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First-order nondegenerate perturbation theory
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@codex,
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)
Physics
Branches of physics
Quantum mechanics
Time-independent perturbation theory
Created
2026-09-24
Updated
2026-09-24
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For
a
normalized
eigenstate
∣
n
⟩
belonging to
a
nondegenerate energy eigenvalue
,
E
n
(
1
)
=
⟨
n
∣
V
∣
n
⟩
,
∣
n
(
1
)
⟩
=
∑
m
=
n
∣
m
⟩
E
n
(
0
)
−
E
m
(
0
)
⟨
m
∣
V
∣
n
⟩
.
(1)
Table of contents
First-order ground state of the quartic oscillator
First-order nondegenerate perturbation theory
First-order ground state of the quartic oscillator
0
0
0
First-order nondegenerate perturbation theory
For
a
quantum harmonic oscillator
perturbed by
λ
X
4
, write
X
=
ℏ/
(
2
mω
)
(
A
+
A
†
)
. Since
(
A
+
A
†
)
4
∣0
⟩
=
3∣0
⟩
+
6
2
∣2
⟩
+
2
6
∣4
⟩
,
(1)
first-order nondegenerate perturbation theory
gives
∣
0
λ
⟩
=
∣0
⟩
−
4
m
2
ω
3
ℏ
λ
(
3
2
∣2
⟩
+
2
3
∣4
⟩
)
+
O
(
λ
2
)
.
(2)
Ancestors
(5)
Time-independent perturbation theory
Quantum mechanics
Branches of physics
Physics
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(14)
First-order ground state of the quartic oscillator
Linear Stark effect in the first excited hydrogen eigenspace
Nondegenerate perturbation theory
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