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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]].
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| ==Introduction== | |
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| 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.
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| 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 (disambiguation)|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]].<ref>{{Citation
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| | year = 1990
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| | title = Eigensolution of the Coulomb Hamiltonian via supersymmetry
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| | journal = American Journal of Physics
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| | publisher = AAPT
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| | volume = 58
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| 5| pages = 487–491
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| | url = http://link.aip.org/link/?AJP/58/487/1
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| | doi = 10.1119/1.16452
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| |bibcode = 1990AmJPh..58..487V }}</ref> Oddly enough, this approach is analogous to the way [[Erwin Schrödinger]] first solved the hydrogen atom.<ref>{{Citation | last = Schrödinger
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| | first = Erwin
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| | year = 1940
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| | title = A Method of Determining Quantum-Mechanical Eigenvalues and Eigenfunctions
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| | journal = Proceedings of the Royal Irish Academy
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| | publisher = Royal Irish Academy
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| | volume = 46
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| | pages = 9–16
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| | first = Erwin
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| | year = 1941
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| | title = Further Studies on Solving Eigenvalue Problems by Factorization
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| | journal = Proceedings of the Royal Irish Academy
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| | publisher = Royal Irish Academy
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| | volume = 46
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| |bibcode = }}</ref> Of course, he did not ''call'' his solution supersymmetric, as SUSY was thirty years in the future.
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| 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. | |
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| SUSY quantum mechanics involves pairs of [[Hamiltonian (quantum mechanics)|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.
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| 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.
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| ==The SUSY QM superalgebra==
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| In fundamental quantum mechanics, we learn that an algebra of operators is defined by [[commutator|commutation]] relations among those operators. For example, the canonical operators of position and momentum have the commutator [''x'',''p'']=''i''. (Here, we use "[[natural unit]]s" 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:
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| :<math>\{A,B\} = AB + BA.</math>
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| 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 <math>\mathcal{H}</math> and a set of ''N'' self-adjoint operators ''Q<sub>i</sub>.'' We shall call this system ''supersymmetric'' if the following anticommutation relation is valid for all <math>i,j = 1,\ldots,N</math>: | |
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| :<math>\{Q_i,Q^\dagger_j\} = \mathcal{H}\delta_{ij}.</math>
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| If this is the case, then we call ''Q<sub>i</sub>'' the system's ''supercharges.''
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| == Example == | |
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| 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<sup>†</sup>'' then transforms a spin down particle into a spin up particle; the operators are normalized such that the anticommutator {''b'',''b''<sup>†</sup>}=1. And of course, ''b''<sup>2</sup>=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
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| :<math>Q_1=\frac{1}{2}\left[(p-iW)b+(p+iW^\dagger)b^\dagger\right]</math> | |
| :<math>Q_2=\frac{i}{2}\left[(p-iW)b-(p+iW^\dagger)b^\dagger\right]</math>
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| Note that ''Q<sub>1</sub>'' and ''Q<sub>2</sub>'' are self-adjoint. Let the [[Hamiltonian (quantum mechanics)|Hamiltonian]]
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| :<math>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)</math>
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| where ''W<nowiki>'</nowiki>'' is the derivative of ''W''. Also note that {''Q<sub>1</sub>,Q<sub>2</sub>''}=0. This is nothing other than ''N = 2'' supersymmetry. Note that <math>\Im\{W\}</math> acts like an electromagnetic [[vector potential]].
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| 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, ''Q<sub>1</sub>'' and ''Q<sub>2</sub>'' maps "bosonic" states into "fermionic" states and vice versa.
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| Let's reformulate this a bit:
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| Define
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| :<math>Q=(p-iW)b</math>
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| and of course,
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| :<math>Q^\dagger=(p+iW^\dagger)b^\dagger</math>
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| :<math>\{Q,Q\}=\{Q^\dagger,Q^\dagger\}=0</math> | |
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| and
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| :<math>\{Q^\dagger,Q\}=2H</math>
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| 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]] <nowiki>[,}</nowiki> 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]].
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| Then, x and p are bosonic operators and b, <math>b^\dagger</math>, Q and <math>Q^\dagger</math> are fermionic operators.
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| Let's work in the [[Heisenberg picture]] where x, b and <math>b^\dagger</math> are functions of time.
