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There are different ways of describing quantum mechanics. Each has its individual strength and weaknesses, but in terms of observable predictions they are all equivalent. This situation is analogous to how we can describe classical mechanics in terms of Newtonian mechanics, Lagrangian mechanics, Hamiltonian mechanics or Koopmann-von-Neumann mechanics.
The transition from classical mechanics to quantum mechanics is known as quantization.
The most famous descriptions of quantum mechanics are
Both Heisenberg's matrix mechanics and Schrödinger's wave mechanics are formulations both belong to the description known as canonical quantum mechanics. The relevant mathematical stage for both formulations is Hilbert space. The connection between them lies in the identification of Heisenberg's infinite matrices $p_j$ and $q^i$ ($i,j=1,2,3$), representing the momentum and position of a particle moving in $\mathbb{R}^3$, with Schrödinger's operators $-i\hbar\partial/\partial x^j$ and $x^i$ (seen as a multiplication operator) on the Hilbert space $\mathcal H=L^2(\mathbb{R}^3)$, respectively. The key to this identification lies in the canonical commutation relations $$ [p_i,q^j]=-i\hbar \delta^j_i. $$ We usually call these two formulations the "Heisenberg picture" and the "Schrödinger picture", since, both descriptions are actually equivalent. In some sense, the transformation between them is "just a basis change in Hilbert space"^{1)}.
Instead of using a Hilbert space, we can use the corresponding configuration space and its associated tangent bundle. In this formulation, our main object is the Lagrangian. This Lagrangian formulation of quantum mechanics is known as path integral quantum mechanics.
Another possibility is to use the corresponding phase space. Our main interest in phase space is the Hamiltonian and the algebraic structure of quantum mechanics is given by the Moyal bracket. The Moyal bracket is the appropriate deformation of the Poisson bracket, which governs the phase space formulation of classical mechanics, i.e. Hamiltonian mechanics. This way of describing quantum mechanics is simply known as phase space formulation of quantum mechanics.
Finally, we can also focus on the individual trajectories in real space ($\mathbb{R}^3$) of the objects in our system. This formulation of quantum mechanics is known as Bohmian mechanics and is analogous to the Newtonian formulation of classical mechanics.
There is no general consensus as to what the fundamental principles of quantum mechanics are and what it really "means". While almost any physicist can do calculations^{2)} in quantum mechanics, the stories that are told about what we really do when we perform these calculations vary wildly. For example, a common question is whether a particle in quantum mechanics already has well-defined properties before we measure it or if they only take on definite values as soon as we measure them.
The thing is that experimentally outcomes stay the same no matter which interpretation we believe in^{3)}. In this sense, discussions about the interpretation of quantum mechanics are mostly a matter of taste.
Important notions regarding the interpretation of quantum mechanics are
The standard (orthodox) interpretation of quantum mechanics is presented in almost every textbook and known as the Copenhagen interpretation.
According to this interpretation, particles do not possess specific dynamical properties (momentum, position, angular momentum, energy, etc.) until we perform a measurement.
The wave function is interpreted statistically and it collapses once we measure it. Therefore, if we immediately repeat a measurement, we will get the same result again.
Regarding the question, whether a particle already has a definite momentum etc. before we measure it, the Copenhagen interpretation states that
"observations not only disturb what has to be measured, they produce it!" - Pascual Jordan.
In contrast, hidden variable interpretations which are also called realist interpretations, state that
“the position of the particle was never indeterminate, but was merely unknown to the experimenter.” - Bernard d'Espagnat.
A third popular interpretation is called the agnostic interpretation states that it makes no sense to ask such a question since how can we discuss anything that we can never measure. By definition, a property like momentum is undetermined until we measure it and a discussion about its value before measurement makes no sense:
"One should no more rack one’s brain about the problem of whether something one cannot know anything about exists all the same, than about the ancient question of how many angels are able to sit on the point of a needle." - Wolfgang Pauli
An amazing discussion of the Copenhagen interpreation and how it came about can be found in Quantum Dialogue by Mara Beller.
There are dozens of other interpretations of what quantum mechanics really means:
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“If you are not confused by quantum mechanics, then you haven’t really understood it.” Niels Bohr
“I think I can safely say that nobody understands quantum mechanics.” Richard Feynman
A solid understanding of classical mechanics is certainly helpful to understand quantum mechanics. The standard formulation of quantum mechanics makes use of the Hamiltonian, and hence an understanding of Hamiltonian mechanics is necessary.
Mathematics
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:
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.
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:
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:
Advanced
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:
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.
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:
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:
The most important approximation schemes in quantum mechanics are
To describe scattering processes in quantum mechanics additional tools are needed, especially
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In the beginning quantum mechanics was only a set of heuristic rules derived from experimental observations like the infrared catastrophe in the black-body radation. This set of heuristic rules is known as the "Old Quantum Theory"^{6)}.
Historically, Heisenberg's matrix mechanics^{7)} was the first complete formulation of quantum mechanics. Heisenberg's main focus was the noncommutative structure of the quantum mechanical algebra of observables.
Soon after, Schrödinger developed his wave mechanics^{8)} whose main focus is the classical geometric structure of configuration space and all about wave functions.
Only one year afterward, Neumann^{9)} unified the two approaches by introducing the abstract concept of a Hilbert space. Schrödinger's wave functions are vectors living in Hilbert space and Heisenberg’s observables are linear operators acting on these vectors.
Heisenberg's matrix mechanics was developed into its current complete full set of equations by Born, Jordan, and Heisenberg ^{10)}
Around the same time, Dirac discovered independently the same structure^{11)}).
The spectrum of the Hydrogen atom was first calculated by Pauli^{12)}.
Only some time afterward - in 1926 - Erwin Schrödinger introduced the concept of the wave function and also calculated the hydrogen spectrum. Since in principle, quantum mechanics was already complete before Schrödinger's famous paper it is reasonable to ask: "So, what did Schrödinger do, in his 1926 paper?"
With hindsight, he took a technical and a conceptual step. The technical step was to change the algebraic language of the theory, unfamiliar at the time, into a familiar one: differential equations. This brought ethereal quantum theory down to the level of the average theoretical physicist. The conceptual step was to introduce the notion of “wave function” ψ, soon to be evolved into the notion of “quantum state” ψ, endowing it with heavy ontological weight."Space is blue and birds fly through it" by Carlo Rovelli
Since Heisenberg and Co. were much earlier than Schrödinger it also makes sense to ask why ultimately Schrödinger "won". Conventionally Students are introduced to quantum mechanics by starting with Schrödinger's "wave mechanics".
Heisenberg lost the political battle against Schrödinger, for a number of reasons. First, all this was about “interpretation” and for many physicists this wasn’t so interesting after all, once the equations of quantum mechanics begun producing wonders. Differential equations are easier to work with and sort of visualise, than non-commutative algebras. Third, Dirac himself, who did a lot directly with non-commutative algebras, found it easier to make the calculus concrete by giving it a linear representation on Hilbert spaces, and von Neumann followed: on the one hand, his robust mathematical formulation of the theory brilliantly focused on the proper relevant notion: the non-commutative observable algebra, on the other, the weight given to the Hilbert space could be taken by some as an indirect confirmation of the ontological weight of the quantum states. Fourth, and most importantly, Bohr —the recognised fatherly figure of the community— tried to mediate between his two brilliant bickering children, by obscurely agitating hands about a shamanic “wave/particle duality”. "Space is blue and birds fly through it" by Carlo Rovelli
Recommended Resources
For many more questions and answers see: http://www.mat.univie.ac.at/~neum/physfaq/physics-faq.html