Partition function of multi-particle canonical ensemble

Question

According to the orthogonality of function basis, Why can't the partition function be written directly as the following form \begin{align} \begin{split} Z & = \frac{1}{N!} \sum_{p_{1},p_{2}, \cdots ,p_{N}} \int { \rm{d}}^{3N} r \Psi_{p}(1,2, \cdots,N) e^{ - \frac{ \beta}{2m}(p_{1}^{2} + p_{2}^{2}+ \cdots + p_{N}^{2})} \Psi_{p}^{*}(1,2, \cdots,N) \\ & = \frac{1}{N!} \sum_{p_{1},p_{2}, \cdots ,p_{N}} e^{ - \frac{ \beta}{2m}(p_{1}^{2} + p_{2}^{2}+ \cdots + p_{N}^{2})} \\ \end{split} \end{align} In fact , this form is only an approximation at high temperature.If the first-order approximation has been considered, then there will be an interaction potential.

Answer 1

This is more a physics question. The reason you can't use directly the second form is that $\Psi_p$ might not be an eigenfunction of the $e^{-\frac\beta{2m}(p_1^2+p_2^2+...p_N^2)}$ operator. It is an eigenfunction if the wavefunction is completely separable, such as in the case of independent, non-interacting, particles.