- #1
Jezuz
- 31
- 0
Hi.
Does anyone know if it is possible to start from the thermal density matrix
[tex] \hat \rho_T = \frac{e^{-\hat H_0/kT}}{\mathrm{Tr}e^{-\hat H_0/kT}} [/tex]
and from that derive that the single particle density matrix can be written as
[tex] \rho(p ; p') = \delta_{p,p'} f(\epsilon_p) [/tex]
just by using the antisymmetry of fermions?
I have tried to start with the many particle density matrix
[tex] \rho^{(N)}(p_1, p_2 ,... p_N ; p'_1, p'_2 ,... p'_N) =
\left< p_1 \wedge p_2 ... p_N \right| \hat \rho_T
\left| p'_1 \wedge ... p'_N \right> [/tex]
and traced out all but one particle momenta, but this gives me quite difficult sums that are dependent on each other.
Here [tex] \hat H_0 = \sum_{i=1}^N \hat p^2_i /2m [/tex] where [tex] N [/tex] is the number of fermions. [tex] \epsilon_p = p^2/2m [/tex] and [tex] f [/tex] is the Fermi-Dirac distribution function. The states [tex] \left| p_1 \wedge p_2 ,... p_N\right> [/tex] are totally antisymmetric states.
Does anyone know if it is possible to start from the thermal density matrix
[tex] \hat \rho_T = \frac{e^{-\hat H_0/kT}}{\mathrm{Tr}e^{-\hat H_0/kT}} [/tex]
and from that derive that the single particle density matrix can be written as
[tex] \rho(p ; p') = \delta_{p,p'} f(\epsilon_p) [/tex]
just by using the antisymmetry of fermions?
I have tried to start with the many particle density matrix
[tex] \rho^{(N)}(p_1, p_2 ,... p_N ; p'_1, p'_2 ,... p'_N) =
\left< p_1 \wedge p_2 ... p_N \right| \hat \rho_T
\left| p'_1 \wedge ... p'_N \right> [/tex]
and traced out all but one particle momenta, but this gives me quite difficult sums that are dependent on each other.
Here [tex] \hat H_0 = \sum_{i=1}^N \hat p^2_i /2m [/tex] where [tex] N [/tex] is the number of fermions. [tex] \epsilon_p = p^2/2m [/tex] and [tex] f [/tex] is the Fermi-Dirac distribution function. The states [tex] \left| p_1 \wedge p_2 ,... p_N\right> [/tex] are totally antisymmetric states.