Gauge Transformations for a Driven Quantum Particle in an Infinite Square Well

In this paper, we present a method for studying systems in the modified formulation of quantum mechanics known as Snyder space, proposed by Snyder (1947 Phys. Rev. 71 38–41). Snyder space predicts a modified commutation algebra for position and momentum operators. The method described in this paper introduces operators satisfying the canonical commutation relations and relates them to the position and momentum operators of Snyder space, effectively mapping a problem in Snyder space into a similar problem in standard quantum mechanics. The method is applied to the simple harmonic oscillator (SHO) in one and two dimensions as well as to the one-dimensional infinite square well. The energy spectra are calculated perturbatively for the SHO. We also find an exact spectrum for the one-dimensional infinite square well potential. These results are shown to agree with similar results found elsewhere in the literature.

Generalized uncertainty principle and self-adjoint operators

We extend the standard treatment of the asymmetric infinite square well to include solutions that have zero curvature over part of the well. This type of solution, both within the specific context of the asymmetric infinite square well and within the broader context of bound states of arbitrary piecewise-constant potential energy functions, is not often discussed as part of quantum mechanics texts at any level. We begin by outlining the general mathematical condition in one-dimensional time-independent quantum mechanics for a bound-state wavefunction to have zero curvature over an extended region of space and still be a valid wavefunction. We then briefly review the standard asymmetric infinite square well solutions, focusing on zero-curvature solutions as represented by energy eigenstates in position and momentum space.

The construction of Dirac delta type potentials has been achieved with the use of the theory of self adjoint extensions of non-self adjoint formally Hermitian (symmetric) operators. The application of this formalism to investigate the possible self adjoint extensions of the one dimensional kinematic operator $K=-d^2/dx^2$ on the infinite square well potential is quite illustrative and has been given elsewhere. This requires the definition and use of four independent real parameters, which relate the boundary values of the wave functions at the walls. By means of a different approach, that fixes matching conditions at the origin for the wave functions, it is possible to define a perturbation of the type $a\delta(x)+b\delta'(x)$, thus depending on two parameters, on the infinite square well. The objective of this paper is to investigate whether these two approaches are compatible in the sense that perturbations like $a\delta(x)+b\delta'(x)$ can be fixed and determined using the first approach.

A class of highly symmetric ,nonrelativistic, Euclidean invariant, model scalar field theories are examined assuming the existence of field and momentum operators that satisfy the canonical commutation relations (CCR). The high degree of symmetry that we assume permits explicit determination of every relevant CCR representation. These consist of a two-parameter family of unitarily inequivalent representations, some of which are irreducible while the others are reducible. It is demonstrated that only those models that are analogs of the free field can be encompassed within the irreducible representations. Hence every model with interaction—including an analog of the relativistic λφ4 theory—requires a reducible CCR representation. For the reducible representations, we determine every relevant Hamiltonian operator possessing the required high degree of symmetry. These Hamiltonians, as well as the generators of space translations, cannot be expressed (solely) as functions of the field and momentum operators, which is characteristic of any system with a unique ground state and reducible CCR representation. Nevertheless, it is demonstrated that these Hamiltonians, as well as the generators of space translations, fulfill the ``weak correspondence principle,'' in which the expectation value of a quantum generator, such as the Hamiltonian operator, in a suitably labeled overcomplete family of states is identified with the associated classical generator, such as the classical Hamiltonian. Our principal results depend on the existence and make extensive use of the countably infinite number of degrees of freedom existing in a field theory. Entirely analogous results apply to related models defined in a finite spacial volume since they still have an infinite number of degrees of freedom.

Escape rate under the influence of a high-frequency field has appeared as a theme of topical interest in different variants of the kinetic model of chemical and biological phenomena over more than two decades. The escape rate of a quantum system modulated by a rapid, timeperiodic external force is studied here in the presence of environmental dissipation with the effect of the foundational process of the intricate interplay of the temperature, field and the system parameters on the escape rate profile with the help of the modified Langevin equation obtained by systematically expanding in terms of (1/ω) via perturbative methods using the Kapitza−Landau time scale and quantum gauge transformation in the frame of the Floquet theorem. Present formalism, which is classical in appearance yet quantum in nature, provides the full time-independent activated escape rate that often plays a pivotal role in different processes and phenomena ranging from diffusion in condensed phases to biological and chemical reactions. This work, therefore, furnishes avenues of controlling such escape rates in realistic systems of arbitrary complexity and generality.

We construct families of squeezed quantum states on an interval (depending on boundary conditions, we interpret the interval as a circle or as the infinite square potential well) and obtain estimates of position and momentum dispersions for these states. A particular attention is paid to the possibility of proper localization of a particle in nanoscale space domains. One of the constructed family of squeezed states is based on the theta function. It is a generalization of the known coherent and squeezed states on the circle. Also we construct a family of squeezed states based on truncated Gaussian functions and a family of wave packets based on the discretization of an arbitrary continuous momentum probability distribution. The problem of finiteness of the energy dispersion for the squeezed states in the infinite well is discussed. Finally, we perform the limit of large interval length and the semiclassical limit. As a supplementary general result, we show that an arbitrary physical quantity has a finite dispersion if and only if the wave function of a quantum system belongs to the domain of the corresponding self-adjoint operator. This can be regarded as a physical meaning of the domain of a self-adjoint operator.

