A solid understanding of calculus and a rudimentary understanding of linear algebra is essential. You need to know what derivatives, integrals, and Taylor expansions are and how to multiply matrices + what eigenvalues/eigenvectors are. Moreover you should know how to solve ordinary differential equations. Since quantum mechanics is all about probabilities a basic understanding of probability theory is a must-have.

Essential Math:

• Linear Algebra
• Vector Calculus
• Differential Equations
• Probability Theory

The state of a system is described by an object called wave function. The time evolution of states is determined by the Schrödinger equation. Observables are described by operators. Eigenvalues of these operators are possible measurement outcomes. By acting with an operator on the wave function we can calculate the probability for different measurement outcomes. Some observables cannot be determined at the same time with arbitrary precision. For example, we can't determine the position of a particle and its momentum at the same time with arbitrary precision. This is called an uncertainty relation. Because we describe systems in probabilistic terms, the wave function must be normalized. This means that free particles must be described in terms of wave packets, because plane waves cannot be normalized.

Essential Concepts:

• Wave mechanics
• Schrödinger equation
• Expectation values
• Wave packets
• Quantum operators
• Uncertainty relation

The most important experiment that encodes most mysteries of quantum mechanics is the double slit experiment. To quote Feynman: "We choose to examine a phenomenon which is impossible, absolutely impossible, to explain in any classical way, and which has in it the heart of quantum mechanics. In reality, it contains the only mystery."

Essential Experiments:

• Double-slit experiment

To really understand how quantum mechanics works in practice it is crucial to understand a few canonical examples. The particle in a box examples demonstrates nicely the quantization of energy levels and the particle in a potential well how classically impossible things become possible in quantum mechanics (tunneling).

Essential Problems:

• Particle in a box
• Particle in a potential well

To grasp the deeper structure of quantum mechanics a solid understanding of group theory is crucial. The most important aspect of group theory for quantum mechanics is representation theory. Moreover, to understand what is really going on in many calculations some knowledge of functional analysis and complex analysis are essential.

Essential Math:

• Group theory
• Functional analysis
• Complex analysis, especially complex integration

The machinery of quantum mechanics is nicely exposed by using the Dirac notation. The wave mechanical description can then be understood as just one special case. The state of a system is then no longer described by a wave function but by an abstract vector in Hilbert space. One of the most important observables is angular momentum and the closely related "internal angular momentum", called spin. For many real-world problems perturbation theory is crucial, because almost no problem in quantum mechanics can be solved exactly. To prepare for quantum field theory, which is mostly about scattering theory, learning the basics in the quantum mechanical context makes sense. Moreover, to understand some of the subtler aspects of quantum mechanics and to see that there is a different but equally powerful formulation, getting some understanding of the path integral formulation is a smart idea.

Essential Concepts:

• Dirac notation
• Hilbert space
• Angular momentum theory
• Spin
• Perturbation theory
• Scattering theory
• Path integral formulation

One of the most important subtle aspects of quantum mechanics, spin, is best understood by having a look at the famous Stern-Gerlach experiment. The role of another crucial notion, called gauge potentials, is exposed by the Ahoronov-Bohm experiment.

Essential Experiments:

• Stern-Gerlach experiment
• Aharonov-Bohm experiment

To understand the many advanced concepts of quantum mechanics getting a solid understanding of the harmonic oscillator is absolutely crucial. One of the triumphs of quantum mechanics is the correct description of the energy levels of the hydrogen atom. Thus, calculating them, including spin-orbit corrections etc., is something every serious student of quantum mechanics should be able to do.

Essential Problems:

• Harmonic oscillator
• Hydrogen atom