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In theoretical physics, supersymmetric quantum mechanics is an area of research where mathematical concepts from high-energy physics are applied to the field of quantum mechanics.

Introduction

Understanding the consequences of supersymmetry has proven mathematically daunting, and it has likewise been difficult to develop theories that could account for symmetry breaking, i.e., the lack of observed partner particles of equal mass. To make progress on these problems, physicists developed supersymmetric quantum mechanics, an application of the supersymmetry (SUSY) superalgebra to quantum mechanics as opposed to quantum field theory. It was hoped that studying SUSY's consequences in this simpler setting would lead to new understanding; remarkably, the effort created new areas of research in quantum mechanics itself.

For example, as of 2004 students are typically taught to "solve" the hydrogen atom by a laborious process which begins by inserting the Coulomb potential into the Schrödinger equation. After a considerable amount of work using many differential equations, the analysis produces a recursion relation for the Laguerre polynomials. The final outcome is the spectrum of hydrogen-atom energy states (labeled by quantum numbers n and l). Using ideas drawn from SUSY, the final result can be derived with significantly greater ease, in much the same way that operator methods are used to solve the harmonic oscillator. Oddly enough, this approach is analogous to the way Erwin Schrödinger first solved the hydrogen atom. Of course, he did not call his solution supersymmetric, as SUSY was thirty years in the future—but it is still remarkable that the SUSY approach, both older and more elegant, is taught in so few universities.

The SUSY solution of the hydrogen atom is only one example of the very general class of solutions which SUSY provides to shape-invariant potentials, a category which includes most potentials taught in introductory quantum mechanics courses.

SUSY quantum mechanics involves pairs of Hamiltonians which share a particular mathematical relationship, which are called partner Hamiltonians. (The potential energy terms which occur in the Hamiltonians are then called partner potentials.) An introductory theorem shows that for every eigenstate of one Hamiltonian, its partner Hamiltonian has a corresponding eigenstate with the same energy (except possibly for zero energy eigenstates). This fact can be exploited to deduce many properties of the eigenstate spectrum. It is analogous to the original description of SUSY, which referred to bosons and fermions. We can imagine a "bosonic Hamiltonian", whose eigenstates are the various bosons of our theory. The SUSY partner of this Hamiltonian would be "fermionic", and its eigenstates would be the theory's fermions. Each boson would have a fermionic partner of equal energy—but, in the relativistic world, energy and mass are interchangeable, so we can just as easily say that the partner particles have equal mass.

SUSY concepts have provided useful extensions to the WKB approximation. In addition, SUSY has been applied to non-quantum statistical mechanics through the Fokker-Planck equation, showing that even if the original inspiration in high-energy particle physics turns out to be a blind alley, its investigation has brought about many useful benefits.
The SUSY QM superalgebra

In fundamental quantum mechanics, we learn that an algebra of operators is defined by commutation relations among those operators. For example, the canonical operators of position and momentum have the commutator [x,p]=i. (Here, we use "natural units" where Planck's constant is set equal to 1.) A more intricate case is the algebra of angular momentum operators; these quantities are closely connected to the rotational symmetries of three-dimensional space. To generalize this concept, we define an anticommutator, which relates operators the same way as an ordinary commutator, but with the opposite sign:

$$\{A,B\} = AB + BA.$$

If operators are related by anticommutators as well as commutators, we say they are part of a Lie superalgebra. Let's say we have a quantum system described by a Hamiltonian \mathcal{H} and a set of N self-adjoint operators Qi. We shall call this system supersymmetric if the following anticommutation relation is valid for all i,j = 1,\ldots,N:

$$\{Q_i,Q^\dagger_j\} = \mathcal{H}\delta_{ij}.$$

If this is the case, then we call Qi the system's supercharges.
Example

Let's look at the example of a one-dimensional nonrelativistic particle with a 2D (i.e., two states) internal degree of freedom called "spin" (it's not really spin because "real" spin is a property of 3D particles). Let b be an operator which transforms a "spin up" particle into a "spin down" particle. Its adjoint b† then transforms a spin down particle into a spin up particle; the operators are normalized such that the anticommutator {b,b†}=1. And of course, b2=0. Let p be the momentum of the particle and x be its position with [x,p]=i. Let W (the "superpotential") be an arbitrary complex analytic function of x and define the supersymmetric operators

$$Q_1=\frac{1}{2}\left[(p-iW)b+(p+iW^\dagger)b^\dagger\right]$$
$$Q_2=\frac{i}{2}\left[(p-iW)b-(p+iW^\dagger)b^\dagger\right]$$

Note that Q1 and Q2 are self-adjoint. Let the Hamiltonian

$$H=\{Q_1,Q_1\}=\{Q_2,Q_2\}=\frac{(p+\Im\{W\})^2}{2}+\frac{{\Re\{W\}}^2}{2}+\frac{\Re\{W\}'}{2}(bb^\dagger-b^\dagger b)$$

where W' is the derivative of W. Also note that {Q1,Q2}=0. This is nothing other than N = 2 supersymmetry. Note that $$\Im\{W\}$$ acts like an electromagnetic vector potential.