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.

UDC 517.98 Dedicated with great pleasure to Prof. Leonid P. Nizhnik on the occasion of his 90th birthday. We construct and present a detailed study of self-adjoint operators of momentum and their perturbations by nonlocal potentials on the entire axis with one limit point and on a finite interval. The domain of definition of these perturbed self-adjoint operators is described by extended homogeneous boundary conditions, which contain not only boundary values of the function but also linear functionals of functions in the entire domain of their definition. The spectral properties of exactly solvable models of this kind are investigated.

We analyze the effect of impurity on the work output and efficiency of quantum Otto and quantum Carnot heat cycles, modeled as a single quantum particle in an infinite square well potential, which is the working substance. We solve this quantum mechanical system perturbatively up to first and second order in strength of the impurity for strong- and weak-coupling regimes, respectively. We derive the analytical expressions of work and efficiency for the strong-coupling regime to the first order in the strength parameter. The threshold value of the strength parameter in weak coupling is obtained up to which the numerical result agrees with the perturbative result for a repulsive and attractive impurity. To our surprise, an embedded impurity unlocks new operational phases in the system, such as a quantum heat engine, quantum refrigerator, and quantum cold pump. In addition, the efficiency of the quantum Otto heat engine is seen to reach Carnot efficiency for some parameter regimes. The cooling power and coefficient of performance of the quantum refrigerator and quantum cold pump are nontrivially affected by the impurity.

On the ladder operators and nonclassicality of generalized coherent state associated with a particle in an infinite square well

This work develops and rigorously analyzes reduced‐order modeling (ROM) techniques for the time‐dependent Schrödinger equation (TDSE), with the goal of efficiently capturing essential quantum dynamics at significantly reduced computational cost. Three major ROM frameworks–Proper Orthogonal Decomposition (POD), Dynamic Mode Decomposition (DMD), and Reduced Basis Methods (RBM) are explored and compared–in the context of quantum wavefunction evolution. Comprehensive mathematical formulations are presented, including projection‐based Galerkin approximations, a priori and a posteriori error estimates, stability analyses, and convergence guarantees. Numerical experiments are conducted for canonical quantum systems such as the infinite square well, harmonic oscillator, and tunneling through potential barriers, as well as a time‐dependent controlled two‐level system. It is demonstrated that ROMs can achieve orders‐of‐magnitude dimensionality reduction while maintaining high fidelity with full‐order model (FOM) solutions. Furthermore, the framework is extended to higher‐dimensional problems, nonlinear potentials, and multi‐particle systems, with applications in quantum control and entanglement dynamics. Visualization of ROM accuracy through mesh, surface, and isosurface plots, as well as convergence studies, confirms the robustness of the proposed methods. To support reproducibility and further research, all MATLAB code used to generate the numerical experiments is made publicly available via GitHub. These results establish ROM as a powerful tool for real‐time simulation, control, and optimization in computational quantum mechanics.

In the course of one's studies, it may be desired to approximate a particle constrained to a curved surface, such as a sheet of graphene or a fullerene, but without modelling the lattice of Coulomb potentials. This can be done using the machinery of differential geometry to formulate a Schrödinger equation for a particle constrained so by applying an infinitely narrow infinite square well potential normal to the surface. The purpose of this paper is to clarify two mistakes that might be made when attempting to formulate this equation. These mistakes are: (1) incorrectly assuming that the Laplacian of the surface is the same as the Laplacian of the space, with the constrained term removed; and (2) neglecting to include the effective potential energy due to the curvature of the surface. It is also noted that the constrained approach produces results that, if the resulting Schrödinger equation is separable, will produce one constant of separation instead of the traditional two. The emergence of the second quantum number is demonstrated and, hence, the apparent inconsistency is resolved. The prolate spheroid is given as a non-trivial example because it highlights the issues presented.

We discuss at first in this paper the Gauge equivalence among several u‐linear Hamiltonian operators and present explicitly the associated Gauge transformation of Bäcklund type among them. We then establish the sufficient and necessary conditions for the linear superposition of the discussed u‐linear operators and matrix differential operators with constant coefficients of arbitrary order to be Hamiltonian, which interestingly shows that the resulting Hamiltonian operators survive only up to the third differential order. Finally, we explore a few illustrative examples of integrable hierarchies from Hamiltonian pairs embedded in the resulting Hamiltonian operators.

We are taught that gauge transformations in classical and quantum mechanics do not change the physics of the problem. Nevertheless, here we discuss three broad scenarios where under gauge transformations: (i) conservation laws are not preserved in the usual manner; (ii) non-gauge-invariant quantities can be associated with physical observables; and (iii) there are changes in the physical boundary conditions of the wave function that render it non-single-valued. We give worked examples that illustrate these points, in contrast to general opinions from classic texts. We also give a historical perspective on the development of Abelian gauge theory in relation to our particular points. Our aim is to provide a discussion of these issues at the graduate level.