Roadmap to studying PDEs for analyzing Quantum Physics better

Question

I am studying the basics of Quantum Physics (involving the characteristics of Schrodinger's Wave equation without actually analyzing it rigorously mathematically) this semester. I was wondering, however, if I did study it in the context of a mathematical background, it would be a great help.

I wish to know what the prerequisites for studying PDEs and Probability Distributions are. I have already studied the following topics:

1. Single-Variable Calculus
2. Partial Differentiation
3. Vectors
4. ODEs (just the basics)
5. Combinatorics

I would be grateful to learn what roadmap I should follow for learning to analyze and solve PDEs and also Probability Distributions. Also, kindly list a few textbooks to follow for the same. Thanks

Answer 1

I recommend do more ODEs, first step. You mention "just the basics". This is the first step to upgrade. Take a normal ODE text (not the snippet of ODEs within a calculus book). Make sure you are familiar with all of it, including the later chapters on series solutions, etc. Do some of the harder homework problems with Legendre polynomials (shows up in solution of hydrogen atom). Get exposed to Bessel functions and Hermite polynomials as well.

In general, you actually won't need deep understanding of several different PDEs for quantum course, since you just work with one or two of them (and turn them into ODEs when solving them). Also, you don't need really probability theory or even a computational probability course. As long as you understand the concept of a distribution (intuitively, not formally) that is plenty.

All that said, PDEs and probes/shafts (probability and statistics) are great courses themselves. But for much, much more than just support of your quantum course.

Realistically, you don't need deep extra math course prep for your quantum course. Just to stay up to speed on what you are doing in that course.