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arXiv:2606.12444

Effective Geometry and Position-Dependent Mass in Dual-$q$ Quantum Mechanics

A. Boumali, A. Makhlouf

quant-phhep-thSubmitted May 27, 2026• Announced May 28, 2026• 0 citations
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Abstract

This work investigates the deformed-derivative formalism introduced by Borges, with emphasis on the relation between the linear operator $D_{(q)}$ and its nonlinear dual counterpart $D^{(q)}$. Directly inserting the dual derivative into the kinetic term leads to a nonlinear Schrödinger equation and obscures the usual interpretation of superposition and probability. We show that this nonlinearity can be removed by a simultaneous transformation of the coordinate and of the wave function. The transformed problem is an ordinary linear Schrödinger equation in a deformed coordinate, and its representation in the physical coordinate is equivalent to a Hermitian position-dependent-mass (PDM) Hamiltonian. In this formulation, the deformation parameter $q$ determines both the effective mass profile and the associated metric. The formalism is applied to the free particle, the infinite square well, the rectangular barrier, and the harmonic oscillator in the weak-deformation regime. Comparison with the nonadditive-translation approach of Costa Filho \emph{et al.} shows that the Borges dual-$q$ framework provides an alternative route to the same effective geometric structure. For $q<1$, the effective confinement length is reduced, which raises the bound-state spectrum and enhances tunneling; for $q>1$, the effective length is increased, which lowers the spectrum and suppresses tunneling relative to the undeformed limit $q=1$.

