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Discrete Fourier Transform

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Discrete Bargmann transform.

Kenan Uriostegui

Journal of the Optical Society of America. A, Optics, Image Science, and Vision

|September 11, 2019

Summary

This study introduces a discrete Bargmann transform for finite functions using the su(2) harmonic oscillator model. The new transform accurately reconstructs functions and preserves key properties, enabling new applications in quantum mechanics.

Area of Science:

Quantum mechanics
Mathematical physics
Harmonic oscillator models

Background:

The Bargmann transform is a powerful tool in quantum mechanics for analyzing functions.
Discrete and finite versions are needed for computational applications.
The su(2) harmonic oscillator model provides a framework for discrete quantum systems.

Purpose of the Study:

To define and analyze a discrete Bargmann transform for discrete and finite functions.
To explore its properties within the su(2) finite harmonic oscillator model.
To establish its accuracy and utility for computational quantum mechanics.

Main Methods:

Definition of a discrete Bargmann transform using coherent states.
Application to discrete and finite functions within the su(2) harmonic oscillator model.

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Analysis of the inverse transform and its error bounds.

Main Results:

The discrete Bargmann transform was successfully defined over a finite complex plane mesh.
The inverse transform accurately reconstitutes functions with an average error of 10^-7.
Key properties of the continuous Bargmann transform were shown to hold for the discrete version, including self-reproduction of functions and correspondence of dynamics under fractional transforms.

Conclusions:

The developed discrete Bargmann transform is a viable and accurate tool for analyzing finite quantum systems.
It preserves essential mathematical properties, making it suitable for computational studies.
This work extends the applicability of Bargmann transforms to discrete and finite settings in quantum mechanics.