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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