Temperature as perturbation in quantum mechanics

TL;DR

This paper develops a temperature-dependent quantum mechanics framework using perturbation theory, analyzing how low temperatures influence quantum systems and revealing effects like residual tunneling probabilities.

Contribution

It introduces a generalized, self-consistent Hamiltonian incorporating temperature effects, extending quantum mechanics to low-temperature regimes with novel insights.

Findings

Corrected Hamiltonians and energy spectra for typical systems at low temperatures

Identified residual tunneling probability due to thermal coupling

Discussed thermal environment effects on wavefunction properties

Abstract

The perturbative approach was adopted to develop a temperature-dependent version of non-relativistic quantum mechanics in the limit of low-enough temperatures. A generalized, self-consistent Hamiltonian was therefore constructed for an arbitrary quantum-mechanical system in a way that the ground-state Hamiltonian turned out to be just a limiting case at absolute zero. The weak-coupling term connecting the system of interest and its immediate environment was accordingly treated as the perturbation. Applying the obtained generalized Hamiltonian to some typical quantum systems with exact zero-temperature solutions, including the free particle in a box, the free particle in vacuum, and the harmonic oscillator, up to the first order of self-consistency, therefore corrected their associated Hamiltonians, energy spectrums, and wavefunctions to be consistent with the low-temperature limit. Further investigation revealed some kind of quantum tunneling effect by a residual probability for the free particle in a box, as a chief consequence of thermally coupling to the reservoir. The possible effects of thermal environment on the main properties of the wavefunctions were also thoroughly examined and discussed.