Module 5 lecture 3 part 2 Canonical Commutation Relations

Module 5 lecture 3 part 2, Physics 253, Fall 2020 semester, Georgetown University

In this video, we motivate the canonical commutation relation from the fact that atomic spectra have sharp lines in them. This uses 5 ideas: (1) the Planck-Einstein relation, that the energy of a photon is hbar times its angular frequency; (2) the Bohr ideas that atoms have steady states with definite energy, and the transition between two states produces a photon with energy given by the energy difference between the states; (3) the Heisenberg idea, that the matrix element of an operator sandwiched between these initial and final states varies harmonically with the frequency of light that is emitted; (4) the Leibniz rule for the commutator of the product of operators and (5) the fact that the total energy is given by the kinetic energy plus the potential energy.

What we can then argue is that if the commutator of position with momentum is given by a number, then that number must be i times hbar. There are a few places where we have some hand-waving arguments. This is why it is a motivation, not a rigorous derivation.

This video does show that the canonical commutation can be physically motivated and does not need to be simply postulated in quantum mechanics.