Supersymmetric Quantum Mechanics

An Introduction

Asim Gangopadhyaya, Jeffry Mallow;Constantin Rasinariu

• Publisher: WSPC
• Year: 2017
• eBook ISBN: 9789813221062
• Edition: 2
• Topic: Physical Sciences
• Subtopic: Nuclear Physics

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We have written this book in order to provide a single compact source for undergraduate and graduate students, as well as for professional physicists who want to understand the essentials of supersymmetric quantum mechanics. It is an outgrowth of a seminar course taught to physics and mathematics juniors and seniors at Loyola University Chicago, and of our own research over a quarter of a century.

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--> Contents:

• Introduction
• Quantum Mechanics and Clues to SUSYQM
• Operator Formalism in Quantum Mechanics
• Supersymmetric Quantum Mechanics
• Shape Invariance
• Supersymmetry and Its Breaking
• Potential Algebra
• Special Functions and SUSYQM
• Isospectral Deformations
• Generating Additive Shape Invariant Potentials
• Singular Potentials in SUSYQM
• WKB and Supersymmetric WKB
• Dirac Theory and SUSYQM
• Natanzon Potentials
• The Quantum Hamilton-Jacobi Formalism and SUSYQM
• The Phase Space Quantum Mechanics Formalism and SUSYQM
• Solutions to Problems

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--> Readership: Undergraduates, graduate and academics in physics. -->
Keywords:SUSYQM;Supersymmetry;Quantum MechanicsReview:

Reviews of the First Edition:

“This work is appropriate for anyone with a solid background in upper-division undergraduate mathematics and physics … The problems, which are scattered throughout the chapters, were very well chosen.”

CHOICE

“The book under review provides, to undergraduate and graduate students of physics as well as to professional physicists, a basic theoretical background of SUSY QM. This book can be recommended reading for people interested in SUSY QM, the present book has been written keeping in mind junior and senior researchers that can understand quickly these subjects through a lot of examples, figures and problems (with solutions).”

Zentralblatt MATH
Key Features:

• It is the only text on this topic that is specifically designed for and tested on undergraduates
• It contains an accessible presentation of the ab initio method of generating all of the traditional Shape Invariant potentials
• Connects Quantum Mechanics and Supersymmetric Quantum Mechanics throughout

Chapter 1

Introduction

Obtaining exact solutions to the Schrödinger equation (and its relativistic counterpart, the Dirac equation) has always been a focus of quantum mechanical studies.

One feature of interest was the factorization method for obtaining solutions. A hamiltonian can be rewritten as a product of two factors, usually called “raising and lowering operators.” The method replaces the need to directly solve the Schrödinger equation, a second order differential equation, with the capability to solve a first order equation. Schrödinger himself noticed this, and provided a way to factorize the hamiltonian for the hydrogen atom and other potentials in 1941.¹ A decade later, Infeld and Hull generalized this to numerous other systems² (a set of systems now known as “shape invariant potentials”). All of these observations turned out to be hidden manifestations of an underlying symmetry, subsequently explained by supersymmetric quantum mechanics.

The next impetus came in the early 1980s, when elementary particle physicists attempting to find an underlying structure for the basic forces of nature, proposed the existence of “shadow” partner particles to the various known (or conjectured) elementary particles. Thus, to the photon is conjectured the photino; to the quark-binding gluon, the gluino; to the as-yet-unobserved graviton, the gravitino; and to the W particle, the wino (pronounced wee-no!). These particle partnerships and their interrelationships comprise what is called supersymmetry.

Supersymmetry with its new mathematical apparatus soon led to investigations of partnerships in other areas of physics. One was ordinary quantum mechanics itself. First employed as a so-called “toy model” of field theory,³ supersymmetric quantum mechanics, based on the notion of “partner potentials” derivable from an underlying “superpotential,” was born. It was soon found to have value in its own right, with application to the resolution of numerous questions in quantum mechanics and the posing of various interesting new ones.

In this chapter, we shall illustrate how certain unusual features of some well-known examples of the Schrödinger equation (and one less well-known) suggest underlying symmetries. In Chapter 2, we will work out the examples the conventional way, by direct solution of the Schrödinger equation. In later chapters, these examples will be revisited and solved more elegantly, to illustrate the ideas of supersymmetric quantum mechanics.

But first and foremost, let us replace the cumbersome “supersymmetric quantum mechanics” with the acronym SUSYQM, pronounced “Suzy-Cue-Em.”

The time dependent Schrödinger equation is

with H(r, t) the hamiltonian of the system.

All of the problems that we shall consider will involve time-independent one-dimensional potentials: either V(x) in Cartesian coordinates, or V(r), a radially symmetric potential in spherical coordinates. For any time-independent potential V(r), the time dependence and space dependence can be separated:

This yields

where E is a constant, since r and t are independent variables.

Problem 1.1. Obtain Eq. (1.2).

Solving the time dependent part

we obtain f(t) = e^−iEt/ħ, while the space-dependent part gives the time-independent Schrödinger equation:

From the wave-mechanical definition of linear momentum: p ≡ −iħ∇, the first term on the left is the quantum extension of the kinetic energy p²/2m, the second term is the extension of the potential energy; thus, E must be the total energy of the system. In one dimension,

Let us consider the first example: the infinite square well in one dimension, V(x) = 0, 0 < x < L infinite elswhere. The energy eigenvalues are found to be

where m is the particle’s mass and L is the width of the well.

Now let us look at another problem, less well-known: the potential

Why we chose this particular form will become evident when we compare it to the infinite well. The energy eigenvalues turn out to be

We notice something remarkable about the energy spectrum. It is virtually identical to that of the infinite well, except for the starting value of n = 0 rather than 1, the shift from n² to (n + 2)², and the constant shift −1. These distinctions are independent of the physics: we could shift the infinite well’s bottom from V = 0 to V =

; we could just as easily start the infinite well count at n = 0. Then the energy spectrum for the infinite well would be given by

Thus, for some reason these two very different potentials yield virtually identical spectra. This may be a signal that there is a hidden symmetry connecting them.

Our next example is the harmonic oscillator:

The energy turns out to be

The full solution of the Schrödinger equation for the harmonic oscillator, which we shall work out in Chapter 2, is quite cumbersome. This is surprising, given how symmetric the hamiltonian is: quadratic in both momentum

and position x. It would be nice if we could factor it into two linear terms. Unfortunately, the fact that x and

do not commute seems to thwart this program. But this factorization approach is more powerful for determining the eigenenergies than is the direct solution of the Schrödinger equation. Indeed, it will prove to be the first step toward SUSYQM. We shall work it out in Section 3.4.

We have seen that the in...

Table of contents

• Cover Page
• Title
• Copyright
• Dedication
• Preface
• Contents
• 1. Introduction
• 2. Quantum Mechanics and Clues to SUSYQM
• 3. Operator Formalism in Quantum Mechanics
• 4. Supersymmetric Quantum Mechanics
• 5. Shape Invariance
• 6. Supersymmetry and its Breaking
• 7. Potential Algebra
• 8. Special Functions and SUSYQM
• 9. Isospectral Deformations
• 10. Generating Additive Shape Invariant Potentials
• 11. Singular Potentials in SUSYQM
• 12. WKB and Supersymmetric WKB
• 13. Dirac Theory and SUSYQM
• 14. Natanzon Potentials
• 15. The Quantum Hamilton-Jacobi Formalism and SUSYQM
• 16. The Phase Space Quantum Mechanics Formalism and SUSYQM
• 17. Solutions to Problems
• Bibliography
• Index