So far we have computed path integrals for potentials v which are linear in the variable x. Pdf continued fractions and the harmonic oscillator. Harmonic oscillator, coherent states, and feynman path. A comment on path integral for the quantum harmonic oscillator. This is the procedure illustrated by feynman in his book quantum mechanics and path integrals. The fermionic harmonic oscillator the prototype for a fermion. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or functional integral, over an infinity of quantum mechanically possible trajectories to compute a quantum amplitude. The path integral approach to quantum mechanics lecture. Path integral representation of fractional harmonic oscillator. The study of this system using path integrals can serve to introduce a formulation of quantum. We have encountered the harmonic oscillator already in sect.
Note that for the particular case at hand, the simple harmonic oscillator, the leading order quadratic term is in fact the whole story, and so gives the exact result. An harmonic oscillator is a particle subject to a restoring force that is proportional to the displacement of the particle. Among the examples where path integrals have been carried out exactly, the generalized harmonic oscillator ghoisaspecial one, asfeynman andhibbs have pointed out2. Pdf path integral harmonic therefore, we expect that the main contribution to the path integral comes from paths. Browse other questions tagged quantummechanics homeworkandexercises harmonicoscillator pathintegral wickrotation or ask your own question. It will be shown how to compute the density matrix for the harmonic oscillator. The motivation for writing these notes was that i found the typical treatment of quantum mechanics path integrals in a quantum eld theory text to be too brief to be digestible there are some exceptions, while monographs on path integrals are usually far too detailed to allow one to get anywhere in a reasonable amount of time.
The quantum statistical mechanical propagator for a harmonic oscillator with a timedependent force constant, m. We then discuss a variety of applications, including path integrals in multiplyconnected spaces, euclidean path integrals and statistical mechanics, perturbation theory in quantum mechanics and in. Questions tagged pathintegral physics stack exchange. In section 2 of this paper we show that the quantum statistical mechanical path integral density matrix for a harmonic oscillator with a timedependent force constant. A simple evaluation of a determinant in a path integral. Since their introduction,1 feynman path integrals have be come a powerful method of calculation for quantum me chanical problems. They also clarify various aspects of quantum mechanics, such as the.
The harmonic oscillator to get acquainted with path integrals we consider the harmonic oscillator for which the path integral can be calculated in closed form. This is generally true not for all kinds of quantum systems and not just for the harmonic oscillator. In this contribution, the quantum mechanics of the gho will be discussed based on the authors works in recent years 3. It is hoped that our results may provide insights for the generalization of path integral approach to fractional derivative quantum fields. Lecture 10 notes pdf path integral formulation of quantum mechanics, the propagator, path integrals. The exact propagator for a harmonic oscillator with timedependent mass and frequency is found by the schwinger method and a path integral with a generalized canonical transformation. Harmonic oscillator in 2d and 3d, and in polar and spherical coordinates. The calculation of a onedimensional quantum harmonic oscillator propagator using the path integral has been reconsidered for more simplicity and more pedagogical signi. Noncommutative harmonic oscillator at finite temperature. Is it possible to evaluate path integral for harmonic oscillator directly by evaluating the integral for every time slice up to the last fixed time slice. Harmonic oscillator excited states numerical solution.
Path integrals as an example of how this can be used consider a particle which at time t 0 has the form. Quantum mechanical path integral in phase space and class of harmonic oscillators with varied frequencies chapter pdf available in journal of modern physics 0704 june 2016 with 51 reads. We focus on the evaluation of a determinant resulting from the action integral in the discretized form of the path integral. My own point of view is that the path integral approach to quantum theories is simultaneously more intuitive, more fundamental, and more exible than the standard operator state description, but i do not intend to get into an argument about this.
Lattice monte carlo study of the harmonic oscillator in the. We apply the method to the free particle and quantum harmonic oscillator, investigate the euclidean path integral, and discuss other applications. Derivation of the harmonic oscillator propagator using the. Coherent state path integral for the harmonic oscillator and. The full propagator, including both the prefactor and the classical. Introduction a bit of theory numerical approach to evaluate path integrals results summary lattice monte carlo study of the harmonic oscillator in the path integral formulation desy summer student programme zeuthen 2012 aleksandra s lapik university of silesia, poland willian m. For bosons occupying only a finite number of quantum states. We present the path integral formulation of quantum mechanics and demonstrate its equivalence to the schr. The harmonic oscillator coherent state path integral is evaluated exactly at the discrete level, and its relation with various regularizations is. Feb 11, 2015 to describe the quantum evolution of several identical and thus indiscernible quantum particles, the path integral formulation has to be generalized. Thus, we can use eulerlagrangian equation to solve this problem. The motivation for writing these notes was that i found the typical treatment of quantum mechanics path integrals in a quantum field theory text.
Monte carlo simulations of harmonic and anharmonic. We present a purely analytical method to calculate the propagator for the quantum harmonic oscillator using feynmans path integral. Lattice monte carlo study of the harmonic oscillator in. On the other hand, it is well known that the dynamics of a free, relativistic, spinless massive particle is best described by a scalar field which is equivalent to an infinite number of harmonic oscillators. The method of path integrals pis was developed by richard feynman. One is that the application might be statistical mechanics.
