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

Undergraduate quantum mechanics focuses on teaching through a wavefunction approach with the representation in position space. 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 the series solution of the harmonic oscillator to 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 understanding. In work at Georgetown University and with edX, we have been teaching a class focused on an operator-first 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 Leibnitz 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). 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. I will summarize the experiences we have had with this approach and describe what resources are available for others interested in trying the approach in their classroom. Together, we can help modernize quantum instruction, which is desperately in need of modernization for the second quantum revolution.