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Physical Review A

Derivation of the modified Schrödinger equation for a particle with a spatially varying mass through path integrals

Kai C. Yung, Jick H. Yee

Published Jul 1, 1994• 40 citations
Publisher

Abstract

This paper describes our first-principles derivation of the Hamiltonian for the calculation of the bound states for a particle with a spatially varying mass in an arbitrary potential well. Our approach is based on the Feynman path-integral technique and results in a Hermitian form of the Schr\"odinger equation that conserves current at the boundary. We deduced the correct operator ordering for the Hamiltonian and found a correctional term that can be viewed as an effective potential.

DOI
10.1103/physreva.50.104

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This paper describes our first-principles derivation of the Hamiltonian for the calculation of the bound states for a particle with a spatially varying mass in an arbitrary potential well. Our approach is based on the Feynman path-integral technique and results in a Hermitian form of the Schr"odinger equation that conserves current at the boundary. We deduced the correct operator ordering for the Hamiltonian and found a correctional term that can be viewed as an effective potential.
With the consideration of spherical symmetry for the potential and mass function, one-dimensional solutions of nonrelativistic Schrödinger equations with spatially varying effective mass are successfully extended to arbitrary dimensions within the frame of recently developed elegant nonperturbative technique, where the BenDaniel-Duke effective Hamiltonian in one dimension is assumed like the unperturbed piece, leading to well-known solutions, whereas the modification term due to possible use of other effective Hamiltonians in one dimension and, together with the corrections coming from the treatments in higher dimensions, are considered as an additional term like the perturbation. Application of the model and its generalization for the completeness are discussed.
With the consideration of spherical symmetry for the potential and mass function, one-dimensional solutions of non-relativistic Schrodinger equations with spatially varying effective mass are successfully extended to arbitrary dimensions within the frame of recently developed elegant non-perturbative technique, where the BenDaniel-Duke effective Hamiltonian in one-dimension is assumed like the unperturbed piece, leading to well-known solutions, whereas the modification term due to possible use of other effective Hamiltonians in one-dimension and, together with, the corrections coming from the treatments in higher dimensions are considered as an additional term like the perturbation. Application of the model and its generalization for the completeness are discussed.
A path-integral formulation of quantum mechanics is investigated which is closely related to that of Feynman. It differs from Feynman's formulation in that it involves the Hamiltonian function of the canonically conjugate coordinates and momenta. The classical limit yields the variational principle: $\ensuremath{\delta}ff(p\ifmmode\ddagger\else\textdaggerdbl\fi{}\ensuremath{-}N)dt=0$. A path-integral formula is also obtained for the energy eigenstate projection operator associated with the time-independent Schr\"odinger equation. The classical limit of the projection operator formula yields a modified form of the well-known variational principle for the phase-space orbit of given energy. Relativistically covariant Hamiltonian variational principles are analyzed and lead naturally to a relativistic scalar wave equation which involves a proper time variable which is canonically conjugate to the mass in the same manner as the ordinary time variable is conjugate to the energy in nonrelativistic quantum theory.
A systematic procedure to study one-dimensional Schrodinger equation with a position-dependent effective mass (PDEM) in the kinetic energy operator is explored. The conventional free-particle problem reveals an interesting situation in that, in the presence of a mass background, formation of bound states is signalled. We also discuss coordinate-transformed, constant-mass Schrodinger equation, its matching with the PDEM form and the consequent decoupling of the ambiguity parameters. Many exact results are derived in this way.
In the context of quantum mechanics reformulated in a modified Hilbert space, we can formulate the Feynman’s path integral approach for the quantum systems with position-dependent mass particle using the formulation of position-dependent infinitesimal translation operator. Which is similar a deformed quantum mechanics based on modified commutation relations. An illustration of calculation is given in the case of the harmonic oscillator, the infinite square well and the inverse square plus Coulomb potentials.
The Schrödinger equation with a position-dependent mass (SEPDM) is employed in many areas of quantum physics. Exact solutions for the SEPDM lie at the center of interest of the professional public because it helps us to understand the behavior of quantum particles in the cases in which their mass varies spatially. For this purpose, we used the mass function represented by a quartic polynomial and a quadratic potential function, which extends the current class of exact solutions of the SEPDM. The exact analytical solution of the problem is expressed as a linear combination of local Heun functions. Heun’s equation contains many parameters, resulting in its general nature. We studied how limit changes in some of these parameters will affect the solution of the SEPDM. The obtained solutions are particularly suitable for the transfer matrix method and solutions of scattering problems; this is demonstrated by the calculation of bound states.