Variational Principle for a Schrödinger Equation with Non-Hermitian Hamiltonian and Position-Dependent Mass

Abstract

A classical field theory for a Schrödinger equation with a non-Hermitian Hamiltonian describing a particle with position-dependent mass has been recently advanced by Nobre and Rego-Monteiro (NR) [Phys. Rev. A 88 (2013) 032105]. This field theory is based on a variational principle involving the wavefunction Ψ(x,t) and an auxiliary field Φ(x,t). It is here shown that the relation between the dynamics of the auxiliary field φ(x,i) and that of the original wavefunction Ψ(x,t) is deeper than suggested by the NR approach. Indeed, we formulate a variational principle for the aforementioned Schrödinger equation which is based solely on the wavefunction Ψ(x,t). A continuity equation for an appropriately defined probability density, and the concomitant preservation of the norm, follows from this variational principle via Noether´s theorem. Moreover, the norm-conservation law obtained by NR is reinterpreted as the preservation of the inner product between pairs of solutions of the variable mass Schrödinger equation.