Introduction to Quantum Statistical Mechanics

The objective of the following discussion is to calculate the partition function Z of a single molecule that populates several states with a distribution that is appropriate for a canonical ensemble. In the language of quantum mechanics  

Each state of a time-varying system is described by a complex time-dependent wavefunction (r, time), where r represents a set of generalized spatial coordinates. 

The time-dependent expectation value of a thermodynamic observable (i.e., property) is illustrated in terms of the Hamiltonian operator. The result provides an estimate of the total energy of the system, whose classical thermodynamic analog is the internal energy  

Equilibrium thermodynamic properties, such as internal energy, are calcn-lated from the time-dependent expectation value of the total system energy as follows  

Analogous to the construction of molecular orbital wavefunctions based on linear combinations of atomic orbitals, t/f for a single molecule is expressed in terms of a set of time-independent orthonormal basis functions t , (r)  

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