Constraints and spectra of a deformed quantum mechanics (original) (raw)
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Symmetry, Integrability and Geometry: Methods and Applications, 2007
Two generalizations of Kempf's quadratic canonical commutation relation in one dimension are considered. The first one is the most general quadratic commutation relation. The corresponding nonzero minimal uncertainties in position and momentum are determined and the effect on the energy spectrum and eigenfunctions of the harmonic oscillator in an electric field is studied. The second extension is a function-dependent generalization of the simplest quadratic commutation relation with only a nonzero minimal uncertainty in position. Such an uncertainty now becomes dependent on the average position. With each function-dependent commutation relation we associate a family of potentials whose spectrum can be exactly determined through supersymmetric quantum mechanical and shape invariance techniques. Some representations of the generalized Heisenberg algebras are proposed in terms of conventional position and momentum operators x, p. The resulting Hamiltonians contain a contribution proportional to p 4 and their p-dependent terms may also be functions of x. The theory is illustrated by considering Pöschl-Teller and Morse potentials.
Scattering and Bound States of a Deformed Quantum Mechanics
arXiv (Cornell University), 2012
We construct the exact position representation of a deformed quantum mechanics which exhibits an intrinsic maximum momentum and use it to study problems such as a particle in a box and scattering from a step potential, among others. In particular, we show that unlike usual quantum mechanics, the present deformed case delays the formation of bound states in a finite potential well. In the process we also highlight some limitations and pitfalls of low-momentum or perturbative treatments and thus resolve two puzzles occurring in the literature.
Feinberg-Horodecki Exact Momentum States of Improved Deformed Exponential-Type Potential
Journal of Applied Mathematics and Physics, 2020
We obtain the quantized momentum eigenvalues, P n , and the momentum eigenstates for the space-like Schrodinger equation, the Feinberg-Horodecki equation, with the improved deformed exponential-type potential which is constructed by temporal counterpart of the spatial form of these potentials. We also plot the variations of the improved deformed exponentialtype potential with its momentum eigenvalues for few quantized states against the screening parameter.
Feinberg-Horodocki exact momentum states of improved deformed exponential-type potential
2020
We obtain the quantized momentum eigenvalues, P n , and the momentum eigenstates for the space-like Schrodinger equation, the Feinberg-Horodecki equation, with the improved deformed exponential-type potential which is constructed by temporal counterpart of the spatial form of these potentials. We also plot the variations of the improved deformed exponentialtype potential with its momentum eigenvalues for few quantized states against the screening parameter.
Shape invariance and the exactness of the quantum Hamilton–Jacobi formalism
Physics Letters A, 2008
Quantum Hamilton-Jacobi Theory and supersymmetric quantum mechanics (SUSYQM) are two parallel methods to determine the spectra of a quantum mechanical systems without solving the Schrödinger equation. It was recently shown that the shape invariance, which is an integrability condition in SUSYQM formalism, can be utilized to develop an iterative algorithm to determine the quantum momentum functions. In this paper, we show that shape invariance also suffices to determine the eigenvalues in Quantum Hamilton-Jacobi Theory. Quantum Hamilton-Jacobi Theory and supersymmetric quantum mechanics (SUSYQM) are two very different methods that give eigenvalues for quantum mechanical systems without solving the Schrödinger differential eigenvalue equation. Supersymmetric quantum mechanics is a generalization of Dirac's ladder operator method 4 for the harmonic oscillator. This method consists of factorizing Schrödinger's second order differential operator into two first order differential operators that play roles analogous to ladder operators. If the interaction of a quantum mechanical system is described by shape invariant potentials [1, 2], SUSYQM allows one to generate all eigenvalues and eigenfunctions through algebraic methods. Another formulation of quantum mechanics, the Quantum Hamilton-Jacobi (QHJ) formalism, was developed by Leacock and Padgett [3] and independently by Gozzi [4]. It was made popular by a series of papers by Kapoor et. al. [5]. In this formalism one works with the quantum momentum function (QMF) p(x), which is related to the wave function ψ through the relationship p(x) = −ψ′(x)/ψ(x), where prime denotes differentiation with respect to x. Our definition of QMF's differs by a factor of i ≡ √ −1 from that of ref. [3, 5, 6], where they define p(x) = −i ψ ′ (x) ψ(x) ; we use p(x) = − ψ ′ (x) ψ(x). It was shown, on a case by case basis, that the singularity structure of the function p(x) determines the eigenvalues of the Hamiltonian [3, 4, 5] for all known solvable potentials. Kapoor and his collaborators have shown that the QHJ formalism can be used not only to determine the eigenvalues of the Hamiltonian of the system, but also its eigenfunctions [7]. They have also used QHJ to analyze Quasi-exactly solvable systems where only an incomplete set of the eigenspectra can be derived analytically and also to study periodic potentials [8]. It is important to note that all cases worked out in Refs. [3, 5] satisfied the integrability condition known as the translational shape invariance for which the SUSYQM method always gave the exact result.