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| Then,
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| :<math>[Q,x\}=-ib</math>
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| :<math>[Q,b\}=0</math>
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| :<math>[Q,b^\dagger\}=\frac{dx}{dt}-i\Re\{W\}</math>
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| :<math>[Q^\dagger,x\}=ib^\dagger</math>
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| :<math>[Q^\dagger,b\}=\frac{dx}{dt}+i\Re\{W\}</math>
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| :<math>[Q^\dagger,b^\dagger\}=0</math>
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| This is nonlinear in general: ''i.e.,'' x(t), b(t) and <math>b^\dagger(t)</math> do not form a linear SUSY representation because <math>\Re\{W\}</math> isn't necessarily linear in ''x.'' To avoid this problem, define the self-adjoint operator <math>F=\Re\{W\}</math>. Then,
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| :<math>[Q,x\}=-ib</math>
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| :<math>[Q,b\}=0</math>
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| :<math>[Q,b^\dagger\}=\frac{dx}{dt}-iF</math>
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| :<math>[Q,F\}=-\frac{db}{dt}</math>
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| :<math>[Q^\dagger,x\}=ib^\dagger</math>
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| :<math>[Q^\dagger,b\}=\frac{dx}{dt}+iF</math>
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| :<math>[Q^\dagger,b^\dagger\}=0</math>
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| :<math>[Q^\dagger,F\}=\frac{db^\dagger}{dt}</math>
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| and we see that we have a linear SUSY representation.
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| Now let's introduce two "formal" quantities, <math>\theta</math>; and <math>\bar{\theta}</math> with the latter being the adjoint of the former such that
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| :<math>\{\theta,\theta\}=\{\bar{\theta},\bar{\theta}\}=\{\bar{\theta},\theta\}=0</math>
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| and both of them commute with bosonic operators but anticommute with fermionic ones.
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| Next, we define a construct called a [[superfield]]:
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| :<math>f(t,\bar{\theta},\theta)=x(t)-i\theta b(t)-i\bar{\theta}b^\dagger(t)+\bar{\theta}\theta F(t)</math>
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| ''f'' is self-adjoint, of course. Then,
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| :<math>[Q,f\}=\frac{\partial}{\partial\theta}f-i\bar{\theta}\frac{\partial}{\partial t}f,</math> | |
| :<math>[Q^\dagger,f\}=\frac{\partial}{\partial \bar{\theta}}f-i\theta \frac{\partial}{\partial t}f.</math>
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| Incidentally, there's also a U(1)<sub>R</sub> symmetry, with p and x and W having zero R-charges and <math>b^\dagger</math> having an R-charge of 1 and b having an R-charge of -1.
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| ==Shape Invariance==
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| Suppose W is real for all real x. Then we can simplify the expression for the Hamiltonian to
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| :<math>H = \frac{(p)^2}{2}+\frac{{W}^2}{2}+\frac{W'}{2}(bb^\dagger-b^\dagger b)</math>
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| There are certain classes of superpotentials such that both the bosonic and fermionic Hamiltonians have similar forms. Specifically
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| :<math> V_{+} (x, a_1 ) = V_{-} (x, a_2) + R(a_1)</math>
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| where the a's are parameters. For example, the hydrogen atom potential with angular momentum l can be written this way.
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| :<math> \frac{-e^2}{4\pi \epsilon_0} \frac{1}{r} + \frac{h^2 l (l+1)} {2m} \frac{1}{r^2} - E_0</math>
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| This corresponds to <math>V_{-}</math> for the superpotential
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| :<math>W = \frac{\sqrt{2m}}{h} \frac{e^2}{2 4\pi \epsilon_0 (l+1)} - \frac{h(l+1)}{r\sqrt{2m}}</math>
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| :<math>V_+ = \frac{-e^2}{4\pi \epsilon_0} \frac{1}{r} + \frac{h^2 (l+1) (l+2)} {2m} \frac{1}{r^2} + \frac{e^4 m}{32 \pi^2 h^2 \epsilon_0^2 (l+1)^2}</math>
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| This is the potential for l+1 angular momentum shifted by a constant. After solving the <math>l=0</math> ground state, the supersymmetric operators can be used to construct the rest of the bound state spectrum.
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| ==See also==
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| *[[Supersymmetry algebra]]
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| *[[Superalgebra]]
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| ==References==
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| {{reflist}}
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| [http://inspirehep.net/search?p=find+t+superymmetric+or+supersymmetry+and+quantum+mechanics References from INSPIRE-HEP]
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| {{DEFAULTSORT:Supersymmetric Quantum Mechanics}}
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| [[Category:Quantum mechanics]]
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| [[Category:Supersymmetry]]
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