  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.
