Trying to learn Newtonian mechanics without a priori or concurrently learning Calculus is not a good idea, in my opinion. (I mean here Calculus taught in a non-rigorous, applied way; I don’t mean rigorous Mathematical Analysis). Indeed, non-rigorous Calculus is extremely helpful and insight-giving in doing Newtonian mechanics. It’s not surprising that both disciplines were invented concurrently by the same guy, Isaac Newton.
Analogously, I think that trying to learn Quantum Mechanics (QM) without a priori or concurrently learning Group theory (GT) is a terrible idea. (Again, I mean here GT taught in a non-rigorous, applied way). Non-rigorous GT is extremely helpful and insight-giving in doing QM. GT and QM were not invented concurrently by the same physicist, but they could easily have been. Mathematicians like Schur, Frobenius, Young, Galois, etc., had already developed GT to a high degree before QM came along, but there is no doubt that the invention of QM has spurred the invention of many new tools in GT.
As one can see from the Wikipedia “Timeline of Quantum Mechanics”, most of the fundamentals of QM were discovered during the miraculous decade 1920 to 1930. Note, for example, that in
1923 – Louis de Broglie extends wave–particle duality to particles,
1930 – Dirac’s textbook “Principles of Quantum Mechanics” was published
Note also that Hermann Weyl (1885-1955) came out with his book “The Theory of Groups and Quantum Mechanics” (available as a Dover edition) in 1930 and Eugene Wigner (1902-1995) with his book “Group Theory and its Application to the Quantum Mechanics of Atomic Spectra” in 1931. So already during the miraculous decade, and certainly long thereafter, Weyl and Wigner were common household “group-pests” who never lost a chance to point out to anyone within earshot how closely related QM and GT are.
As you can see, the observation that a knowledge of GT greatly helps one to understand QM is an observation that is almost as old as QM itself, and this observation has been widely publicized by some of the patron saints of QM.
I learned QM from some books that I think are very good (for example, the 2 volume set by Cohen-Tannoudji, Diu and Laloe), but, unfortunately, those books don’t use GT explicitly. Then I learned a bit of GT from some books that I think are pretty lousy. Either because of the shortcomings of my first GT books or due to my own shortcomings, or both, it took me a long time to feel comfortable with GT. The tide first started to turn for me when I read the two volume set by Elliot and Dawber entitled “Symmetries in Physics”, two books which I find truly wonderful and excellent. Using the Dawber and Elliot books as my “base camp”, I was able to explore GT further and read other more recent and specialized GT texts. You might prefer different books than I do. The field of GT is so vast, I think no single book or author can cover everything in detail and satisfy the topic selection and stylistic tastes of everyone.
Nowadays, I feel much more comfortable with GT, and I feel compelled to analyze in GT language everything that I do in QM, and I mean everything, be that physical chemistry, AMO (atomic-molecular-optical) physics, condensed matter physics, nuclear physics, particle physics, or relativity. It’s amazing how widely applicable GT is in physics.
Of course, finding applications of GT to quantum computation and quantum Bayesian networks is high in my agenda. My latest patent, number 12 in this list, is about GT. Two previous blog posts of mine on GT are