Temperature Dependent Schrodinger Equation and a Particle in a Box

Recently, there have been a number of articles in the literature describing Schrodinger type equations  modified to account for temperature. In general, these are of the form of the regular Schrodinger equation with an extra potential term dependent on temperature and the wavefunction. It should be noted that temperature dependent Hartree Fock theory for a nucleus includes temperature induced changes to the potential felt by a single particle and so the single particle levels calculated are modified by temperature. Temperature dependent Hartree Fock, however, is not the same as the new Schrodinger type equations presented in the literature. In this note, we consider the case of a single particle in a box and consider dU=TdS, where U is the energy, T the temperature and S Shannon’s entropy in momentum space. We obtain a value for fp , the momentum distribution, by varying fp and use a Fourier transform to find the wavefunction. We then compare the result to predictions of equations in the literature.