![]()
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...
