Quantum Particle in a Box and Entropy Extremization

In (1), a quantum particle in a box with infinite potential walls is described in terms of a p (momentum) distribution involving all p’s from minus infinite to infinite. The spatial wavefunction which is roughly of the form sin(kx) involves k which is the root mean square momentum. The energy E is an average energy which satisfies Integral dp p*p/2m fp*fp in one dimension. The condition of a zero wavefunction at the walls is also enforced. Here fp is the Fourier transform of W(x), the wavefunction, and is of the form  cos(ap-pi/2 n)/ (E-p*p/2m) or sin(ap-pi/2 n)/(E-p*p/2m), where n is the energy level (1). In this note, we point out that fp may be obtained from a variational principle which uses W(a)=0 as a constraint. Next, we argue that there may somehow be a temperature in the wall which forces an average energy of Eave. Normally, one would treat such a system as a Fermi-Dirac gas or Bose-Einstein gas. Recently, however, (2), it has been suggested one may extremize Shannon’s spatial entropy to treat such a system. Here we extremize Shannon’s momentum entropy with the W(a)=0 constraint incorporated and compare with the case of (2).