Second quantization

We want to obtain an expression for a one-body operator which conserves the number of particles. Here we study the one-body operator for the kinetic energy plus an eventual external one-body potential. The action of this operator on a particular \( n \)-body state with its pertinent expectation value has already been studied in coordinate space. In coordinate space the operator reads $$ \begin{equation} \hat{H}_0 = \sum_i \hat{h}_0(x_i) \tag{31} \end{equation} $$ and the anti-symmetric \( n \)-particle Slater determinant is defined as $$ \Phi(x_1, x_2,\dots ,x_n,\alpha_1,\alpha_2,\dots, \alpha_n)= \frac{1}{\sqrt{n!}} \sum_p (-1)^p\hat{P}\psi_{\alpha_1}(x_1)\psi_{\alpha_2}(x_2) \dots \psi_{\alpha_n}(x_n). $$