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Applied and numerical harmonic analysis

Quantum Mechanical Path Integrals

John R. Klauder

Published Oct 28, 2010• 0 citations
Publisher

Abstract

The fundamental equation of quantum mechanics for a single particle is generally taken to be a Schrödinger equation of the form $$i\hbar\frac{\partial}{\partial t}\psi(x, t) =\left[-\frac{{\hbar}^{2}}{2m} \frac{{\partial}^{2}}{{\partial}{x}^{2}}+{V (x, t)}\right] \psi(x, t),$$ subject to some initial condition such as $$\lim_{t \rightarrow t^\prime} \psi(x, t)= \psi (x, t^\prime).$$

DOI
10.1007/978-0-8176-4791-9_6

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The fundamental equation of quantum mechanics for a single particle is generally taken to be a Schrödinger equation of the form iℏ∂∂tψ(x,t)=[−ℏ22m∂2∂x2+V(x,t)]ψ(x,t),i\hbar\frac{\partial}{\partial t}\psi(x, t) =\left[-\frac{{\hbar}^{2}}{2m} \frac{{\partial}^{2}}{{\partial}{x}^{2}}+{V (x, t)}\right] \psi(x, t),iℏ∂t∂​ψ(x,t)=[−2mℏ2​∂x2∂2​+V(x,t)]ψ(x,t), subject to some initial condition such as lim⁡t→t′ψ(x,t)=ψ(x,t′).\lim_{t \rightarrow t^\prime} \psi(x, t)= \psi (x, t^\prime).limt→t′​ψ(x,t)=ψ(x,t′).
These lectures are intended as an introduction to the technique of path integrals and their applications in physics. The audience is mainly first-year graduate students, and it is assumed that the reader has a good foundation in quantum mechanics. No prior exposure to path integrals is assumed, however. The path integral is a formulation of quantum mechanics equivalent to the standard formulations, offering a new way of looking at the subject which is, arguably, more intuitive than the usual approaches. Applications of path integrals are as vast as those of quantum mechanics itself, including the quantum mechanics of a single particle, statistical mechanics, condensed matter physics and quantum field theory. After an introduction including a very brief historical overview of the subject, we derive a path integral expression for the propagator in quantum mechanics, including the free particle and harmonic oscillator as examples. We then discuss a variety of applications, including path integrals in multiply-connected spaces, Euclidean path integrals and statistical mechanics, perturbation theory in quantum mechanics and in quantum field theory, and instantons via path integrals. For the most part, the emphasis is on explicit calculations in the familiar setting of quantum mechanics, with some discussion (often brief and schematic) of how these ideas can be applied to more complicated situations such as field theory.
The conventional approach to quantum mechanics begins with the classical Hamiltonian and changes observables to non-commuting operators. Generally, the quantum mechanics based on the Schrodinger equation is preferred for a system of particles. But path integral formulation of quantum mechanics finds applications in more complicated situations, particularly, in quantum field theory. The purpose of solving the Schrodinger equation lies in finding the time evolution operator which is able to generate the time translation of the given system. Richard Feynman noticed that it was necessary to perform integrals over all space variables at every instant of time to find the transition amplitudes. Dissipative phenomena with a quantum origin manifest themselves in many different situations and hence the study of dissipative systems and their quantization is of a great theoretical and practical value. Many dissipative systems can be modelled as a linearly damped harmonic oscillator.
In this thesis we are investigating a different formalism of non-relativistic quantum mechanics called the path integral formalism.It is a generalization of the classical least action principle.The introduction to this subject begins with the construction of the path integral in terms of the idea of probability amplitudes whose absolute square gives the probability of finding a system in a particular state.Then we show that if the Lagrangian is a quadratic form one needs only to calculate the classical action besides from a time-dependent normalization constant to find the explicit expression of the path integral.We look in to the subject of two kinds of slit-experiments: The square slit, the single-and the double-Gaussian slit.Also, the propagator for constrained paths is calculated and applied to the Aharonov-Bohm effect, which shows that the vector potential defined in classical electrodynamics have a physical meaning in quantum mechanics.It is also shown that the path integral formulation is equivalent to the Schrödinger description of quantum mechanics, by deriving the Schrödinger equation from the path integral.Further applications of the path integral are discussed.