Roadmap for Quantum Mechanics
In the context of beginners’ pedagogy, the venerable quantum mechanics gets a bad rap because of the abundance of pop-sci articles ranging between misleading to downright incorrect. Luckily the existence of Griffiths’ introductory quantum mechanics textbook means that you can learn practical, fundamental quantum mechanics without picking up misinformation. Once you’ve pinned down the basics (say, until time-dependent perturbation theory), I suggest immediately moving to Sakurai’s textbook or Littlejohn’s notes for 221AB, and learning everything again. I’m serious - it’s essential to redo it rigorously, but the introductory content from Griffiths’ will allow you to rapidly grasp the material in a more advanced textbook.
Definite Prerequisites:
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Multivariable calculus: You need to be pretty comfortable with differentiating and integrating functions of multiple variables, since that’s what wavefunctions are. It goes without saying that prior to this you should be fluid with Calc I and Calc II. Periodic functions and Fourier transforms also play a crucial role in quantum mechanics - you should be able to visualise all of these notions very well, the intuition will take you far. Also it leads up to:
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Differential equations: Separation of variables, First-order, Second-order homogenous linear ODEs, and some basic understanding of PDEs. Why? Because the Schrödinger equation, a cornerstone of QM, is literally a linear PDE. You’ll get to learn cool analytical techniques to solve a variety of systems along the way.
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Matrices and vectors: You should know how to multiply matrices, diagonalise them, all that jazz. A lot of the time, quantum mechanics actually operates in infinite-dimensional Hilbert spaces, and operators essentially turn into “infinite-dimensional matrices”. Luckily, many (but not all!) notions from finite-dimensional linear algebra carry over fairly smoothly, so you need not have learnt anything about these mysterious creatures while jumping in.
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Classical mechanics: Can’t “go quantum” before you have a handle on the physics of ordinary objects! I’m not saying you need to be intimately familiar with infinite pulley diagrams; more important is a brief background in the Hamiltonian formalism of classical mechanics. It’s best to go through this a few times, so that even if all you remember is “Hamiltonian = time translations” while coming into QM, revisiting those notions in a different light should concretise your understanding.
Learn it along the way (or before, for a better understanding):
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Functional analysis: The big brother of linear algebra. Operator theory, Hilbert spaces, C* algebras, spectral theory, distributions, von Neumann. Forbiddingly dry if presented wrong, but illuminating and powerful when done right. Best served with a smattering of quantum mechanical “paradoxes” arising from trying hopelessly to abstract the properties of finite-dimensional matrices to continuous spaces.
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Electrodynamics: Probably not the most crucial one here, to be honest. Knowledge of Maxwell’s equations and interactions of electric and magnetic fields with matter is just about it: a lot of electrodynamics consists of tools for problem solving, and gaining a qualitative understanding takes a lot less effort than poring through Jackson.
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Path integrals: essentially a (rather large) offshoot of Green’s functions. Some basic measure theory can’t go amiss here, and it is very helpful for quantum field theory later on.
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Lie groups/algebras and representation theory: This is a deep, beautiful subject, and one of my favourite areas in mathematics. It illuminates the otherwise impenetrable concept of spin; ladder operators, particle classicification, angular momentum, simplifying systems using symmetry, multiparticle systems, “generators” of symmetries, Clifford algebras - the list goes on and on.
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Classical symplectic geometry: It’s always fun to see the correspondence between ordinary Poisson brackets and commutators. More seriously, ideas like Hamiltonian vector fields, moment maps and the symplectic form play a key role in demystifying quantum mechanics. A lot of enthusiasts also go the other way, first learning quantum mechanics and then the Hamiltonian formulation. But if you can delay this gratification, then I definitely suggest doing the converse. Also see: Dirac’s quantization of gauge systems (if you learn from Dirac’s textbook, this is the foundation stone anyway)
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NumPy, MATLAB or Mathematica! Very few people are capable of whipping up illuminating visualisations of complex (heh) quantum phenomena. Even creating a animation of tunneling through a barrier and the Fourier transform of the wavefunction is fantastic fun, and it’ll be very powerful to have these tools at your disposal for clarity and visual understanding