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International Journal of Advanced Research

THE ORDERING AMBIGUITY AND POSITION DEPENDENT MASS FOR NON- RELATIVISTIC QUANTUM PARTICLE WITH INTER-DIMENSIONAL (D=1)

A. Alaakol

Published Oct 27, 2016• 0 citations
Publisher

Abstract

By revisiting free particle v(x)=0 endowed with position dependent mass m(x)=1/?(1+(?x)^2)?^(2 ) has d=1 (the inter-dimensional). Starting with the kinetic energy operator, we will try to solve the equation
DOI
10.21474/ijar01/1876

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In the literature (1), the Schrodinger equation: {p1p1/2m + p2p2/2m + V(x1-x2)} W(x1,x2,t) = id/dt partial W(x1,x2,t) may be transformed using P=p1+p2 and (p=m1p2-m2p1)/M and x=x1=x2 where M=m1+m2 ((1)). This leads to a bound Schrodinger equation in x and a “center of mass” equation” in P i..e Hcm =PP/2M which depends on m1 and m2, but not binding energy. . In a previous note, we argued that a free particle’s classical action A (relativistic or nonrelativistic) may be obtained by setting v=x/t and treating x and t as independent (2). This leads to dA/dx partial=p and dA/dt partial = -E. One may write eigenvalue equations: -id/dx partial exp(iA) = pexp(iA) and id/dt partial exp(iA) = E exp(iA). A solution is A=-Et+px with E,t,p and x independent. -Et+px is a Lorentz invariant and this form is used in both relativistic and nonrelativistic quantum mechanics (although E and p have different values). We argue that a free particle rest mass mo may actually involve internal quantum physics, yet the “system” is replaced with a parameter mo and exp(-iEt+px) used to describe it. One may try to apply this approach to a bound system, but exp(-iEt+px) is a statistical probability like object. Thus one needs a statistical theory. We have suggested using an ensemble of exp(-iEt+px) with E=En for all p to indicate a resonance. Then: { Sum over p pp/2m a(p)exp(ipx) } / {Sum over p a(p)exp(ipx) } + V(x) = En ((2)) This is the Schrodinger bound equation. We argue that W=Sum over p a(p)exp(ipx) should be real and this implies that Sum over p a(p) p exp(ipx) = 0 i.e. average momentum for this type of average is 0. As a result, one may create a 4-vector using { 0, mcc + En} and use the Lorentz boost approach to find the momentum and energy in a moving frame. One may then take the nonrelativistic limit to find kinetic energy i.e. the full center of mass kinetic energy. We argue that this is equivalent to considering exp(-iEt+ipx) for the overall bound system which is now translating in space. We argue that ((2)) is a different approach to ((1)) in that binding energy, which may be significant with respect to mcc, should be part of the overall energy of the translating bound system which is called the “center of mass” system in the literature.
The problem of d-dimensional Schrödinger equations with a position-dependent mass is analyzed in the framework of first-order intertwining operators. With the pair (H,H1) of intertwined Hamiltonians one can associate another pair of secondorder partial differential operators (R,R1), related to the same intertwining operator and such that H (resp. H1) commutes with R (resp. R1). This property is interpreted in superalgebraic terms in the context of supersymmetric quantum mechanics (SUSYQM). In the two-dimensional case, a solution to the resulting system of partial differential equations is obtained and used to build a physically-relevant model depicting a particle moving in a semi-infinite layer. Such a model is solved by employing either the commutativity of H with some second-order partial differential operator L and the resulting separability of the Schrödinger equation or that of H and R together with SUSYQM and shape-invariance techniques. The relation between both approaches is also studied.
The interest in physical systems with position-dependent mass (PDM) is always topical. In this work we present a new way to construct kinetic energy operators (KEO’s) for particles endowed with a position varying mass within the Schrödinger equation. Taking recent developments of non-Hermitian quantum mechanics into account, we start with a non-Hermitian Hamiltonian and we recall that it is quasi-Hermitian and can be always associated to a Hermitian partner. The latter is then used along with the formalism of supersymmetric quantum mechanics (SUSY QM) to generate exactly solvable models on the basis of shape invariance condition. Some known and new concrete examples are considered as illustrations of the proposed scheme.