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Progress in mathematical physics

Path Integral for the Harmonic Oscillator

Floyd L. Williams

Published Jan 1, 2003• 2 citations
Publisher

Abstract

So far we have computed path integrals for potentials V which are linear in the variable x: V(x) = cx for some constant c. Consider now a quadratic potential: V(x) = cx2. Specifically, consider V(x) =C9; 2 mx 2/2 where m is the mass of a simple harmonic oscillator with frequency v = ω/2π. As discussed in Chapter 2 the equation of motion is 1 $$x(t) = A\cos (\omega t - \delta ) = A\sin (\omega t + \Phi ),$$

DOI
10.1007/978-1-4612-0009-3_13

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So far we have computed path integrals for potentials V which are linear in the variable x: V(x) = cx for some constant c. Consider now a quadratic potential: V(x) = cx2. Specifically, consider V(x) =C9; 2 mx 2/2 where m is the mass of a simple harmonic oscillator with frequency v = ω/2π. As discussed in Chapter 2 the equation of motion is 1 x(t)=Acos⁡(ωt−δ)=Asin⁡(ωt+Φ),x(t) = A\cos (\omega t - \delta ) = A\sin (\omega t + \Phi ),x(t)=Acos(ωt−δ)=Asin(ωt+Φ),
A practical method is developed for calculating numerically one-dimensional path integrals for an arbitrary potential V(x)V(x)V(x). For the oscillator potential V(x)=kx22V(x)=\frac{k{x}^{2}}{2}V(x)=2kx2​ the well-known analytic solution is obtained. To illustrate the numerical convergence of this method the path integral for the Ginzburg-Landau potential, $V(x)=\ensuremath{-}\frac{A{x}^{2}}{2}+\frac{B{x}^{4}}{4}$, is calculated over a range of the positive constants AAA and BBB and compared with numerical solutions of the Schr\"odinger equation.
A simple harmonic oscillator, or oscillator in short, is one of the most important systems in physics as well as in quantum mechanics. The path integral and Hamiltonian of a simple harmonic oscillator are leading exemplars for the description of a wide class of physical systems. An oscillator is a theoretical model of great utility, primarily due to its simplicity that allows for the exact derivation of many results. The simple harmonic oscillator can be generalized to the case of infinitely many oscillators, and is also the starting point for analysis of many nonlinear quantum systems. Path integrals for the simple harmonic oscillator are all based on Gaussian integration, which has been discussed in Section 7.2. The harmonic oscillator is described by Gaussian path integrals , and is a bedrock of the general theory of path integration. Moreover, all perturbation expansions of nonlinear path integrals about the oscillator path integral, which includes the semi-classical expansion, are based on results of Gaussian integration. The properties of the harmonic oscillator are analyzed in the coordinate space representation, and its path integral as well as its correlation functions are analyzed. All formulas and derivations are for Euclidean time. In Sections 11.1 and 11.2 the Hamiltonian and state space of the oscillator are studied and the correlator is derived using state space methods. In Section 11.2 the infinite time oscillator is introduced and in Sections 11.3–11.5 the path integral for the oscillator is evaluated.
The exact solutions to the time-dependent Schrodinger equation for a harmonic oscillator with time-dependent mass and frequency were derived in a general form. The quantum mechanical propagator was calculated by the Feynman path integral method, while the wave function was derived from the spectral representation of the obtained propagator. It was shown that the propagator and the wave function depended on the s solution of a classical oscillator, in which the amplitude and phase satisfied the auxiliary equations. To demonstrate the derivation of the solution from our auxiliary equations, exponential and periodic functions of mass with constant frequency were imposed to evaluate the propagator and wave function for the Caldirola-Kanai and pulsating mass oscillators, respectively.