Effective Hamiltonian for piecewise flat potentials and masses

TL;DR

This paper derives an effective Hamiltonian for systems with position-dependent mass and flat potentials, ensuring physical consistency by imposing transmission properties, and identifies the most suitable Hamiltonian form for such systems.

Contribution

The paper proposes a specific form of the Hamiltonian for position-dependent mass systems with flat potentials, resolving ordering ambiguities and improving physical accuracy.

Findings

The Hamiltonian form $H_{flat}$ effectively describes flat potential and mass systems.

Imposing transmission coefficient conditions reduces ordering ambiguity.

The derived Hamiltonian aligns with physical expectations for high-energy limits.

Abstract

We consider a class of Hermitian Hamiltonians with position-dependent mass $H=((m^alpha)p(m^beta)p(m^alpha))/2+\V$ with $2(alpha)+\beta=-1$. We apply these Hamiltonians to different piecewise flat potentials and masses (step, barrier, well and multibarrier). To raise the ordering ambiguity we impose that the transmission coefficient tends to the unity as the energy increases indefinitely. We arrive at the conclusion that the form $H_flat=(m^(-1/4)pm^(-1/2)pm^(-1/4))/2+\V$ of the effective Hamiltonian is the most adequate to describe such flat potentials and masses systems.