The usual quantum mechanical cluster expansion in the theory of imperfect gases involves the trace of operators of the form $\mathrm{exp}(\ensuremath{-}\frac{{H}_{n}}{\mathrm{kT}})$, where ${H}_{n}$ is the $n$-body Hamiltonian. It is shown here that this expansion is identical with the recently derived cluster expansion in terms of the collision lifetime matrices ${\mathrm{Q}}_{n}(E)$, in which the characteristic terms involve the expressions $\mathrm{Tr}[\mathrm{exp}(\ensuremath{-}\frac{E}{\mathrm{kT}}){\mathrm{Q}}_{n}(E)]$.