- #1
KFC
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In quantum harmonic oscillator, we define the so called number operator as
[tex]\hat{N} = \hat{a}^\dagger\hat{a}[/tex]
Apply [tex]\hat{N}[/tex] to the state with n number of particles, it gives
[tex]\hat{N}|n\rangle = n |n\rangle[/tex]
so
[tex]\langle n| \hat{N}|n\rangle = \langle n| n |n\rangle = n[/tex]
But in other textbook about statistical mechanics, it gives
[tex]\langle n |\hat{N}|n\rangle = \bar{n}[/tex]
Why these two results are not the same? For later one, it seems to consider something related to the statistics, but how?
I still have another question by the trace, in harmonic oscillator, the density operator is given by
[tex]
\hat{\rho} = \sum_n |n\rangle\langle n|
[/tex]
But sometimes, for a specific state, says [tex]|\varphi\rangle[/tex], the density operator just
[tex]
\hat{\rho} = |\varphi\rangle\langle \varphi|
[/tex]
why there is no summation? When do we need to consider the summation?
[tex]\hat{N} = \hat{a}^\dagger\hat{a}[/tex]
Apply [tex]\hat{N}[/tex] to the state with n number of particles, it gives
[tex]\hat{N}|n\rangle = n |n\rangle[/tex]
so
[tex]\langle n| \hat{N}|n\rangle = \langle n| n |n\rangle = n[/tex]
But in other textbook about statistical mechanics, it gives
[tex]\langle n |\hat{N}|n\rangle = \bar{n}[/tex]
Why these two results are not the same? For later one, it seems to consider something related to the statistics, but how?
I still have another question by the trace, in harmonic oscillator, the density operator is given by
[tex]
\hat{\rho} = \sum_n |n\rangle\langle n|
[/tex]
But sometimes, for a specific state, says [tex]|\varphi\rangle[/tex], the density operator just
[tex]
\hat{\rho} = |\varphi\rangle\langle \varphi|
[/tex]
why there is no summation? When do we need to consider the summation?
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