Physics Research: Quantum Mechanics

During my attendance at Wentworth Institute of Technology, my team and I studied the implications of wave functions that are their own Fourier Transform. Our paper was submitted to the American Journal of Physics for publication and is pending review.

Context to the Fourier Transform

The Fourier transform is a powerful tool used in mathematics and physics with applications in signal processing, image editing, and many more. By changing the domain of the function (sometimes from time to frequency) breaks a continuous signal down into individual intrinsic frequencies which can then be processed.

Some special functions (the Gaussian, hyperbolic secant, etc.) are their own Fourier transforms that differ only by an amplitude and angular scaling coefficient.

Our research investigates the implications of a quantum wave with an initial state described by a function that is it's own Fourier transform.

Quantum Nuances

My role on the project was entirely theoretical. I worked out some of the mathematics regarding the scaling coefficients and Fourier transform properties.

Findings: We found that if a wave function is characterized by these special functions, then the wave function always minimizes the Heisenberg Uncertainty Principle without invoking the Cauchy-Schwartz Inequality, a typical starting point of uncertainty proofs.

The image (above) shows the final generalized uncertainty of a wave function that is its own Fourier transform as a function of the corresponding expectation values.

This image (above) is the final solution of the wave function within the harmonic oscillator potential, a function of the respective nth value Hermite polynomial. Implementing a clever calculus of variations technique, the solutions are referred to as the eigenfunctions of the Fourier transform, and the ground state solution is the Gaussian which pertains to the Fourier transform properties.

Publications

Our team submitted our findings to the American Journal of Physics for publication and is in the second stage of the reviewing process.

Citation: Zengel, N. Devitto, N. Hillyer, J. Rodden, and V. Vu, “The uncertainty principle and quantum wave functions that are their own Fourier transforms,” (Submitted, 2023)