Where are the particles when the box is hot?

We consider the known problem of a particle in a one-dimensional box, i.e. in an infinite potential well. Specifically, we evaluate the probability density for finding a particle in a certain position when the system is in contact with a reservoir at a temperature T not equal 0. We observe that, when the T is increased, the density tends to become uniform in the box, except in a 'boundary layer' close to the walls. Then, we consider two particles in the box, discriminating between bosons and fermions. The particles are first analyzed without accounting for the spin. Evaluating the thermalized probability density in this way, results in a clear difference between bosons and spinless fermions. Next, we consider the complete wave function for spin-1/2fermions, and the probability density changes significantly with respect to the spinless case. This shows that disregarding the electron spin for the sake of simplicity may lead to misleading results.