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Undergraduate quantum mechanics focuses on teaching through a wavefunctionapproach in the position-space representation. This leads to a differentialequation perspective for teaching the material. However, we know that abstractrepresentation-independent approaches often work better with students, bycomparing student reactions to learning the series solution of the harmonicoscillator versus the abstract operator method. Because one can teach all ofthe solvable quantum problems using a similar abstract method, it brings up thequestion, which is likely to lead to a better student understanding? In work atGeorgetown University and with edX, we have been teaching a class focused on anoperator-forward viewpoint, which we like to call operator mechanics. Itteaches quantum mechanics in a representation-independent fashion and allowsfor most of the math to be algebraic, rather than based on differentialequations. It relies on four fundamental operator identities -- (i) the Leibnizrule for commutators; (ii) the Hadamard lemma; (iii) theBaker-Campbell-Hausdorff formula; and (iv) the exponential disentanglingidentity. These identities allow one to solve eigenvalues, eigenstates andwavefunctions for all analytically solvable problems (including some not oftenincluded in undergraduate curricula, such as the Morse potential or thePoschl-Teller potential). It also allows for more advanced concepts relevantfor quantum sensing, such as squeezed states, to be introduced in a simplerformat than is conventionally done. In this paper, we illustrate the threeapproaches of matrix mechanics, wave mechanics, and operator mechanics, we showhow one organizes a class in this new format, we summarize the experiences wehave had with teaching quantum mechanics in this fashion and we describe how itallows us to focus the quantum curriculum on more modern 21st century topicsappropriate for the

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