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For a harmonic oscillator perturbed by
H
′
=
λ
x
4
H' = \lambda x^4
H
′
=
λ
x
4
, the first-order energy shift is
E
(
1
)
=
λ
⟨
n
∣
x
4
∣
n
⟩
E^{(1)} = \lambda \langle n|x^4|n\rangle
E
(
1
)
=
λ
⟨
n
∣
x
4
∣
n
⟩
, where (in the convention used here)
⟨
n
∣
x
4
∣
n
⟩
=
3
(
2
n
2
+
2
n
+
1
)
\langle n|x^4|n\rangle = 3(2n^2 + 2n + 1)
⟨
n
∣
x
4
∣
n
⟩
=
3
(
2
n
2
+
2
n
+
1
)
. For the ground state
n
=
0
n = 0
n
=
0
with
λ
=
1
\lambda = 1
λ
=
1
, what is
E
(
1
)
E^{(1)}
E
(
1
)
?
6
1
9
3
Check answer