Let's also call the spin down state "bosonic" and the spin up state "fermionic". This is only in analogy to quantum field theory and should not be taken literally. Then, Q1 and Q2 maps "bosonic" states into "fermionic" states and vice versa.

Let's reformulate this a bit:

Define

$$Q=(p-iW)b$$

and of course,

$$Q^\dagger=(p+iW^\dagger)b^\dagger$$

$$\{Q,Q\}=\{Q^\dagger,Q^\dagger\}=0$$

and

$$\{Q^\dagger,Q\}=2H$$

An operator is "bosonic" if it maps "bosonic" states to "bosonic" states and "fermionic" states to "fermionic" states. An operator is "fermionic" if it maps "bosonic" states to "fermionic" states and vice versa. Any operator can be expressed uniquely as the sum of a bosonic operator and a fermionic operator. Define the supercommutator [,} as follows: Between two bosonic operators or a bosonic and a fermionic operator, it is none other than the commutator but between two fermionic operators, it is an anticommutator.

Then, x and p are bosonic operators and b, b^\dagger, Q and Q^\dagger are fermionic operators.

Let's work in the Heisenberg picture where x, b and b^\dagger are functions of time.

Then,

$$[Q,x\}=-ib$$
$$[Q,b\}=0$$
$$[Q,b^\dagger\}=\frac{dx}{dt}-i\Re\{W\}$$
$$[Q^\dagger,x\}=ib^\dagger$$
$$[Q^\dagger,b\}=\frac{dx}{dt}+i\Re\{W\}$$
$$[Q^\dagger,b^\dagger\}=0$$

This is nonlinear in general: i.e., x(t), b(t) and b^\dagger(t) do not form a linear SUSY representation because \Re\{W\} isn't necessarily linear in x. To avoid this problem, define the self-adjoint operator F=\Re\{W\}. Then,

$$[Q,x\}=-ib$$
$$[Q,b\}=0$$
$$[Q,b^\dagger\}=\frac{dx}{dt}-iF$$
$$[Q,F\}=-\frac{db}{dt}$$
$$[Q^\dagger,x\}=ib^\dagger$$
$$[Q^\dagger,b\}=\frac{dx}{dt}+iF$$
$$[Q^\dagger,b^\dagger\}=0$$
$$[Q^\dagger,F\}=\frac{db^\dagger}{dt}$$

and we see that we have a linear SUSY representation.

Now let's introduce two "formal" quantities, \theta; and \bar{\theta} with the latter being the adjoint of the former such that

$$\{\theta,\theta\}=\{\bar{\theta},\bar{\theta}\}=\{\bar{\theta},\theta\}=0$$

and both of them commute with bosonic operators but anticommute with fermionic ones.

Next, we define a construct called a superfield:

$$f(t,\bar{\theta},\theta)=x(t)-i\theta b(t)-i\bar{\theta}b^\dagger(t)+\bar{\theta}\theta F(t)$$

f is self-adjoint, of course. Then,

$$[Q,f\}=\frac{\partial}{\partial\theta}f-i\bar{\theta}\frac{\partial}{\partial t}f,$$
$$[Q^\dagger,f\}=\frac{\partial}{\partial \bar{\theta}}f-i\theta \frac{\partial}{\partial t}f.$$

Incidentally, there's also a U(1)R symmetry, with p and x and W having zero R-charges and $$b^\dagger$$ having an R-charge of 1 and b having an R-charge of -1.
References

^ Valance, A.; Morgan, T. J.; Bergeron, H. (1990), "Eigensolution of the Coulomb Hamiltonian via supersymmetry", American Journal of Physics (AAPT) 58 (5): 487–491, Bibcode 1990AmJPh..58..487V, doi:10.1119/1.16452
^ E.Shrödinger, Proc.R.Irish Acad. A46,9 (1940), A46, 183(1941)