Both ˆa†ˆa and ˆaˆa† are Hermitian, how do we know which one represents the particle number?
Answer
We require that the number operator have the following property:
ˆn|0⟩=0.
We know that
ˆa|0⟩=0
and we know that
ˆa|1⟩=|0⟩
and we know that
ˆa†|0⟩=|1⟩.
Thus, it follows that
ˆaˆa†≠ˆn
since
ˆaˆa†|0⟩=ˆa|1⟩≠0.
Now, it remains to be shown that
ˆa†ˆa=ˆn.
Can you take it from here?
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