Operator Mechanics: A new form of quantum mechanics without waves or matrices

Quantum mechanics was created with thematrix mechanics of Heisenberg, Born, and Jordan. Schroedinger'swave mechanics shortly followed and allowed for simpler and morepowerful calculations. Both Pauli and Dirac introduced a formulation ofquantum mechanics based on operators and commutation relations, butit was never fully developed in the 1920's. Instead, Schroedinger formulatedthe operator approach with his factorization method, which later wasadopted by the high-energy community as supersymmetric quantum mechanics. In this talk, I will explain how one can formulate nearly all ofquantum mechanics algebraically by a proper use of thetranslation operator on top of Schroedinger's factorization method. Iwill give examples of how one can compute spherical harmonics algebraically,how one can find harmonic oscillator wavefunctions, and will even describean operator-based derivation of the wavefunctions of Hydrogen. I will endwith a proposal for a novel way to teach quantum mechanics, focusing firston conceptual ideas related to superposition, projective measurements, andentanglement. Then developing more conventional topics like spin, harmonicoscillator, angular momentum, interacting spin models, central potentials,particles in a box and so on. This is the subject of a book in progressentitled Quantum Mechanics without Calculus.