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The Quantum Mechanics of Supersymmetry

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The Quantum Mechanics of Supersymmetry The Quantum Mechanics of Supersymmetry A Senior Project presented to the Faculty of the Physics Department California Polytechnic State University, San Luis Obispo In Partial Fulfillment of the Requirements for the Degree Bachelor of Science by Joshua Gearhart May, 2012 ©2012 Joshua William Gearhart I. Introduction Most, if not all, physicists are familiar with the idea of a grand unified theory (GUT). The general idea behind a GUT is to have a single framework that succinctly describes all of the fundamental interactions of nature: gravity, electromagnetism (EM), strong nuclear, weak nuclear. The current dominating model of physics is the Standard Model (SM), which unifies the EM, strong, and weak interactions with a reasonable level of success, but has no mechanism to explain gravity. It is precisely this problem that modern GUTs seek to resolve. Unfortunately, there are about as many GUTs as there are physicists in the field. That being said, there appear to be some recurring elements in many such theories. One of the more intriguing aspects to many GUTs is the idea of supersymmetry (SUSY). The fundamental idea behind SUSY is that for every elementary particle, there exists a “superpartner” particle that differs in spin by a half unit. As a reminder, all elementary particles possess an intrinsic angular momentum characterized by their spin number. Particles with integer spin (s = 0, 1, 2, …) are called bosons and particles with half-integer spin (s = 1/2, 3/2, 5/2, …) are called fermions. Although they will not be discussed in detail here, bosons and fermions possess many unique and interesting properties due to their spins and are experimentally easy to differentiate. SUSY is particularly interesting then because it states that every boson has a fermion superpartner and vice versa. In more technical terms, one would say that SUSY relates bosonic and fermionic degrees of freedom. SUSY was first used in string theory to unify the bosonic and fermionic sectors but turned out to have many other interesting properties when other groups started to apply it to their theories. Of particular importance was the generalization of gravity to incorporate SUSY. In such theories where gravity is invariant under SUSY, i.e. there is some gravitational parameter that 2 remains constant under supersymmetric transformations, a supergravity theory results. These theories are among the leading candidates for physics beyond the SM and thus represent some of the first steps towards a GUT. In the following, the mechanisms and algebras typically associated with supersymmetry will be applied to rudimentary quantum mechanics. The level of the paper will be kept to that of a student who has taken, or will soon take, an undergraduate quantum mechanics course. Since this analysis will be kept at such a level, no direct conclusions will be drawn on the nature of supersymmetry as it applies to quantum mechanics or vice versa. Rather, it will serve as an intellectual exercise that will shed some insight into the underlying mathematical structure of supersymmetry. Some familiarity with undergraduate quantum mechanics is required so a brief summary/review is provided here. If the reader feels sufficiently comfortable in this regard, then the remainder of this introduction can be skipped without loss of continuity. One of the most basic ideas involved in both quantum mechanics and classical mechanics is the Hamiltonian. In its simplest interpretation, the Hamiltonian is the total energy of a given system, i.e., it is the sum of a system’s potential and kinetic energies. Ē Ɣ ͎ ƍ ͐ (1) In equation 1, T is the kinetic energy and V is the potential energy. In a classical system, both T and V would be calculated based on the dynamics of the system. In quantum mechanics, both T and V must be calculated using a branch of mathematics called operator theory. A proper treatment of operator theory as it applies to quantum mechanics can be found in any respectable undergraduate text (e.g. David Griffiths’ Introduction to Quantum Mechanics) and the reader is encouraged to gain a deeper understanding in such a text. Concisely speaking, operators are used 3 as a sort of query applied to a system. For example, in order to calculate the kinetic energy of the ^ system, one would apply the kinetic energy operator, notated by T (read “t-hat”). The object that is operated on in quantum mechanics is called the wave function, denoted by Ψ or ψ, and is a function of both position and time. Ψ typically represents a particle, but it is important to note that Ψ also contains information about the environment the particle exists in. For