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For the quantum harmonic oscillator with
x
=
(
a
+
a
†
)
/
2
x = (a + a^\dagger)/\sqrt{2}
x
=
(
a
+
a
†
)
/
2
in natural units, compute the matrix element
⟨
2
∣
x
2
∣
0
⟩
\langle 2| x^2 |0\rangle
⟨
2∣
x
2
∣0
⟩
. (Use
a
∣
n
⟩
=
n
∣
n
−
1
⟩
a|n\rangle = \sqrt{n}|n-1\rangle
a
∣
n
⟩
=
n
∣
n
−
1
⟩
and
a
†
∣
n
⟩
=
n
+
1
∣
n
+
1
⟩
a^\dagger|n\rangle = \sqrt{n+1}|n+1\rangle
a
†
∣
n
⟩
=
n
+
1
∣
n
+
1
⟩
.)
🔬 try it before you answer
1/2 = 0.5
2
\sqrt{2}
2
≈ 1.414
1
1
/
2
1/\sqrt{2}
1/
2
≈ 0.707
Check answer