Physics Research International

Volume 2015 (2015), Article ID 895134, 8 pages

http://dx.doi.org/10.1155/2015/895134

## Common Pedagogical Issues with De Broglie Waves: Moving Double Slits, Composite Mass, and Clock Synchronization

NASA Johnson Space Center, Houston, TX 77058, USA

Received 28 September 2015; Accepted 9 November 2015

Academic Editor: Martin Kröger

Copyright © 2015 Robert L. Shuler. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper addresses gaps identified in pedagogical studies of how misunderstanding of De Broglie waves affects later coursework and presents a heuristic for understanding the De Broglie frequency of composite. De Broglie’s little known derivation is reviewed with a new illustration based on his description. Simple techniques for reference frame independent analysis of a moving double slit electron interference experiment are not previously found in any literature and cement the concepts. Points of similarity and difference between De Broglie and Schrödinger waves are explained. The necessity of momentum, energy, and wavelength changes in the electrons in order for them to be vertically displaced in their own reference frame is shown to be required to make the double slit analysis work. A relativistic kinematic analysis of De Broglie frequency is provided showing how the higher De Broglie frequency of moving particles is consistent with Special Relativity and time dilation and that it demonstrates a natural system which obeys Einstein’s clock synchronization convention of simultaneity and no other. Students will be better prepared to identify practical approaches to solving problems and to think about fundamental questions.

#### 1. Introduction

At some point the undergraduate physics student and many engineering or chemistry students as well will take a one semester course in quantum mechanics (QM). When the student encounters things like De Broglie wavelengths and double slit experiments, no mention is made of how to analyze the problem from any other reference frame, not even with a simple Galilean transformation relationship. Yet any student that actually looks at and thinks about will feel puzzled. How can a wavelength that varies in that way with momentum and thus velocity be a real physical thing? How could one analyze it from another reference frame? The wavelength has an inverse dependence on velocity and becomes infinite in the particle’s own reference frame.

Studies of the wave-particle knowledge of university students completing a modern physics course suggest that only the most insightful students will get as far as suggested above, to become puzzled. Many bring incorrect ideas from misunderstanding optics to bear on electrons, such as thinking photons “bend at the slit edges” in diffraction, and compound their misunderstanding by “failure to recognize the De Broglie wave is not an inherent property of an electron but varies with momentum,” thinking that “diffraction and interference are independent of velocity,” and believing that “equations that apply to the wavelength of light apply to De Broglie wavelength” [1].

An experienced instructor doubtless will feel that, in an already crowded syllabus, it is best to solve each problem in the simplest reference frame. For the case of electron interference or diffraction, this is the frame of the slit(s). But the instructor is already clear on the concepts and with a junior or senior class by now skilled in memorizing formulae and passing exams, the misconceptions from earlier courses may persist. This paper attempts to show that, in the case of De Broglie wavelength, a short excursion into another reference frame may be precisely the exercise that will allow students to identify and correct misconceptions quickly and effectively.

Actually De Broglie struggled with relativity while developing his theory [2]. But De Broglie started with* frequency* and the idea of a* nonpropagating* wave that had the same phase everywhere. The problem was that this was incompatible with Special Relativity, and to fix the problem he applied the Lorentz transform and found that, in frames other than the particle’s frame, a phase wave and associated wavelength appear as a result. It is similar in some ways to the idea that the magnetic field is due to the effects of relativity on the electrostatic field, which has been known for more than a century [3] and taught to undergraduates for nearly half a century, popularized in Purcell’s 1963 textbook [4].

For a simple example of the puzzle of applying De Broglie waves in certain reference frames, consider a double slit experiment from the point of view of the electrons as shown in Figure 1. The resulting pattern will be the same regardless of who is moving. But how can this be if the wavelength becomes infinite?