PHAS Colloquia

A notable quasi-relativistic wave equation and its relation to the Schrödinger, Klein-Gordon, and Dirac equations

by Luis Grave de Peralta (Department of Physics, Texas Tech University)

Tuesday, 22 September 2020 from to (America/Chicago)
Speaker: Dr. Luis Grave de Peralta (Physics & Astronomy, Texas Tech University)

Title: A notable quasi-relativistic wave equation and its relation to the Schrödinger, Klein-Gordon, and Dirac equations.

Abstract: In this talk, I will discuss interesting theoretical results found when I got curious about the following Schrödinger-like but quasi-relativistic wave equation: (Please find the image of equation (1) in the material below). 
𝑖ℏ 𝜕/𝜕𝑡 𝜓=−ℏ^2/(𝛾_𝑉+1)𝑚 ∇^2 𝜓+𝑈𝜓, 𝛾_𝑉=1/√(1−𝑉^2/𝑐^2 ) (1)
In Eq. (1), 𝑚 is the particle’s mass, ℏ is the Plank constant (ℎ) divided by 2𝜋, 𝑈 is a scalar potential, and 𝛾_𝑉 is the well-known relativistic parameter depending on the square of the particle speed (𝑉^2). Eq. (1) describes a quantum particle with a relativistic relation between the particle’s linear momentum and its kinetic energy. Nevertheless, due to the striking similitude between Eq. (1) and the Schrödinger equation, Eq. (1) can be solved following the same mathematical steps required for solving the Schrödinger equation in most of the problems often included in Introductory Quantum Mechanics courses. This includes a free particle [1], confinement of a quantum particle in box [1, 3-4], reflexion by a sharp quantum potential [3], tunnel effect [3], and the quasi-relativistic description of Hydrogen-like atoms [2, 4- 5]. This equation then allows for a smooth introduction of special relativity concepts and ideas in Introductory Quantum Mechanics courses. Moreover, Eq. (1) fits well in between the number-of-particles-conserving Schrödinger equation and the fully relativistic quantum mechanics, where the number of particles is not conserved. Eq. (1) describes a quantum particle with energies up to mc^2. The number of particles is constant at these quasi-relativistic energies; thus, this energy range includes all Chemistry.

You can find what you are looking for when you are motivated, curious...and you have nothing else to do. I will discuss the existing relationships between Eq. (1) and the well-known relativistic Klein-Gordon and Dirac equations [3, 6]. If I have time, I will tell you an inspirational story...once upon a time in America, a free-thinker, rebellious, and currently old Physics professor was motivated to work on something new...but too old to be considered fashionable in these days. This happened while he was working with a young but extremely bright Ph. D. student, who came to study in America attracted by the reputation of one of our own, Prof. Bill Poirier...

References: (downloadable from section “Physics: selected works” at

1. “Natural extension of the Schrödinger equation to quasi-relativistic speeds,” L. Grave de Peralta, Journal of Modern Physics 11, 196-213 (2020).
2. “Quasi-relativistic description of Hydrogen-like atoms,” L. Grave de Peralta, Journal of Modern Physics 11, 788-802 (2020).
3. “Quasi-relativistic description of a quantum particle moving through one-dimensional piecewise constant potentials,” L. Grave de Peralta, Results in Physics 18, 103318 (2020).
4. “Did Schrödinger have other options?” L. Grave de Peralta, European Journal of Physics, (in press, 2020).
5. “Exact quasi-relativistic wave functions for Hydrogen-like atoms,” L. Grave de Peralta, Scientific Reports (2020).
6. “A notable quasi-relativistic wave equation and its relation to the Schrödinger, Klein-Gordon, and Dirac equations,” L. Grave de Peralta, Canadian Journal of Physics (under review, 2020).

About the speaker: Dr. Luis Grave de Peralta has been on the faculty of the Department of Physics and Astronomy at Texas Tech University since 2007. He received his MS in Physics from Oriente University at Cuba in 1982 and his PhD in Electrical Engineering from Texas Tech University in 2000. Dr. Luis is a member of the TTU Nanotechnology Center since its creation. He had advised 28 Masters and 8 Ph. D. graduates. Dr. Luis had published more than 75 scientific papers in archival peer reviewed journals. He was a recipient of a 2012-2015 NSF-CARRER Award and a USA National Academy of Science Medal for his Human Rights advocacy. Dr. Luis is the author of several books about the political realities in Cuba, Luis’ birth motherland. Luis says USA and specifically (Lubbock) Texas is his motherland by choice. He is proud to be an American. His primary research interests concentrate on nanophotonics, fourier optics, photonic and plasmonic crystals, and subwavelength resolution microscopy. This includes theoretical description and the experimental characterization of topological photonic crystals, and the development of advanced phase-recovery microscopy techniques (dual-space microscopy).


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