How the factorization method provides a quantum-classical correspondence that eases the transition for students

The 1930 quantum textbook by Born and Jordan entitled Elementare Quantenmechanik illustrated a connection between quantum and classical systems that has been forgotten for almost 100 years. In this talk, I will illustrate how this quantum-classical correspondence relates decoupling of classical equations of motion into the operators that factorize quantum Hamiltonians. I will use the harmonic oscillator and the Kepler/hydrogen problem as examples for how this works. This relationship makes the transition from classical to quantum much clearer than using standard approached with DeBroglie waves or Poisson-brackets to commutators. One of the interesting consequences of this approach is that the quantum potential differs from the classical potential by a quantum correction that is simple to determine. The quantum and classical ladder operators have identical equations of motion, which is how the correspondence works in this approach. It typically yields different equations of motion than the Ehrenfest theorem does. We believe this correspondence will aid students in understanding the interconnected nature of the quantum and classical worlds in a new light.