This is a homogeneous system and the one-particle wave functions are given by plane wave functions normalized to a volume \( \Omega \) for a box with length \( L \) (the limit \( L\rightarrow \infty \) is to be taken after we have computed various expectation values) $$ \psi_{\mathbf{k}\sigma}(\mathbf{r})= \frac{1}{\sqrt{\Omega}}\exp{(i\mathbf{kr})}\xi_{\sigma} $$ where \( \mathbf{k} \) is the wave number and \( \xi_{\sigma} \) is a spin function for either spin up or down $$ \xi_{\sigma=+1/2}=\left(\begin{array}{c} 1 \\ 0 \end{array}\right) \hspace{0.5cm} \xi_{\sigma=-1/2}=\left(\begin{array}{c} 0 \\ 1 \end{array}\right). $$