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Journal of Mathematical Physics

An integral transform related to quantization

Ingrid C. Daubechies, Alex Grossmann

Published Aug 1, 1980• 52 citations
Publisher

Abstract

We study in some detail the correspondence between a function f on phase space and the matrix elements. (Qf)(a,b) of its quantized Qf between the coherent states ‖ a〉 and ‖ b〉. It is an integral transform: Qf(a,b) = ℱ{a,b ‖ v}f(v) dv, which resembles in many ways the integral transform of Bargmann. We obtain the matrix elements of Qf between harmonic oscillator states as the Fourier coefficients of f with respect to an explicit orthonormal system.

DOI
10.1063/1.524702

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We study in some detail the correspondence between a function f on phase space and the matrix elements. (Qf)(a,b) of its quantized Qf between the coherent states ‖ a〉 and ‖ b〉. It is an integral transform: Qf(a,b) = ℱ{a,b ‖ v}f(v) dv, which resembles in many ways the integral transform of Bargmann. We obtain the matrix elements of Qf between harmonic oscillator states as the Fourier coefficients of f with respect to an explicit orthonormal system.
It is shown that the representation matrices of the quantum mechanical group in the coherent state basis can be used as bases for the expansion of arbitrary square integrable functions on the group. Although this basis is nonorthogonal the expansion can be inverted in a manner similar to an orthogonal basis. Next a coherent state ‘‘wave function’’ is defined in analogy with the Schrödinger wave function and the above matrix elements are used to determine the action of the group in the space of the coherent state wave functions. The function space in this realization is Bargmann’s Hilbert space of analytic functions. It is shown that the mixed basis matrix element between the coordinate and coherent state bases is essentially the integral kernel of Bargmann. The transpose of this matrix element yields the kernel for the inversion of the transform. In this construction the Bargmann transform appears as the unitary transformation connecting the Schrödinger and coherent state wave functions. This considerably simplifies Bargmann’s original arguments and yields the integral transform as well as its inversion formula in a simple way.
The Hamiltonian of the harmonic oscillator is usually defined as a differential operator, but an integral representation can be obtained by using the coherent state quantization. The finite frame quantization is a finite counterpart of the coherent state quantization and it allows us to define a finite oscillator by starting from the integral representation of the harmonic oscillator. Our purpose is to investigate the oscillator obtained in this way, and to present a possible application to the discrete fractional Fourier transform.
Abstract Coherent states, in Wigner representation of quantum mechanics which is classical in structure with Wigner function playing the role of classical phase space probability distribution, and standard quantum mechanical expression for average values of corresponding Weyl quantum operators, general expression for matrix elements of the operator corresponding to phase of the oscillator by Weyl procedure, are obtained in the |n〉 basis. The general expression for matrix elements in the same basis of any other operator is also obtained. As the only mathematical technique necessary in the proposed procedure is a simple change of variables to polar coordinates in the corresponding integrals, this way to introduce Weyl phase operator, and some other operators in Weyl quantization, greatly simplifies necessary derivations and calculations.