Questions tagged path integral ask question path integral formulation due to feynman is a major formulation of quantum mechanics along with matrix mechanics due to heisenberg and pauli, wave mechanics due to schrodinger, and variational mechanics due to dirac. This is generally true for all kinds of quantum systems and not just for the harmonic oscillator. With this we may solve any dynamics and thermodynamics for t i problem in harmonic oscillator. The path integral approach to quantum mechanics lecture notes for quantum mechanics iv. Quantum mechanical path integral in phase space and class of. The quantum harmonic oscillator is the quantum mechanical analog of the classical harmonic oscillator. Lecture 9 notes pdf spin12 in an ac field, resonant drive, offresonant drive, path integral formulation of quantum mechanics. As for harmonic oscillator, its lagrangian is l 1 2 mv 2 21 2 mw x2. For example, for the euclidean harmonic oscillator with evolutionkernel 6. Coherent state path integral quantization of quantum field theory 8. Jahan department of physics, amirkabir university of technology p. Because an arbitrary smooth potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics. Coherent state path integral quantization of quantum field theory. Quantum mechanical path integral in phase space and class.
Feynmans approach to teaching the subject of quantum mechanics evolved somewhat away from the initial path integral approach. A comment on path integral for the quantum harmonic oscillator kiyoto hira dated. May 8, 20 an elementary derivation of quantum harmonic oscillator propagator using the path integral have been reconsidered to modify the calculation, which used to be, for more pedagogical signi cance. This is not, of course, true for general quantum systems. Furthermore, it is one of the few quantum mechanical systems for which an exact. We present the path integral formulation of quantum mechanics and demon. The harmonic oscillator expansion about the classical path last updated. We allow for an arbitrary timedependent oscillator strength and later include a time dependent external force. Harmonic oscillator energies and eigenfunctions derived from the propagator. Coherent state path integral for the harmonic oscillator. In principle, one should be able to construct the quantum theory for such a system by the path integral approach using this action. Hilke content classical stochastic dynamics brownian motion random walk quantum dynamics free particle particle in a potential driven harmonic oscillator semiclassical approximation statistical description imaginary time quantum dissipative systems introduction path integrals are used in.
Pdf quantum mechanical path integral in phase space and. In the case of bosons, it is based on the coherent states holomorphic formalism and the hilbert space of analytic entire functions. Motivations for a path integral formulation of qm essentially the hamiltonian operator, and can be handled more easily. Path integral 1 feynmans path integral formulation feynmans formulation of quantum mechanics using the socalled path integral is arguably the most elegant. The simple harmonic oscillator plays a prominent role in most undergraduate quantum mechanics courses. Connection with quantum harmonic oscillator in this nal part of our paper, we will show the connection of hermite polynomials with the quantum harmonic oscillator. Though the details of the calculation are involved, the general approach uses only matrix diagonalization and wellknown integrals, techniques which an advanced undergraduate should understand. Path integral for the quantum harmonic oscillator using elementary methods s.
Path integrals in statistical mechanics the feynman path integral formulation reveals a deep and fruitful interrelation between quantum mechanics and statistical mechanics. Pdf path integral for the quantum harmonic oscillator using. The path integral approach to quantum mechanics lecture notes for quantum mechanics iv riccardo rattazzi may 25, 2009. Path integral for the harmonic oscillator springerlink.
The motivation for writing these notes was that i found the typical treatment of quantum mechanics path integrals in a quantum eld theory text to be too brief to be digestible there are some exceptions, while monographs on path integrals are usually far too detailed to. Phase space path integral approach to harmonic oscillator. Path integral for the relativistic particle and harmonic. The path integral approach to quantum mechanics universitat bern. The path integral approach to quantum mechanics lecture notes. Box 1587544, tehran, iran received on 2 november, 2007 we use the path integral approach to a twodimensional noncommutative harmonic oscillator to derive the. You will see that feynman invented the path integral with the hope of replacing quantum field theory with particle quantum mechanics. Because an arbitrary potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics. We study quantum harmonic and anharmonic oscillators in the path integral formulation, using the metropolis monte carlo method to generate paths to calculate the lowest two energy levels and the ground state probability densit. We believe that path integrals have great beauty in the simplicity of their basic formulation. Pdf path integral for the quantum harmonic oscillator. Quantum mechanics is written in terms of classical paths, making the connection between the two. The computation of the path integral for a particle of spin s put in a constant magnetic field is carried out using harmonic oscillator coherent states and spin coherent states, with a careful analysis of infinitesimal terms in 1n where n is the number of time slices appearing in the lagrangian.
The path integral formulation is a description in quantum mechanics that generalizes the action principle of classical mechanics. The calculation of a onedimensional quantum harmonic oscillator propagator using the path integral has been reconsidered for more simplicity and more pedagogical significance. Objectively, the strongest points in favour of the path integral appoach are that. Path integral for the quantum harmonic oscillator using. Harmonic oscillator ground state numerical solution. It is easy to evaluate the greens function using path integral approach by evaluating classical action and using functional calculus method. Questions tagged pathintegral ask question path integral formulation due to feynman is a major formulation of quantum mechanics along with matrix mechanics due to heisenberg and pauli, wave mechanics due to schrodinger, and variational mechanics due to dirac. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or functional integral, over an infinity of quantum mechanically possible trajectories to compute a quantum amplitude this formulation has proven crucial to the. The harmonic oscillator expansion about the classical path. Coherent state path integral quantization of quantum field. Such path integrals are frequently encountered in semiclassical path integral evaluations and having exact analytical. The same problem with a constant frequency has been treated by path integral by many authors for example. Sabir and rajagopalan treated the 10 cases of the strongly pulsating mass and a model of growing mass, the powerlow suppressed harmonic oscillator 11 is also solved. Although our calculation may not bring any additional physical insight to the harmonic oscillator problem in the frame of the path integral, in this paper we try to simplify the path integral calculation for a quantum harmonic oscillator.