  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
