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Axioms

An Asymmetric Model Position Dependent Mass: Quantum Mechanical Study

Biswanath Rath, Pravanjan Mallick, Jihad Asad, Rania T. Wannan, Rabab Jarrar, Hussein Shanak

Published Mar 23, 2023• 3 citations
Publisher

Abstract

We propose an asymmetric model mass and study its quantum mechanical behaviour on different potentials such as harmonic oscillator potential, double well potential, Gaussian single well potential and triangular single well model potential. It is observed from our study that the model asymmetric mass works well for weak coupling preserving the symmetric phase portrait. However, the dominance of asymmetric feature of the mass in the system clearly visible for higher values of the constant associated with the mass. Though, both position dependent mass and potential have significant role in controlling the spectral feature of the system, one may dominate over other for certain cases.

DOI
10.3390/axioms12040318

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We propose an asymmetric model mass and study its quantum mechanical behaviour on different potentials such as harmonic oscillator potential, double well potential, Gaussian single well potential and triangular single well model potential. It is observed from our study that the model asymmetric mass works well for weak coupling preserving the symmetric phase portrait. However, the dominance of asymmetric feature of the mass in the system clearly visible for higher values of the constant associated with the mass. Though, both position dependent mass and potential have significant role in controlling the spectral feature of the system, one may dominate over other for certain cases.
One of the most important scientific contributions of Professor Marcos Moshinsky has been his study on the harmonic oscillator in quantum theory vis à vis the standard Schrödinger equation with constant mass [1].However, a simple description of the motion of a particle interacting with an external environment such as happen in compositionally graded alloys consist of replacing the mass by the so-called effective mass that is in general variable and dependent on position.Therefore, honoring in memoriam Marcos Moshinsky, in this work we consider the position-dependent mass Schrodinger equations (PDMSE) for the harmonic oscillator potential model as former potential as well as with equi-spaced spectrum solutions, i.e. harmonic oscillator isospectral partners.To that purpose, the point canonical transformation method to convert a general second order differential equation (DE), of Sturm-Liouville type, into a Schrödinger-like standard equation is applied to the PDMSE.In that case, the former potential associated to the PDMSE and the potential involved in the Schrödinger-like standard equation are related through a Riccati-type relationship that includes the equivalent of the Witten superpotential to determine the exactly solvable positions-dependent mass distribution (PDMD) m(x).Even though the proposed approach is exemplified with the harmonic oscillator potential, the procedure is general and can be straightforwardly applied to other DEs.
We study the generalized harmonic oscillator that has both the position-dependent mass and the potential depending on the form of mass function in a more general framework. The explicit expressions of the eigenvalue and eigenfunction for such a system are given, they have the same forms as those for the usual harmonic oscillator with constant mass. The coherent state and its properties for the system with PDM are also discussed. We give the corresponding effective potentials for several mass functions, the systems with such potentials are isospectral to the usual harmonic oscillator.
The solution of one--dimensional asymmetric quantum harmonic oscillator is presented. The asymmetry can be realized, for example, by using two springs, one spring is glued with the mass, and the second spring is freely connected with the mass in the equilibrium point and it is located inside or outside the first spring which acts on the mass only from the contact point on the right. We study the spectrum of a quantum harmonic oscillator, which has a spring constant $ k_-$ to the left of the equilibrium position and a spring constant $ k_ +$ to the right of the equilibrium position. In the presented case the contact point of the second string is the equilibrium point of the first string. The explicit form of eigenfunctions, the way to calculate the eigenvalues and the properties of the eigenfunctions are discussed.
We revisit the problem of the deformed oscillator with position-dependent mass [da Costa et al., J. Math. Phys. 62, 092101 (2021)] in the classical and quantum formalisms by introducing the effect of the mass function in both kinetic and potential energies. The resulting Hamiltonian is mapped into a Morse oscillator by means of a point canonical transformation from the usual phase space (x, p) to a deformed one (xγ, Πγ). Similar to the Morse potential, the deformed oscillator presents bound trajectories in phase space corresponding to an anharmonic oscillatory motion in classical formalism and, therefore, bound states with a discrete spectrum in quantum formalism. On the other hand, open trajectories in phase space are associated with scattering states and continuous energy spectrum. Employing the factorization method, we investigate the properties of the coherent states, such as the time evolution and their uncertainties. A fast localization, classical and quantum, is reported for the coherent states due to the asymmetrical position-dependent mass. An oscillation of the time evolution of the uncertainty relationship is also observed, whose amplitude increases as the deformation increases.
We find quantum systems having finite distinct discrete energy levels can reflect infinite distinct energy levels under suitable form of position dependent mass systems. A model example has been investigated considering the fractional Harmonic Oscillator [6].In addition to this,we also notice the same behaviour in other quantum models. In all the cases, we find finite distinct discrete levels becoming infinite distinct discrete levels under position dependent mass quantum systems without the change in the potential energy.