Universality of the quantum boundary layer for a maxwellian gas

For an ideal gas confined in a rectangular domain, it has been shown that the density is not homogenous even in thermodynamic equilibrium and it goes to zero within a layer near to the boundaries due to the wave character of particles. This layer has been called the quantum boundary layer (QBL). In literature, an analytical expression for the thickness of QBL has been given for only a rectangular domain since both energy eigenvalues and eigenfunctions of the Schrödinger equation can analytically be obtained for only a rectangular domain. In this study, ideal Maxwellian gases confined in spherical and cylindrical domains are considered to investigate whether the thickness of QBL is independent of the domain shape. Although the energy eigenvalues are the roots of Bessel functions and there is no analytical expression giving the roots, the thickness of QBL is expressed analytically by considering the density distributions and using some simplifications based on the numerical calculations. It is found that QBL has the same thickness for the domains of different shapes. Therefore, QBL seems to have a universal thickness independent of the domain shape for an ideal Maxwellian gas.