Learning about – Quantum Mechanics – the introductionCredits for image original: Eric Davidsson.
Starting out with quantum theory there are a couple of mathematical prerequisites. These form the language and the tools to work with the subject. Most central are some ideas from Calculus, Linear Algebra, and Complex Numbers. There is also a differential equation at the core of the theory, so techniques from Partial Differential Equations can enter into derivations. Fourier Analysis is another area that finds frequent applications in quantum theory.
Even without the mathematical background, you can enjoy this inspiring introduction to the topic, made by the brilliant creator behind 3Blue1Brown, Grant Sanderson. This time in collaboration with Henry Reich from Minute Physics.
I had the fortune of being taught by Dr Babak Majidzadeh Garjani. His mathematical clarity made the most involved derivation pleasant to follow. Here you can access his condensed guide in mathematics prerequisites for introductory Quantum Mechanics.
Mathematics Companion to QM
Professor Allan Adams is a charismatic teacher whose teaching style makes the introductory MIT course Quantum Physics I very enjoyable.
Video lectures, MIT Quantum Physics I
Quantum Physics II at MIT is also great. In particular the introduction to the Stern-Gerlach experiment. The mathematics of spin is rich enough that it can demand a lot of attention, and perhaps less is spent on its connection to the rest of physics. However, professor Barton Zwiebach has an inclusive approach.
Video lectures, MIT Quantum Physics II
"Professor M" is in fact two edudators from the Monserrat Lab, at the University of Cambridge. They aim to present a rigorous starting point that I feel helps with the conceptual
understanding too. Many videos works well as a standalone watch, when you quickly want to look into some concept without following an entire lecture.
Professor M does Science
Looking Glass Universe is the YouTube channel of Mithuna Yoganathan. With her unique style, she introduces and explains a lot of the quantum weirdness you may have already heard about in popular science. In addition, she presents great discussions about the underlying mathematics.
A great tool for visualising and taking complex superpositions of hydrogen atom eigenfunctions—built by Paul Falstad.