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In theoretical physics, supersymmetric quantum mechanics is an area of research where supersymmetry are applied to the simpler setting of plain quantum mechanics, rather than quantum field theory. Supersymmetric quantum mechanics has found applications outside of high-energy physics, such as providing new methods to solve quantum mechanical problems, providing useful extensions to the WKB approximation, 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 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.^[1] A similar supersymmetric approach can also be used to more accurately find the hydrogen spectrum using the Dirac equation.^[2] Oddly enough, this approach is analogous to the way Erwin Schrödinger first solved the hydrogen atom.^[3][4] 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 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 the form of a modified version of the Bohr-Sommerfeld quantization 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

${\displaystyle H^{\rm {HO}}\psi _{n}(x)={\bigg (}{\frac {-\hbar ^{2}}{2m}}{\frac {d^{2}}{dx^{2}}}+{\frac {m\omega ^{2}}{2}}x^{2}{\bigg )}\psi _{n}(x)=E_{n}^{\rm {HO}}\psi _{n}(x),}$

where ${\displaystyle \psi _{n}(x)}$ is the ${\displaystyle n}$th energy eigenstate of ${\displaystyle H^{HO}}$ with energy ${\displaystyle E_{n}^{HO}}$. We want to find an expression for ${\displaystyle E_{n}^{\rm {HO}}}$ in terms of ${\displaystyle n}$. We define the operators

${\displaystyle A={\frac {\hbar }{\sqrt {2m}}}{\frac {d}{dx}}+W(x)}$

and

${\displaystyle A^{\dagger }=-{\frac {\hbar }{\sqrt {2m}}}{\frac {d}{dx}}+W(x),}$

where ${\displaystyle W(x)}$, which we need to choose, is called the superpotential of ${\displaystyle H^{\rm {HO}}}$. We also define the aforementioned partner Hamiltonians ${\displaystyle H^{(1)}}$ and ${\displaystyle H^{(2)}}$ as

${\displaystyle H^{(1)}=A^{\dagger }A={\frac {-\hbar ^{2}}{2m}}{\frac {d^{2}}{dx^{2}}}-{\frac {\hbar }{\sqrt {2m}}}W^{\prime }(x)+W^{2}(x)}$

${\displaystyle H^{(2)}=AA^{\dagger }={\frac {-\hbar ^{2}}{2m}}{\frac {d^{2}}{dx^{2}}}+{\frac {\hbar }{\sqrt {2m}}}W^{\prime }(x)+W^{2}(x).}$

A zero energy ground state ${\displaystyle \psi _{0}^{(1)}(x)}$ of ${\displaystyle H^{(1)}}$ would satisfy the equation

${\displaystyle H^{(1)}\psi _{0}^{(1)}(x)=A^{\dagger }A\psi _{0}^{(1)}(x)=A^{\dagger }{\bigg (}{\frac {\hbar }{\sqrt {2m}}}{\frac {d}{dx}}+W(x){\bigg )}\psi _{0}^{(1)}(x)=0.}$

Assuming that we know the ground state of the harmonic oscillator ${\displaystyle \psi _{0}(x)}$, we can solve for ${\displaystyle W(x)}$ as

${\displaystyle W(x)={\frac {-\hbar }{\sqrt {2m}}}{\bigg (}{\frac {\psi _{0}^{\prime }(x)}{\psi _{0}(x)}}{\bigg )}=x{\sqrt {m\omega ^{2}/2}}}$

We then find that

${\displaystyle H^{(1)}=\frac{-<!--\; -\; -->{\mathrm{\hslash <!--\; \hslash \; -->}}^2}{}}$Zdroj:https://en.wikipedia.org?pojem=Supersymmetric_quantum_mechanics
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