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arXiv:1507.02457

Revisiting " Morse potential on Quantum Computer for Molecules and Supersymmetric Quantum Mechanics" arxiv:2102.05102v1[quant-physics]

Biswanath Rath

Published Jul 9, 2015• 0 citations
arXivHTMLPDF

Abstract

We find theoretical results on energy eigenvalues and corresponding supersymmetric Hamiltonians reflect contradictory behavior for negative values of A. furthermore the resulting supersymmetric partners potentials can be model scattering states instead of bound states, However following the literature ( Flu 1979);J and A (J. Physics A 2020), we suggest a correct form of super potential , which remains valid for both positive and negative values of constants. A from this in complex space also the eigenvalues remain invariant without discussion of T-symmetry as the previous discussion appears incomplete.

DOI
10.48550/arxiv.1507.02457

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We find theoretical results on energy eigenvalues and corresponding supersymmetric Hamiltonians reflect contradictory behavior for negative values of A. furthermore the resulting supersymmetric partners potentials can be model scattering states instead of bound states, However following the literature ( Flu 1979);J and A (J. Physics A 2020), we suggest a correct form of super potential , which remains valid for both positive and negative values of constants. A from this in complex space also the eigenvalues remain invariant without discussion of T-symmetry as the previous discussion appears incomplete.
In this paper we discuss the Morse potential on a quantum computer. The Morse potential is useful to describe diatomic molecules and has a finite number of bound states which can be measured through spectroscopy. It is also a example of an exactly soluble potential using supersymmetric quantum mechanics. Using the the supersymmetric quantum mechanics formalism one can derive a heirachy of Hamiltonians such that the ground state of the next rung on the heirarchy yeids the first excited state of the hamiltonian below it. Using this method one can determine all the states of the Morse potential by calculating all the ground states of the sequence of Hamiltonians in the heirarchy. We use the IBM QISKit software together with the Variational Quantum Eiegensolver (VQE) algorithm to calculate the ground state and first excited state energy of the Morse potential and find agreement with the exact expression for the bound state energies of the Morse Potential. We analyze different optimizers to study the numerical effect on the calculations. Finally we perform quantum computations for diatomic and triatomic molecules to illustrate the application of these techniques on near term quantum computers and find excellent agreement with experimental data.
Some physical restrictions and their influences on the boundary conditions for a particle in a one-dimensional box are pointed out. We discuss the invariance of the corresponding Hamiltonian under time-reversal T ̂ and its relation with the de-generacy of the energy eigenvalues. We also discuss the effect on the boundary conditions of requiring the invariance of the Hamiltonian under parity P ̂ and si-multaneous space and time reflection P ̂ T ̂. A condition, which depends only on the boundary values of an eigenfunction of the Hamiltonian and its derivative, deter-mines if the corresponding eigenvalue satisfies the inequality E> V (x) = const, V (x) being the potential inside the box. Once these results have been presented, we choose various representative analytically solvable examples of boundary conditions (Hamiltonians) and supersymmetrize the corresponding problems. We find real and complex valued supersymmetric partner potentials among all these systems.
We consider the application of the supersymmetric quantum-mechanical formalism to the Schr\"odinger equation describing a particle characterized by a position-dependent effective mass m(x).m(x).m(x). We show that any one-dimensional quantum system with effective mass has a supersymmetric partner system characterized by the same position dependence of the mass, but with a new potential function. The form of this supersymmetric partner potential V2(x){V}_{2}(x)V2​(x) depends on both the form of the original potential V1(x){V}_{1}(x)V1​(x) and the form of the mass x dependence. We also generalize the concept of shape invariance to the nonconstant mass scenario. As illustrative examples we provide, for a given form m(x)m(x)m(x) of the effective mass, shape-invariant potentials exhibiting (a) harmonic-oscillator-like spectra and (b) Morse-like spectra. In both cases the energy eigenvalues and eigenfunctions can be obtained in algebraic fashion.
This undergraduate thesis contains one dimensional introductory treatment of Supersymmetric Quantum Mechanics (SUSY QM). The factorisation method is used to obtain Partner Hamiltonians, and the notions of partner potentials and superpotential are introduced. It demonstrated how energy spectra and eigenfunctions of partner potentials can be correlated for the cases of unbroken and broken supersymmetry. Generalisations of these cases attributed to Sets of Hamiltonians and shape invariant potentials are introduced. The devel oped formalisation is used to obtain energy eigenfunctions and spectra of partner potentials of some well-known quantum systems, namely the Infinite Square Well and the Free Particle. In these systems, the partner potentials are trigonometric and hyperbolic functions, respectively. A comparison of these two physically distinct but mathematically related partner potentials is provided. A new philosophical interpretation of SUSY QM is introduced, with an ontological discussion of supersymmetric notions and SUSY QM. The current observation problem of SUSY is discussed, and on that basis, two opposing arguments are presented. The method of Falsification by Karl Popper is briefly introduced, and an argument is produced regarding its application to SUSY.
Abstract The interest in the study of physical systems characterized by negative and effective negative masses and their behaviour has been a subject of investigation by several researchers. The focus of the majority of these studies is experimental in nature. Inspired by the same, we have solved the Schrodinger equation for complex Morse potential with negative and complex masses and obtained its exact solution. The normalized eigenfunction and eigenvalues are obtained and the condition for admissibility of the reality of eigenvalue spectrum for the ground state of complex Morse potential has been discussed. This study establishes that it is indeed feasible for such negative masses proposed in the literature for various physical systems to be bound together by complex Morse potential. Further, we propose that atoms with complex and negative masses may bind together under the action of complex Morse-like potentials and form molecular structures.