Should we trade off higher-level mathematics for abstraction to improve student understanding of quantum mechanics?

Abstract

Undergraduate quantum mechanics focuses on teaching through a wavefunction approach in the position-space representation. This leads to a differential equation perspective for teaching the material. However, we know that abstract representation-independent approaches often work better with students, by comparing student reactions to learning the series solution of the harmonic oscillator versus the abstract operator method. Because one can teach all of the solvable quantum problems using a similar abstract method, it brings up the question, which is likely to lead to a better student understanding? In work at Georgetown University and with edX, we have been teaching a class focused on an operator-forward viewpoint, which we like to call operator mechanics. It teaches quantum mechanics in a representation-independent fashion and allows for most of the math to be algebraic, rather than based on differential equations. It relies on four fundamental operator identities -- (i) the Leibniz rule for commutators; (ii) the Hadamard lemma; (iii) the Baker-Campbell-Hausdorff formula; and (iv) the exponential disentangling identity. These identities allow one to solve eigenvalues, eigenstates and wavefunctions for all analytically solvable problems (including some not often included in undergraduate curricula, such as the Morse potential or the Poschl-Teller potential). It also allows for more advanced concepts relevant for quantum sensing, such as squeezed states, to be introduced in a simpler format than is conventionally done. In this paper, we illustrate the three approaches of matrix mechanics, wave mechanics, and operator mechanics, we show how one organizes a class in this new format, we summarize the experiences we have had with teaching quantum mechanics in this fashion and we describe how it allows us to focus the quantum curriculum on more modern 21st century topics appropriate for the