Deformed shape invariance and exactly solvable Hamiltonians with position-dependent effective mass
Journal of Physics A: Mathematical and General, 2005
Known shape-invariant potentials for the constant-mass Schrödinger equation are taken as effective potentials in a position-dependent effective mass (PDEM) one. The corresponding shape-invariance condition turns out to be deformed. Its solvability imposes the form of both the deformed superpotential and the PDEM. A lot of new exactly solvable potentials associated with a PDEM background are generated in this way. A novel and important condition restricting the existence of bound states whenever the PDEM vanishes at an end point of the interval is identified. In some cases, the bound-state spectrum results from a smooth deformation of that of the conventional shape-invariant potential used in the construction. In others, one observes a generation or suppression of bound states, depending on the mass-parameter values. The corresponding wavefunctions are given in terms of some deformed classical orthogonal polynomials.
The Partnership of Potentials in Quantum Mechanics and Shape Invariance
Modern Physics Letters A, 2000
The concept of partnership of potentials is studied in detail and in particular the non-uniqueness due to the ambiguity in the election of the factorization energy and in the choice of the solution of certain Riccati equation. We generate new factorizations from old ones using invariance under parameter transformations. The theory is illustrated with some examples.
2015
The major of physics applications of quantum mechanics is based on the Schrödinger equation, it was most successful in describing physics phenomena’s in 2 and 3 dimensional spaces and in a particular in the central potentials at week energy to study the atoms nuclei, molecules and their spectral behaviours, this theory will correctly described phenomena only when the velocities are small compared to the light [1-19]. The second revolution in the last century it was the standard model, in where the three fundamental forces in nature: electromagnetic, week and strong, are successful unfitted in the framework of gauge field theories. But the last forth forces, gravitation, its out of this model of unification. The hop to get a new gauge theory, in which the four forces at include in this theory, is possible, when the symmetries will be huge, which satisfied by the notion of the noncommutativity of space-time, which is extended to the canonical commutation relations between position coo...
Shape invariance and the exactness of quantum Hamilton-Jacobi formalism
2007
Quantum Hamilton-Jacobi Theory and supersymmetric quantum mechanics (SUSYQM) are two parallel methods to determine the spectra of a quantum mechanical systems without solving the Schrödinger equation. It was recently shown that the shape invariance, which is an integrability condition in SUSYQM formalism, can be utilized to develop an iterative algorithm to determine the quantum momentum functions. In this paper, we show that shape invariance also suffices to determine the eigenvalues in Quantum Hamilton-Jacobi Theory. Quantum Hamilton-Jacobi Theory and supersymmetric quantum mechanics (SUSYQM) are two very different methods that give eigenvalues for quantum mechanical systems without solving the Schrödinger differential eigenvalue equation. Supersymmetric quantum mechanics is a generalization of Dirac's ladder operator method 4 for the harmonic oscillator. This method consists of factorizing Schrödinger's second order differential operator into two first order differential operators that play roles analogous to ladder operators. If the interaction of a quantum mechanical system is described by shape invariant potentials [1, 2], SUSYQM allows one to generate all eigenvalues and eigenfunctions through algebraic methods. Another formulation of quantum mechanics, the Quantum Hamilton-Jacobi (QHJ) formalism, was developed by Leacock and Padgett [3] and independently by Gozzi [4]. It was made popular by a series of papers by Kapoor et. al. [5]. In this formalism one works with the quantum momentum function (QMF) p(x), which is related to the wave function ψ through the relationship p(x) = −ψ′(x)/ψ(x), where prime denotes differentiation with respect to x. Our definition of QMF's differs by a factor of i ≡ √ −1 from that of ref. [3, 5, 6], where they define p(x) = −i ψ ′ (x) ψ(x) ; we use p(x) = − ψ ′ (x) ψ(x). It was shown, on a case by case basis, that the singularity structure of the function p(x) determines the eigenvalues of the Hamiltonian [3, 4, 5] for all known solvable potentials. Kapoor and his collaborators have shown that the QHJ formalism can be used not only to determine the eigenvalues of the Hamiltonian of the system, but also its eigenfunctions [7]. They have also used QHJ to analyze Quasi-exactly solvable systems where only an incomplete set of the eigenspectra can be derived analytically and also to study periodic potentials [8]. It is important to note that all cases worked out in Refs. [3, 5] satisfied the integrability condition known as the translational shape invariance for which the SUSYQM method always gave the exact result.