Question

In page two of the Classical Gases section of David Tong's notes on Statistical Physics, he is deriving the partition function for a classical particle from the quantum particle.

Z1 = ∫ dq (q|e-ẞp²/2me-BV(q)|q) ᏰᏙ

∫ dq e-BV (q) (q|e-ẞp²/2m|q)

I'm a bit confused about how he goes from the first line to the second. How is he able to push the exp(−βV(q^)) term through the exp(−βp^2/2m) term and the bra?

Answer 1

The derivation in David Tong's notes shows that the potential energy term e-βV(q) can be factored out of the quantum mechanical matrix element because it does not depend on momentum, leading to the classical partition function. The resulting equation is a simplification of the quantum mechanical description into a classical one by integrating over all momentum states and taking the classical limit.

Step-by-step explanation:

The equation provided by the student from David Tong's notes on Statistical Physics represents the partition function for a classical particle, which is derived from the quantum mechanical description. The process of simplification from the first line to the second line involves the fact that the potential energy V(q) depends only on position q and not on momentum p. Thus, the exponential term e-βV(q) can be factored out of the quantum mechanical matrix element involving momentum, which results in the separation of the volume integral of the potential from the momentum integral. This is possible because in quantum mechanics, position q and momentum p are represented by non-commuting operators, but for a classical particle in a potential V(q), the potential energy and kinetic energy are separable.

The integral of the momentum operator over states |q>, (q|e-βp²/2m|q), is an integral over all possible momentum states and thus represents a summation over kinetic energies of the system. In the classical limit, these integrals become continuous and directly yield the classical partition function. This procedure illustrates how quantum statistical mechanics provides a path to classical statistical mechanics by taking appropriate limits, in this case the high-temperature limit where quantum effects become negligible.

The reference to the ubiquity of the factor e-kBT in both quantum and classical statistical mechanics highlights the fundamental role temperature plays in the distribution function, whether we are dealing with classical or quantum particles. The partition function encapsulates all the thermodynamic information of a system and is a cornerstone of statistical mechanics.