Quantum microcanonical ensemble

The quantum microcanonical ensemble is the quantum extension of the microcanonical ensemble. Its density matrix is

where $E$ is the energy of the system, $\Delta\ll E$ is a small energy amount to account for small fluctuations, $\ket{\phi_{n}}$ are the eigenstates of the system and $E_{n}$ are the energy eigenvalues.

Properties#

The entropy is the same as the classical case

where $k_{B}$ is the Boltzmann constant.

Energy levels#

The energy level discussion is a bit more complicated than in the classical case. Since energy levels are discrete due to being eigenstates, there is no longer an energy continuum. However, since there are so many particles, each with their own energy, the actual global energy states are incredibly numerous and only differ from one another by tiny amounts. This has two consequences. For one, the energy states are essentially a continuum even though we are dealing with quantum eigenstates. Secondly, $\Delta$ is taken to be small with respect to $E$, but large with respect to the separation of energy levels. As such, the interval $[E,E+\Delta]$ actually contains a number of energy states, from $E_{n}$ to $E_{n+m}$ for some integers $n$ and $m$.

Compared to the general case (see Quantum statistical mechanics), the $\lvert b_{n} \rvert^{2}$ coefficients are all normalized. The counting function $\Gamma(E)$ is

where $\text{Tr}$ is the trace. Notably, it does not suffer from the Gibbs paradox, unlike its classical sibling.