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Multiple choice
With
x
=
(
a
+
a
†
)
/
2
x = (a + a^\dagger)/\sqrt{2}
x
=
(
a
+
a
†
)
/
2
for the harmonic oscillator (natural units), compute the matrix element
⟨
2
∣
x
∣
3
⟩
\langle 2| x |3\rangle
⟨
2∣
x
∣3
⟩
. (Recall
a
∣
n
⟩
=
n
∣
n
−
1
⟩
a|n\rangle = \sqrt{n}|n-1\rangle
a
∣
n
⟩
=
n
∣
n
−
1
⟩
.)
🔬 try it before you answer
3
\sqrt{3}
3
≈ 1.732
2
/
3
\sqrt{2/3}
2/3
≈ 0.816
3
/
2
\sqrt{3/2}
3/2
≈ 1.225
3/2 = 1.5
Check answer