In
theoretical physics
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...
, supersymmetric quantum mechanics is an area of research where
supersymmetry are applied to the simpler setting of plain
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistr ...
, rather than
quantum field theory. Supersymmetric quantum mechanics has found applications outside of
high-energy physics
Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) a ...
, such as providing new methods to solve quantum mechanical problems, providing useful extensions to the
WKB approximation
In mathematical physics, the WKB approximation or WKB method is a method for finding approximate solutions to linear differential equations with spatially varying coefficients. It is typically used for a semiclassical calculation in quantum mecha ...
, and
statistical mechanics.
Introduction
Understanding the consequences of supersymmetry (SUSY) 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 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, students are typically taught to "solve" the
hydrogen
Hydrogen is the chemical element with the symbol H and atomic number 1. Hydrogen is the lightest element. At standard conditions hydrogen is a gas of diatomic molecules having the formula . It is colorless, odorless, tasteless, non-toxic ...
atom by a laborious process which begins by inserting the
Coulomb potential into the
Schrödinger equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...
. After a considerable amount of work using many differential equations, the analysis produces a recursion relation for the
Laguerre polynomials In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834–1886), are solutions of Laguerre's equation:
xy'' + (1 - x)y' + ny = 0
which is a second-order linear differential equation. This equation has nonsingular solutions on ...
. The final outcome is the
spectrum
A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors ...
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. A similar supersymmetric approach can also be used to more accurately find the hydrogen spectrum using the Dirac equation. Oddly enough, this approach is analogous to the way
Erwin Schrödinger
Erwin Rudolf Josef Alexander Schrödinger (, ; ; 12 August 1887 – 4 January 1961), sometimes written as or , was a Nobel Prize-winning Austrian physicist with Irish citizenship who developed a number of fundamental results in quantum theo ...
first solved the hydrogen atom. Of course, he did not ''call'' his solution supersymmetric, as SUSY was thirty years in the future.
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
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
s 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 the form of a modified version of the
Bohr-Sommerfeld quantization
The old quantum theory is a collection of results from the years 1900–1925 which predate modern quantum mechanics. The theory was never complete or self-consistent, but was rather a set of heuristic corrections to classical mechanics. The theory ...
condition. 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.
Example: the harmonic oscillator
The Schrödinger equation for the harmonic oscillator takes the form
:
where
is the
th energy eigenstate of
with energy
. We want to find an expression for
in terms of
. We define the operators
:
and
:
where
, which we need to choose, is called the superpotential of
. We also define the aforementioned partner Hamiltonians
and
as
:
:
A zero energy ground state
of
would satisfy the equation
:
Assuming that we know the ground state of the harmonic oscillator
, we can solve for
as
:
We then find that
:
:
We can now see that
:
This is a special case of shape invariance, discussed below. Taking without proof the introductory theorem mentioned above, it is apparent that the spectrum of
will start with
and continue upwards in steps of
The spectra of
and
will have the same even spacing, but will be shifted up by amounts
and
, respectively. It follows that the spectrum of
is therefore the familiar
.
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
. (Here, we use "
natural unit
In physics, natural units are physical units of measurement in which only universal physical constants are used as defining constants, such that each of these constants acts as a coherent unit of a quantity. For example, the elementary charge ma ...
s" where
Planck's constant is set equal to 1.) A more intricate case is the algebra of
angular momentum
In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
operators; these quantities are closely connected to the rotational symmetries of three-dimensional space. To generalize this concept, we define an ''
anticommutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.
Group theory
The commutator of two elements, ...
,'' which relates operators the same way as an ordinary
commutator, but with the opposite sign:
:
If operators are related by anticommutators as well as commutators, we say they are part of a ''
Lie superalgebra
In mathematics, a Lie superalgebra is a generalisation of a Lie algebra to include a Z2 grading. Lie superalgebras are important in theoretical physics where they are used to describe the mathematics of supersymmetry. In most of these theories, the ...
.'' Let's say we have a quantum system described by a Hamiltonian
and a set of
operators
''.'' We shall call this system ''supersymmetric'' if the following anticommutation relation is valid for all
:
:
If this is the case, then we call ''
'' 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
be an operator which transforms a "spin up" particle into a "spin down" particle. Its adjoint
then transforms a spin down particle into a spin up particle; the operators are normalized such that the anticommutator
. And of course,
. Let
be the momentum of the particle and
be its position with
. Let
(the "
superpotential
In theoretical physics, the superpotential is a function in supersymmetric quantum mechanics. Given a superpotential, two "partner potentials" are derived that can each serve as a potential in the Schrödinger equation. The partner potentials hav ...
") be an arbitrary complex analytic function of
and define the supersymmetric operators
: