Quantum mechanics of nuclear spins is quite straightforward and many. This chapter gives a brief introduction to quantum mechanics. In these lecture notes we will present some general formulas concerning susy qm of first and second order for onedimensional arbitrary systems, and we will illustrate the method through the trigonometric poschlteller potentials. Rittenberg physikalisches institut, university of bonn, d5300 bonn 1, west germany received february 22, 1983 we give a general construction for supersymmetric hamiltonians in quantum mechanics. Another reference is ed wittens supersymmetry and morse theory this classical aspects of this story are really dodgy because we have fermions, so we need odd complex numbers which anticommute and things. This bachelor thesis is an introduction to supersymmetry in one dimensional quantum mechanics. Trends in supersymmetric quantum mechanics request pdf. Another reference is ed wittens supersymmetry and morse theory this classical aspects of this story are really dodgy because we have fermions, so we need odd. Supersymmetric quantum mechanical models are computed by the path integral approach. It is shown that every onedimensional quantum mechanical hamiltonian h 1 can have a partner h 2 such that h 1 and h 2 taken together may be viewed as the components of a supersymmetric hamiltonian. Nuclear physics b244 1984 409420 northholland publishing company supersymmetric quantum mechanics in one, two and three dimensions avinash khare and jnanadeva maharana institute of physics, sachivalaya marg, bhubaneswar 751005, india received 4 july 1983 revised 16 april 1984 we discuss a few supersymmetric quantum mechanical models in one, two. See also the description of max plancks trick in eisberg and resnick, sections 1.
We find that nextended supersymmetry imposes very strong constraints, and for n 4 the hamiltonian is integrable. See also the description of max plancks trick in eisberg and resnick. The infinite square well potential in one dimension has a smooth supersymmetric partner potential which is shape invariant. In the limit, the integrals localize to the zero modes. First employed as a socalled toy model of field theory,3 supersymmetric quantum mechanics, based on the notion of partner potentials derivable from an underlying superpotential, was born. Nevertheless, the wkb results are significantly improved when swkb. Supersymmetric quantum mechanics susy qm is a powerful tool for generating new potentials with known spectra departing from an initial solvable one. In class, weve learned about the supersymmetric, or factorization, method for solving some quantum mechanical potentials.
A brief introduction to the supersymmetric quantum mechanics. Understanding the consequences of supersymmetry has proven mathematically daunting, and it has likewise been difficult to develop theories that could account for. The term supersymmetric hamiltonian is taken to mean a hamiltonian defined in terms of charges that obey the same algebra as that of the generators of supersymmetry in field theory. Hence, models of supersymmetric dark matter and design of analys. Here we will concentrate on the quantum mechanical properties. Although ordinary supersymmetric quantum mechanics sqm was introduced more than 20 years ago, its extension as fractional sqm is still the object of numerous works. Introduction to quantum mechanics pdf download book. Marika taylor abstract supersymmetry is a symmetry that relates bosons to fermions. In this section we give a brief introduction to supersymmetric quantum mechanics susyqm techniques which we will use mostly as reference for the next section. Manoukian the institute for fundamental study naresuan university phitsanulok, thailand issn 186845 graduate texts in physics isbn 9783319338514 doi 10. Introduction supersymmetric quantum mechanics susyqm has been developed as an elegant analytical approach to onedimensional problems. Supersymmetry is a quantum mechanical spacetime symmetry which relates bosons and fermions. In quantum mechanics, particles have wavelike properties.
Introduction to supersymmetric quantum mechanics 2. Lessons from supersymmetric lattice quantum mechanics. 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. In this paper, the supersymmetry with two grassmann variables n 2 in classical mechanics is used to implement the dirac. Supersymmetric quantum mechanics and solvable models. The students should learn the basics of supersymmetric quantum mechanics and understand some of the connections with mathematics, such as cohomology, topological invariants of riemannian manifolds and index theory. Aug, 2015 introduction to quantum mechanicssolo table of content 2 introduction to quantum mechanics classical mechanics gravity optics electromagnetism quantum weirdness history physical laws of radiometry zeeman effect, 1896 discovery of the electron, 1897 plancks law 1900 einstein in 1905 bohr quantum model of the atom 19. Contents preface 2 1 the wave function 3 2 timeindependent schrodinger equation 14 3 formalism 62 4 quantum mechanics in three dimensions 87 5 identical particles 2 6 timeindependent perturbation theory 154 7 the variational principle 196 8 the wkb approximation 219 9 timedependent perturbation theory 236 10 the. In this master thesis we will illustrate some aspects of supersymmetric quantum mechanics in the context of path integrals. We present the general ideas on supersymmetric quantum mechanics susyqm using different representations for the operators in question. Introduction supersymmetric quantum mechanics susyqm is a generalization of the factorization method commonly used for the harmonic oscillator. Full text of flow equation for supersymmetric quantum. Is supersymmetric quantum mechanics an active research field. This thesis gives an insight into the basic formalism of.
Introduction supersymmetric quantum mechanics, which describes a supersymmetric particle moving on a compact riemannian manifold, has been studied by many authors see, for example, 27, 1, 26 and has deep connections to the geometry and topology of the target space m on which the particle moves. Manoukian quantum field theory ii introductions to quantum gravity, supersymmetry and string theory 123 edouard b. Beginning with the factorization of hamiltonian we will develop tools to solve energy spectra for many hamiltonians in a very simple way. Nuclear physics b244 1984 409420 northholland publishing company supersymmetric quantum mechanics in one, two and three dimensions avinash khare and jnanadeva maharana institute of physics, sachivalaya marg, bhubaneswar 751005, india received 4 july 1983 revised 16 april 1984 we discuss a few supersymmetric quantum mechanical models in one, two and three dimensions.
For a brief derivation in the context of quantum eld theories one may consider the referenced literature e. Supersymmetric quantum mechanics for twodimensional disk. On the experimental side, its definitely an active field. D h d \circ d h is the hamiltonian if we regard the. Teller system, fractional supersymmetric oscillator of order k, etc. To make progress on these problems, physicists developed supersymmetric quantum mechanics, an application of the supersymmetry susy.
Quantum teleportation, contrasting the classical and quantum mechanics descriptions, classical physics. Contents 1 introduction 5 2 hamiltonian formulation of supersymmetric quantum me. Supersymmetric quantum mechanics susyqm is a generalization of the. We find that the property of shape invariance is lost in this case. Quantum mechanics was developed in just two years, 1925 and 1926 see here if you want to know why. Path integral approach to supersymmetric quantum mechanics a. In this paper, we study the generalization of this to two dimensions by constructing the supersymmetric partner of the disk billiard. Supersymmetric quantum mechanics of onedimensional. An introduction to supersymmetry in quantum mechanical systems.
The term supersymmetric hamiltonian is taken to mean a hamiltonian defined in terms of charges that obey the same algebra as that of the. An introduction to supersymmetry in quantum mechanical. Solvable nparticle quantum mechanical systems are rare. Devising new approaches to construct shape invariant. In theoretical physics, supersymmetric quantum mechanics is an area of research where mathematical concepts from highenergy physics are applied to the field of quantum mechanics. We require also the existence of a unitary, selfadjoint operator y the grading operator and a selfadjoint operator called the supercharge q such that hq.
Its di cult to see how this method relates to everything else you might have heard about supersymmetry and high energy particle physics. The short introduction destinated mainly for experimentalists to the supersymmetric quantum mechanics is given. Supersymmetric quantum mechanics in this chapter we examine simple toy models of supersymmetric eld theories. The students should learn the basics of supersymmetric quantum mechanics and understand some of the connections with mathematics, such as cohomology, topological invariants of riemannian manifolds and. We give a variety of examples, for oneparticle and for manyparticle systems, in. Furthermore, when supplemented with the concept of shape invariance 5, it has also provided a powerful method of generating exactly solvable quantum mechanical models. We give a general construction for supersymmetric hamiltonians in quantum mechanics. Where a system of quantum mechanics is specified by. Annals of physics 151, 99126 1983 supersymmetric quantum mechanics m. On supersymmetric quantum mechanics archive ouverte hal.
C extended harmonic oscillator and parasupersymmetric. Supersymmetric quantum mechanics in one, two and three. Ground state structure in supersymmetric quantum mechanics. A connection between fractional supersymmetric quantum mechanics and ordinary supersymmetric quantum mechanics is established in this letter. Fractional supersymmetric quantum mechanics as a set of. Path integral formalism of quantum mechanics is rst presented as a tool to do susy calculations. Supersymmetric quantum mechanics an introduction for undergraduates bachelor project physics and astronomy 12 ec may 1st june 30th, 2009 author. A simple but often underappreciated fact is that the worldline theory of any spinning particle is supersymmetric, and hence is supersymmetric quantum mechanics, on the worldline. This is a textbook on quantum mechanics which presents the traditional treatment of the schrodinger equation as well as the dirac equation more briefly. In this case, combining equation 18 with equation 28 yields. Many searches are made on cosmic rays to find dark matter and supersymmetry predicts particles which are good candidates. Introduction to quantum mechanicssolo table of content 2 introduction to quantum mechanics classical mechanics gravity optics electromagnetism quantum weirdness history physical laws of radiometry zeeman effect, 1896 discovery of the electron, 1897 plancks law 1900 einstein in 1905 bohr quantum model of the atom 19.
Supersymmetric quantum mechanics, coherent states, periodic potentials pacs. A connection between fractional supersymmetric quantum mechanics and ordinary supersymmetric quantum mechanics is brie. Finally, we discuss structures more general than supersymmetric quantum mechanics such as parasupersymmetric quantum mechanics in which there is a symmetry between a boson and a parafermion of order p. A brief introduction to the supersymmetric quantum. Fortunately for nmr spectroscopists, the quantum mechanics of nuclear spins is quite straightforward and many useful calculations can be done by hand, quite literally on the back of an envelope. Quantum mechanics is an incomplete theory, for even knowing the wave function, still one cannot determine all the properties of the physical system. These are quantum mechanical systems possessing supersymmetry 1. It is the simplest method to build hamiltonians with prescribed spectra in the. It is designed to be accessible and understandable to someone who has a solid background in quantum mechanics, but no prior knowledge of supersymmetry. Combining the bosonic and fermionic systems together, in total we have. A brief introduction to supersymmetric quantum mechanics november 28, 2007 in class, weve learned about the supersymmetric, or factorization, method for solving some quantum mechanical potentials.
Supersymmetric quantum mechanics of onedimensional systems. Firstly, we present the method for a general potential in sec. It is as well the purpose of this work to achieve enough formality to make the reader easily understand the mathematical framework involved. This thesis gives an insight into the basic formalism of supersymmetric quantum me. Is supersymmetric quantum mechanics an active research. The factorization technique begun by darboux 1 about one. Pdf pt symmetrized supersymmetric quantum mechanics. Lessons from supersymmetric lattice quantum mechanics erich poppitz joel giedt, e. A brief introduction to supersymmetric quantum mechanics.
The resultingsupersymmetric quantum mechanics susy qm revived the study of exactly solvable hamiltonians see e. The case of ordinary supersymmetric quantum mechanics corresponds to k 2. Beginning with the factoriza tion of hamiltonian we will develop tools to solve energy spectra for many hamiltonians in a very simple way. Here we will concentrate on the quantum mechanical properties of supersymmetric quantum systems. A ridiculously short introduction to some very basic.
Supersymmetric quantum mechanics and new potentials. Along the years, supersymmetric quantum mechanics susy qm has been used for studying solvable quantum potentials. At the end we will use all the di erent aspects we looked at to solve the radial equation of the hydrogen atom. Introduction during the past few decades, the two new developments in quantum mechanics, namely pt symmetric quantum mechanics pt qm and supersymmetric quantum mechanics susy qm, have been the subjects of intense theoretical and experimental researches 1,2.
Abstract this bachelor thesis is an introduction to supersymmetry in one dimensional quantum mechanics. This allows us to perform the index computations exactly because of supersymmetric localization, and we will show how the geometry of target space enters the physics of sigma models resulting in the relationship between the supersymmetric model. Electromagnetism and relativity, the origin of quantum physics, hamiltons variational principle and the hamilton jacobi eq, wavepackets, quantum behavior of particles, the amplitude probability, the. Supersymmetric quantum mechanics is defined here by a quadruple 2, h, y, q. The purpose of this paper is to provide an introduction to the topic of supersymmetry. Here there are no technical di culties hiding the essential structures. Ge introduction the techniques based on the factorization method, which try to identify the class of hamiltonians whose spectral problem can be algebraically solved, have attracted for years peoples attention. Supersymmetric quantum mechanics and new potentials here, we use the superalgebra to construct new potentials from exactly sov able potentials. On the other hand, supersymmetric quantum mechanics ssqm has established a nice symmetry between bosons and fermions 4. A supersymmetric generalization of a known solvable quantum mechanical model of n particles.
The data of a system of supersymmetric quantum mechanics may also be formalized in terms of a spectral triple. Quantum mechanics introduction pdf quantum mechanics is the basic tool needed to describe, understand and devise nmr. Feynmans approach to quantum mechanics starts by breaking the time evolution. Therefore, there is some other information, external to quantum mechanics, which together with the wave.
For example, one easily finds by combining previous results that. It was soon found to have value in its own right, with. The introduction by witten 12 of a topological invariant to study dy namical susy. The pair x, h defines a quantum mechanics with a selfadjoint hamiltonian h acting on a hilbert space 2.
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