ
3
ð Þ
i i, etc. as well as E
1
ð Þ
i , E
2
ð Þ
i , E
3
ð Þ
i , etc. are to be determined after the calculation
procedures carried out from now. These states jΨ i i and energies E i result from the
perturbation term λV and represent the deviation from jii and E
0
ð Þ
i of the unperturbed
system.
With the normalization condition we impose the following condition upon jΨ i i
such that
ijΨ i
h
i ¼ 1:
ð5:7Þ
This condition, however, not necessarily means that jΨ i i has been normalized
(vide infra). From (5.4) and (5.7) we have
ijϕ
1
ð Þ
i
D
E
¼ ijϕ
2
ð Þ
i
D
E
¼ Á Á Á ¼ 0:
ð5:8Þ
Let us calculate hΨ i j Ψ i i on condition of (5.7) and (5.8). From (5.5) we have
5.1 Perturbation Method
153
because that state is nondegenerate, and so the problem can be dealt with in parallel
to the cases of the one-dimensional physical systems. We assume that the quantum
sates ji i (i ¼ 0, 1, 2, Á Á Á) constitute the orthonormal eigenvectors such that
j
h j ii ¼ δ ji :
ð5:4Þ
Remember that the notation has already appeared in (2.53); see Chap. 13 as well.
5.1.1 Quantum State and Energy Level Shift Caused by
Perturbation
One of the most important applications of the perturbation method is to evaluate the
shift in quantum state and energy level caused by the applied external field. To this
end, we expand both the quantum sate and energy as a power series of λ. That is, in
(5.1) we expand jΨ i i and E i such that
j Ψ i i ¼j ii þ λ j ϕ
1
ð Þ
i i þ λ
2
j ϕ
2
ð Þ
i i þ λ
3
j ϕ
3
ð Þ
i i þ Á Á Á,
ð5:5Þ
E i ¼ E
0
ð Þ
i þ λE
1
ð Þ
i þ λ
2 E
2
ð Þ
i þ λ
3 E
3
ð Þ
i þ Á Á Á,
ð5:6Þ
where j ϕ
1
ð Þ
i i, j ϕ
2
ð Þ
i i, j ϕ
3
ð Þ
i i, etc. are chosen as correction terms for the state jii that
is associated with the unperturbed system. Once again, we assume that the state ji i
(i ¼ 0, 1, 2, Á Á Á) represents the nondegenerate normalized eigenvector that belongs to
the eigenenergy E
0
ð Þ
i of the unperturbed state. Unknown state vectors j ϕ
1
ð Þ
i i, j ϕ
2
ð Þ
i i,
j ϕ
3
ð Þ
i i, etc. as well as E
1
ð Þ
i , E
2
ð Þ
i , E
3
ð Þ
i , etc. are to be determined after the calculation
procedures carried out from now. These states jΨ i i and energies E i result from the
perturbation term λV and represent the deviation from jii and E
0
ð Þ
i of the unperturbed
system.
With the normalization condition we impose the following condition upon jΨ i i
such that
ijΨ i
h
i ¼ 1:
ð5:7Þ
This condition, however, not necessarily means that jΨ i i has been normalized
(vide infra). From (5.4) and (5.7) we have
ijϕ
1
ð Þ
i
D
E
¼ ijϕ
2
ð Þ
i
D
E
¼ Á Á Á ¼ 0:
ð5:8Þ
Let us calculate hΨ i j Ψ i i on condition of (5.7) and (5.8). From (5.5) we have
5.1 Perturbation Method
153