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Effective Geometry and Position-Dependent Mass in Dual-qqq Quantum Mechanics
This work investigates the deformed-derivative formalism introduced by Borges, with emphasis on the relation between the linear operator D(q)D_{(q)}D(q)​ and its nonlinear dual counterpart D(q)D^{(q)}D(q). Directly inserting the dual derivative into the kinetic term leads to a nonlinear Schrödinger equation and obscures the usual interpretation of superposition and probability. We show that this nonlinearity can be removed by a simultaneous transformation of the coordinate and of the wave function. The transformed problem is an ordinary linear Schrödinger equation in a deformed coordinate, and its representation in the physical coordinate is equivalent to a Hermitian position-dependent-mass (PDM) Hamiltonian. In this formulation, the deformation parameter qqq determines both the effective mass profile and the associated metric. The formalism is applied to the free particle, the infinite square well, the rectangular barrier, and the harmonic oscillator in the weak-deformation regime. Comparison with the nonadditive-translation approach of Costa Filho \emph{et al.} shows that the Borges dual-qqq framework provides an alternative route to the same effective geometric structure. For q<1q<1q<1, the effective confinement length is reduced, which raises the bound-state spectrum and enhances tunneling; for q>1q>1q>1, the effective length is increased, which lowers the spectrum and suppresses tunneling relative to the undeformed limit q=1q=1q=1.
This paper explores the implications of modifying the canonical Heisenberg commutation relations over two simple systems, such as the free particle and the tunnel effect generated by a step-like potential. The modified commutation relations include position-dependent and momentum-dependent terms analyzed separately. For the position deformation case, the corresponding free wave functions are sinusoidal functions with a variable wave vector k(x). In the momentum deformation case, the wave function has the usual sinusoidal behavior, but the energy spectrum becomes non-symmetric in terms of momentum. Tunneling probabilities depend on the deformation strength for both cases. Also, surprisingly, the quantum mechanical model generated by these modified commutation relations is related to the Black–Scholes model in finance. In fact, by taking a particular form of a linear position deformation, one can derive a Black–Scholes equation for the wave function when an external electromagnetic potential is acting on the particle. In this way, the Scholes model can be interpreted as a quantum-deformed model. Furthermore, by identifying the position coordinate x in quantum mechanics with the underlying asset S, which in finance satisfies stochastic dynamics, this analogy implies that the Black–Scholes equation becomes a quantum mechanical system defined over a random spatial geometry. If the spatial coordinate oscillates randomly about its mean value, the quantum particle’s mass would correspond to the inverse of the variance of this stochastic coordinate. Further, because this random geometry is nothing more than gravity at the microscopic level, the Black–Scholes equation becomes a possible simple model for understanding quantum gravity.
Recently Costa Filho et al. [Phys. Rev. A 84, 050102(R) (2011)] have introduced a position-dependent infinitesimal translation operator which corresponds to a position-dependent linear momentum and consequently to a quantum particle with position-dependent effective mass. Although there is no doubt about the novelty of the idea and the formalism, we believe that some aspects of the quantum mechanics in their original work could be enhanced. Here in this Brief Report first we address those points and then an alternative is introduced. Finally we apply the formalism for a quantum particle under a null potential confined in a square well, and the results are compared with those in the paper mentioned above.
Bohmian mechanics and Fisher information for qqq-deformed Schrödinger equation
We discuss the Bohmian mechanics by means of the deformed Schrödinger equation for position dependent mass, in the context of a qqq-algebra inspired by nonextensive statistics. A deduction of the Bohmian quantum formalism is performed by means of a deformed Fisher information functional, from which a deformed Cramér-Rao bound is derived. Lagrangian and Hamiltonian formulations, inherited by the qqq-algebra, are also developed. Then, we illustrate the results with a particle confined in an infinite square potential well. The preservation of the deformed Cramér-Rao bound for the stationary states shows the role played by the qqq-algebraic structure.
An algebraic approach is used to obtain the canonical form of the position-dependent mass Schrödinger equation from where a couple of canonical quantum variables, the q-deformed operators for the position xq, and the hermitian linear momentum operator pq are derived. In this q-deformed coordinate space, the commutator remains invariant namely [xq, pq] = iħ. By taking advantage of these q-deformed variables, one gets to a q-deformed exponential function expq(x) as well as its corresponding q-deformed logarithm function lnq(x). From these q-deformed mathematical relations and from the fact that thermodynamic properties such as the internal energy U, entropy S, free energy F, heat capacity C, and others are related to the partition function Z and ln(Z), it is proposed their generalizations in terms of the q-deformed exponential and q-deformed logarithmic functions. As a result, the structure of Legendre transformations between these statistical properties remains invariant. The usefulness of the proposal is exemplified by considering two specific position-dependent mass distributions. In the same way, other possibilities could be used to generalize the statistical properties straightforwardly.
A Nonlinear qqq-Deformed Schrödinger Equation
We construct a new nonlinear deformed Schrödinger structure using a nonlinear derivative operator which depends on a parameter qqq. This operator recovers Newton derivative when q→1q \rightarrow 1q→1. Using this operator we propose a deformed Lagrangian which gives us a deformed nonlinear Schrödinger equation with a nonlinear kinetic energy term and a standard potential V(x⃗)V(\vec{x})V(x). We analytically solve the nonlinear deformed Schrödinger equation for V(x⃗)=0V(\vec{x}) = 0V(x)=0 and q≃1q \simeq1q≃1. This model has a continuity equation, the energy is conserved, as well as the momentum and also interacts with electromagnetic field. Planck relation remains valid and in all steps we easily recover the undeformed quantities when the deformation parameter goes to 1. Finally, we numerically solve the equation of motion for the free particle in any spatial dimension, which shows a solitonic pattern when the space is equal to one for particular values of qqq.
On using the known equivalence between the presence of a position-dependent mass (PDM) in the Schrödinger equation and a deformation of the canonical commutation relations, a method based on deformed shape invariance has recently been devised for generating pairs of potential and PDM for which the Schrödinger equation is exactly solvable.This approach has provided the bound-state energy spectrum, as well as the ground-state and the first few excited-state wavefunctions.The general wavefunctions have however remained unknown in explicit form because for their determination one would need the solutions of a rather tricky differential-difference equation.Here we show that solving this equation may be avoided by combining the deformed shape invariance technique with the point canonical transformation method in a novel way.It consists in employing our previous knowledge of the PDM problem energy spectrum to construct a constant-mass Schrödinger equation with similar characteristics and in deducing the PDM wavefunctions from the known constantmass ones.Finally, the equivalence of the wavefunctions coming from both approaches is checked.