Canonical Ensemble - Quantum Mechanical Systems

Quantum Mechanical Systems

By applying the canonical partition function, one can easily obtain the corresponding results for a canonical ensemble of quantum mechanical systems. A quantum mechanical ensemble in general is described by a density matrix. Suppose the Hamiltonian H of interest is a self adjoint operator with only discrete spectrum. The energy levels are then the eigenvalues of H, corresponding to eigenvector . From the same considerations as in the classical case, the probability that a system from the ensemble will be in state is, for some constant . So the ensemble is described by the density matrix

\rho = \sum p_n | \psi _n \rangle \langle \psi_n | = \sum C e^{- \beta E_n} | \psi _n \rangle \langle \psi_n|

(Technical note: a density matrix must be trace-class, therefore we have also assumed that the sequence of energy eigenvalues diverges sufficiently fast.) A density operator is assumed to have trace 1, so

which means

Q is the quantum-mechanical version of the canonical partition function. Putting C back into the equation for ρ gives

\rho = \frac{1}{\sum e^{- \beta E_n}} \sum e^{- \beta E_n} | \psi _n \rangle \langle \psi_n| = \frac{1}{ \operatorname{Tr}( e^{- \beta H} ) } e^{- \beta H} .

By the assumption that the energy eigenvalues diverge, the Hamiltonian H is an unbounded operator, therefore we have invoked the Borel functional calculus to exponentiate the Hamiltonian H. Alternatively, in non-rigorous fashion, one can consider that to be the exponential power series.

Notice the quantity

is the quantum mechanical counterpart of the canonical partition function, being the normalization factor for the mixed state of interest.

The density operator ρ obtained above therefore describes the (mixed) state of a canonical ensemble of quantum mechanical systems. As with any density operator, if A is a physical observable, then its expected value is

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