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Apple Academic Press eBooks

Supersymmetry in Quantum Mechanics

Nelson Enrique Bolivar, Gabriel Abellán

Published May 23, 2018• 8 citations
Publisher

Abstract

The problem that usually arises in the study of a quantum system is the integration of the eigenvalue equation of the Hamiltonian, namely the search for eigenstates and the corresponding eigenvalues of energy. The eigenstates, in the representation of the position, are described by eigenfunctions with the same position variable. The relations work modulo a normalization constant. The number of nodes of a function allows to determine its parity: an odd number of nodes characterizes as an antisymmetric function, while a even number of nodes is a symmetric one. The state of a system can be characterized as bosonic or fermionic depending on whether the wave function is even or odd, respectively. The two operators are nilpotent, when applied more than once to an eigenstate, it generates a null ket, regardless of the starting state.

DOI
10.1201/9781351166287-7

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The problem that usually arises in the study of a quantum system is the integration of the eigenvalue equation of the Hamiltonian, namely the search for eigenstates and the corresponding eigenvalues of energy. The eigenstates, in the representation of the position, are described by eigenfunctions with the same position variable. The relations work modulo a normalization constant. The number of nodes of a function allows to determine its parity: an odd number of nodes characterizes as an antisymmetric function, while a even number of nodes is a symmetric one. The state of a system can be characterized as bosonic or fermionic depending on whether the wave function is even or odd, respectively. The two operators are nilpotent, when applied more than once to an eigenstate, it generates a null ket, regardless of the starting state.
We have seen that to determine the possible energies En (energy eigenvalues) which a physical system can have and the corresponding eigenfunctions (wave functions) ψn, we have to set up and solve the time-independent Schro¨dinger equation Hψn = Enψn . (8.1.1) In this equation, the wave function ψn, representing the n-th eigenstate of the system in the coordinate representation, is a function of the continuously varying eigenvalues of the position observables of the system, the number of position observables being equal to number of degrees of freedom of the system. The Hamiltonian operator H, in the coordinate representation, is a differential operator involving the continuously varying eigenvalues of the position observables and the derivatives with respect to them. Since we will be dealing with the operators mostly in the coordinate representation, we shall denote them without the caret.
The postulates of quantum mechanics associate the time-evolution of a system with its time-dependent Schrödinger equation. We start by examining different solutions to this equation for a particle, represented in terms of a Gaussian wave packet, in different scenarios: free, scattered from a potential energy barrier, or trapped in a potential energy well. In some cases, we encounter stationary solutions, in which the probability density does not change in time (a standing wave). These solutions are identified as eigenfunctions of a system Hamiltonian. The properties of the Hamiltonian as a Hermitian operator are introduced, and particularly, the fact that its proper eigenfunctions can compose an orthonormal set, and that the corresponding eigenvalues are real-valued. Learning that all operators that relate to measurables are Hermitian, and that their eigenvalues relate to the measured values, we conclude that the eigenvalues of the energy operator (Hamiltonian) are the energy levels of the quantum system.
This invaluable book provides an elementary description of supersymmetric quantum mechanics which complements the traditional coverage found in the existing quantum mechanics textbooks. It gives physicists a fresh outlook and new ways of handling quantum-mechanical problems, and also leads to improved approximation techniques for dealing with potentials of interest in all branches of physics. The algebraic approach to obtaining eigenstates is elegant and important, and all physicists should become familiar with this. The book has been written in such a way that it can be easily appreciated by
As with the earlier edition, this book provides an accessible introduction to supersymmetric quantum mechanics and its applications in quantum, statistical and solid state physics. Building on the earlier edition it now encapsulates the substantial developments that have been made in supersymmetric quantum mechanics in recent years including conditionally exactly solvable systems, models of classical field theory and the many new interesting aspects within the area of stochastic dynamics, the Pauli and Dirac Equation, coherent states and non-linear algebras. Aimed at graduate students and scientists this book provides a thorough review on supersymmetric quantum mechanics and now includes problems and solutions.
We in this paper study the hermiticity of Hamiltonian and energy spectrum for the SU(1; 1) systems. The Hermitian Hamiltonian can possess imaginary eigenvalues in contrast with the common belief that hermiticity is a suffcient condition for real spectrum. The imaginary eigenvalues are derived in algebraic method with imaginary-frequency boson operators for the Hamiltonian of inverted potential well. Dual sets of mutually orthogonal eigenstates are required corresponding respectively to the complex conjugate eigenvalues. Arbitrary order eigenfunctions seen to be the polynomials of imaginary frequency are generated from the normalized ground-state wave functions, which are spatially non localized. The Hamiltonian including a non-Hermitian interaction term can be converted by similarity transformation to the Hermitian one with an effective potential of reduced slope, which is turnable by the interaction constant. The transformation operator should not be unitary but Hermitian different from the unitary transformation in ordinary quantum mechanics. The effective potential vanishes at a critical value of coupling strength called the exceptional point, where all eigenstates are degenerate with zero eigenvalue and transition from imaginary to real spectra appears. The SU(1; 1) generator S^z\widehat{S}_{z}Sz​ with real eigenvalues determined by the commutation relation of operators, however, is non-Hermitian in the realization of imaginay-frequency boson operators. The classical counterpart of the quantum Hamiltonian with non-Hermitian interaction is a complex function of the canonical variables. It becomes by the canonical transformation of variables a real function indicating exactly the one to one quantum-classical correspondence of Hamiltonians.
Dyson boson mapping of a system of 2n fermions distributed among k single-particle states uses a space generated by two-index ideal bosons. We define a unitary group U(k) to classify this ideal-boson space. The physical subspace is shown to correspond to the antisymmetric irreducible representation [12n{1}^{2n}12n]. This classification enables one to introduce a ``Majorana'' operator S which is a linear combination of one- and two-body Casimir operators of U(k). A zero eigenvalue of S characterizes the physical subspace while positive eigenvalues identify all other irreducible representations that occur. The Hermitian boson image of the Hamiltonian does not connect physical and spurious states because it is a function of the generators of U(k). All the nonphysical eigenvalues and eigenstates of a physical boson Hamiltonian (Dyson boson image of a physical fermion Hamiltonian) which could be Hermitian or non-Hermitian, can be removed without affecting the physical eigenvalues and eigenvectors, by adding a suitable multiple of S to the Hamiltonian.