In the number occupation representation or second quantization we get the following expression for a one-body operator which conserves the number of particles $$ \begin{equation} \hat{H}_0 = \sum_{\alpha\beta} \langle \alpha|\hat{h}_0|\beta\rangle a_\alpha^{\dagger} a_\beta \tag{38} \end{equation} $$ Obviously, \( \hat{H}_0 \) can be replaced by any other one-body operator which preserved the number of particles. The stucture of the operator is therefore not limited to say the kinetic or single-particle energy only.

The opearator \( \hat{H}_0 \) takes a particle from the single-particle state \( \beta \) to the single-particle state \( \alpha \) with a probability for the transition given by the expectation value \( \langle \alpha|\hat{h}_0|\beta\rangle \).