example, a particle in free space will have a different wave function than if that same particle were in a confined box. This is a difficult concept to grasp because there is no macroscopic analog to this ^ scenario. If one wanted to find the kinetic energy of a particle-environment system, T would be ^ applied to ψ. For the sake of conciseness, T is given below without explanation: ^ ^ p 2 T = . (2) 2m ^ Here, p is the momentum operator. (Note: the “squared” notation implies that the operator is ^ applied twice.) p is given in equation 3: ^ p = −ih∇. (3) Notice the similarity to classical mechanics where the kinetic energy T of an object with momentum p is given by the same equation without the hats. Naturally, one would reason that applying the momentum operator to a wave function should return its momentum. But this is only true if ψ satisfies a certain relationship with the momentum operator. That is, ψ must be a ^ ‘momentum eigenstate’. Mathematically, this just means that Ψ and p satisfy a type of equation 4 called an ‘eigenvalue equation’. These types of equations play a fundamental role in quantum mechanics and should be well understood by anyone studying this field. Details on eigenvalue equations as they pertain to quantum mechanics can be found in any respectable undergraduate ^ text and will not be discussed in this paper. The potential energy operator, V , is much more ^ straightforward. V acting on ψ is simply ψ multiplied by the potential field V( r) where r is the ^ position vector. The Hamiltonian operator, H , can now be constructed: ^ ^ ^ (4) H Ɣ T ƍ V . ^ Applying H to the wave function returns the total energy of the system, provided that ψ is an energy eigenstate. If the Hamiltonian is written out in its complete form, the result is the familiar time-independent Schr ӧdinger equation (SE) ^ h 2 Hψ = − ∇ 2ψ +Vψ = Eψ . (5) 2m Note that the first term in the middle expression is equation 2 with the momentum operator expanded out. It is important to recognize that equation 5 has 3 unknowns: the wave function ψ, the energy E, and the potential V. In ordinary algebra, one would be required to know 2 of the variables to determine the third. However in quantum mechanics, there are several boundary conditions that must be met and even forced upon the solutions to the time-independent Schr ӧdinger equation. The result is that, for many attractive potentials, the solutions become quantized. That is, given some potential V, only certain wave functions can exist within that potential. Furthermore, each wave function solution has a specific energy associated with it. The 5 upshot is that not a ll energies are allowed in every potential ; only those energies that are represented by a wave function solution to the SE given that specific V are permitted. This effectively reduced the number of unknowns in the SE to 2. Once calculated, the collection of energy solutions is known as the spectra of solutions for a particular potential. In each spectrum , there is a lowest energy solution called the ground state which is denoted by E 0. The associated wave function is called the ground state wave function a nd is denoted by ψ0. Typically, the potential is given and the solution spectra must be calculated. It is possible to reconstruct the potential given a solution spect rum and the associated wave functions , but this is a much more difficult problem and will not be discussed at length in the following analysis. Provided below is a quick example that will be relevant to later discussion. The canonical example is the one dimensional infinite square well. The potential is plotted in figure 1 below. Figure 1: Infinite Square Well Mathematically, this potenti al is given by equation 6 below: ,0 0 ≤ x ≤ L V (x) = . (6) ∞, x < 0 or x > L 6 Here, L is the length of the well. Now place a particle of mass m within the well. The SE within the well is then given by h 2 d 2 − ψ + 0 = Eψ . (7) 2m dx 2 This is a differential equation that is relatively easy to solve. The solution to this problem can be found in Griffiths or any text on differential equations. The solutions are presented below without proof: 2 πx ψ n = sin (n + )1 , 0 ≤ x ≤ L (8) L L h 2π 2 E = (n + )1 2 . (9) n 2mL 2 Note that n ≥ 0. Equations 8 and 9 represent complete solution spectra for the given infinite well. These wave functions and energies will be built upon later in the analysis. II. Analysis Consider the time-independent Schr ӧdinger equation as shown in equation 5. To simplify the analysis, assume the ground state energy is 0 and that the wave function is dependent only on the x-direction: ^ h 2 ″ H ψ (x) = − ψ (x) +V (x)ψ (x) = .0 (10) 1 0 2m 0 1 0 7 The 1 subscripts on the Hamiltonian and potential terms will make more sense later.
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