My undergraduate advisor said something very interesting to me the other day; it was something like "not knowing quantum mechanics is like never having heard a symphony." I've been meaning to learn quantum for some time now, and after seeing it come up repeatedly in mathematical contexts like Scott Aaronson's blog or John Baez's TWF, I figure I might as well do it now.
Unfortunately, my physics background is a little lacking. I know some mechanics and some E&M, but I can't say I've mastered either (for example, I don't know either the Hamiltonian or the Lagrangian formulations of mechanics). I also have a relatively poor background in differential equations and multivariate calculus. However, I do know a little representation theory and a little functional analysis, and I like q-analogues! (This last comment is somewhat tongue-in-cheek.)
Given this state of affairs, what's my best option for learning quantum? Can you recommend me a good reference that downplays the historical progression and emphasizes the mathematics? Is it necessary that I understand what a Hamiltonian is first?
(I hope this is "of interest to mathematicians." Certainly the word "quantum" gets thrown around enough in mathematics papers that I would think it is.)
Best Answer
It could be just my own personal bias, but I think it is difficult to learn quantum mechanics without first learning classical mechanics. I recommend taking a 1 semester course, either graduate or advanced undergraduate, in classical mechanics and then taking a quantum mechanics course. I also think it would be a mistake to start with an overly mathematically-oriented QM course. You want to learn how physicists think and how they use this stuff to come up with real physical predictions. Otherwise, you're just learning math packaged as "physics". You shouldn't have much trouble later figuring out how to translate the physics back into math. But if you focus too much on the math at the beginning, you make it less likely you'll ever understand the physics.