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arXiv:quant-ph/0211106

Harmonic Oscillator, Coherent States, and Feynman Path Integral

Dae-Yup Song

Published Nov 18, 2002• 1 citation
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Abstract

The Feynman path integral for the generalized harmonic oscillator is reviewed, and it is shown that the path integral can be used to find a complete set of wave functions for the oscillator. Harmonic oscillators with different (time-dependent) parameters can be related through unitary transformations. The existence of generalized coherent states for a simple harmonic oscillator can then be interpreted as the result of a (formal) {\em invariance} under a unitary transformation which relates the same harmonic oscillator. In the path integral formalism, the invariance is reflected in that the kernels do not depend on the choice of classical solutions.

DOI
10.48550/arxiv.quant-ph/0211106

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The Feynman path integral for the generalized harmonic oscillator is reviewed, and it is shown that the path integral can be used to find a complete set of wave functions for the oscillator. Harmonic oscillators with different (time-dependent) parameters can be related through unitary transformations. The existence of generalized coherent states for a simple harmonic oscillator can then be interpreted as the result of a (formal) {\em invariance} under a unitary transformation which relates the same harmonic oscillator. In the path integral formalism, the invariance is reflected in that the kernels do not depend on the choice of classical solutions.
The Feynman path integral for the generalized harmonic oscillator is reviewed, and it is shown that the path integral can be used to nd a complete set of wave functions for the oscillator. Harmonic oscillators with dierent (time-dependent) parameters can be related through unitary transformations. The existence of generalized coherent states for a simple harmonic oscillator can then be interpreted as the result of a (formal) invariance under a unitary transformation which relates the same harmonic oscillator. In the path integral formalism, the invariance is reflected in that the kernels do not depend on the choice of classical solutions.
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.
Definition and formulas for harmonic oscillator coherent states and spin coherent states are reviewed in detail. The path integral formalism and its relation with the partition function of a system are also reviewed. The harmonic oscillator coherent state path integral is evaluated exactly at the discrete level and then used to find its continuum limit using various regularizations. 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 1/N where N is the number of time slices) appearing in the Lagrangian. A mapping of the spin system into a CP1 model is shown explicitly. The theory of a spinless particle in the field of a magnetic monopole and its relation with the spin system are explained. The equivalence of these two models is established up to infinitesimal order by the introduction of an external field correction. This gives a new representation of a coherent state path integral in terms of a more familiar Feynman path integral.
The coherent states for a particle in Kratzer type potentials are constructed by solving Feynman’s path integral. The action of the generalized Kratzer potential is transformed into two harmonic oscillators by Levi-Civita transformation to derive the parametric time coherent states. Green’s function, energy eigenvalues, and the corresponding wave functions for this potential are calculated. The evaluated results are reduced to the modified Kratzer potentials and Kratzer-Fues oscillators which are special cases of the generalized Kratzer potential.