Non-relativistic and relativistic electron liquids confined by a linear potential

摘要

The theory of fermion particle densities in a confining linear scalar potential energy V(x) = gx is considered in both non-relativistic and relativistic quantum mechanics. In the former case., some analytic progress proves possible for weak potentials (small g), by consideration of the canonical density matrix, having the Slater sum as its diagonal element. For larger values of g, some numerical results for the fermion density rho(x) are presented, from essentially the properties of Airy functions.The influence of a non-zero value of the Compton wavelength lambda = h/m(o)c, with m(o) the fermion rest mass, is then considered using relativistic Thomas-Fermi theory. For future work, the treatment of Hiller from the Dirac equation should prove amenable to numerical computation of the relativistic fermion density rho(lambda)(x), for comparison with the Thomas-Fermi results presented here. The possibility of relativistic and non-relativistic densities being related by a difference equation with an interval determined by lambda is briefly referred to.