Sunday, November 24, 2019

quantum mechanics - Why particle number operator hatN is hatadaggerhata rather than hatahatadagger?


Both ˆaˆa 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


ˆaˆaˆn



since


ˆaˆa|0=ˆa|10.


Now, it remains to be shown that


ˆaˆa=ˆn.


Can you take it from here?


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