✎ Quantum Mechanics [The Physics Travel Guide]

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====== Quantum Mechanics ======

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 [[theories:classical_mechanics|classical mechanics]] in terms of [[theories:classical_mechanics:newtonian|Newtonian mechanics]], [[theories:classical_mechanics:lagrangian|Lagrangian mechanics]], [[theories:classical_mechanics:hamiltonian|Hamiltonian mechanics]] or [[theories:classical_mechanics:koopman_von_neumann_mechanics|Koopmann-von-Neumann mechanics]].

The transition from classical mechanics to quantum mechanics is known as [[advanced_tools:quantization|quantization]].

The most famous descriptions of quantum mechanics are

* [[theories:quantum_mechanics:canonical|Canonical quantum mechanics]] $\leftrightarrow$ quantum mechanics in [[basic_tools:hilbert_space|Hilbert space]]
    * Schrödinger's wave mechanics
    * Heisenberg's matrix mechanics
  * [[theories:quantum_mechanics:path_integral|Path-integral quantum mechanics]]  $\leftrightarrow$ quantum mechanics in [[basic_tools:configuration_space|configuration space]]
  * [[theories:quantum_mechanics:phase_space|Phase space quantum mechanics]] $\leftrightarrow$ quantum mechanics in [[basic_tools:phase_space|phase space]]
  * [[theories:quantum_mechanics:bohmian|Bohmian mechanics]] $\leftrightarrow$ quantum mechanics in everyday space

----

Both Heisenberg's matrix mechanics and Schrödinger's wave mechanics are formulations both belong to the description known as **[[theories:quantum_mechanics:canonical|canonical quantum mechanics]]**. The relevant mathematical stage for both formulations is [[basic_tools:hilbert_space|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 [[equations:canonical_commutation_relations|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"((https://en.wikipedia.org/wiki/Heisenberg_picture)).

Instead of using a Hilbert space, we can use the corresponding [[basic_tools:configuration_space|configuration space]] and its associated tangent bundle. In this formulation, our main object is the [[formalisms:lagrangian_formalism|Lagrangian]]. This Lagrangian formulation of quantum mechanics is known as **[[theories:quantum_mechanics:path_integral|path integral quantum mechanics]]**.

Another possibility is to use the corresponding [[basic_tools:phase_space|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. [[theories:classical_mechanics:hamiltonian|Hamiltonian mechanics]]. This way of describing quantum mechanics is simply known as **[[theories:quantum_mechanics:phase_space|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 **[[theories:quantum_mechanics:bohmian|Bohmian mechanics]]** and is analogous to the [[theories:classical_mechanics:newtonian|Newtonian formulation of classical mechanics]].

----

* For an interesting alternative introduction to quantum mechanics, see http://www.scottaaronson.com/democritus/lec9.html.
  * see also https://en.wikipedia.org/wiki/Mathematical_formulation_of_quantum_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((At least in the standard, Hilbert space formulation)) in quantum mechanics, the stories that are told about what we really do when we perform these calculations vary wildly.

The thing is that experimentally outcomes staty the same no matter which interpretation we believe in((This is similar to the statement that it doesn't matter which formulation we use. But here it makes at least some difference since some scenarios can be calculated more easily in a specific formulation.)). In this sense, discussions about the interprtation of quantum mechanics are mostly a matter of taste.

----

The standard 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.

----

There are dozens of other interpretations of what quantum mechanics really means:

* Hidden-Variable Interpretations
    * [[theories:quantum_mechanics:bohmian|Bohmian Mechanics]]
  * Stochastic Interpretation
    * "//It is well known that under certain restrictions, quantum mechanics can be interpreted as a Markov process for classical coordinates. Various authors have derived Schrodinger's equation within the context of classical diffusion theory. These works interpret quantum statistics or quantum indeterminacy as originating from the interaction of nonrelativistic point particles with random forces. These forces must be present even in the vacuum. This view does not conflict with present understanding of nature, since vacuum fluctuations in quantum field theory are known to exist and are well verified experimentally. The random forces experienced by particles in the stochastic interpretation may be viewed as coming about due to random fluctuations of the various fields in the vacuum.//" ((https://aip.scitation.org/doi/abs/10.1063/1.523893))
    * [[http://iopscience.iop.org/article/10.1088/1742-6596/361/1/012011/meta|Review of stochastic mechanics]] by Edward Nelson
  * Dynamical Collapse Interpretations
    * https://plato.stanford.edu/entries/qm-collapse/
  * Quantum Bayesianism
    * [[http://blogs.discovermagazine.com/cosmicvariance/files/2011/11/banks-qmblog.pdf|The Interpretation of Quantum Mechanics]] by Tom Banks 
  * Everett (many-worlds) Interpretation
    * "//Everett's thesis on the many worlds interpretation of quantum mechanics, which is an attempt to reformulate quantum mechanics using Shannon information tools.//"((https://physics.stackexchange.com/questions/43175/uncertainty-principle-for-information))
    * [[http://www.preposterousuniverse.com/blog/2014/06/30/why-the-many-worlds-formulation-of-quantum-mechanics-is-probably-correct/|Why the Many-Worlds Formulation of Quantum Mechanics Is Probably Correct]] by Sean Carroll
  * Relational Quantum Mechanics 
    * [[https://arxiv.org/abs/quant-ph/9801057|What is quantum mechanics trying to tell us?]] by N. David Mermin
    * [[https://arxiv.org/abs/quant-ph/9609002|Relational Quantum Mechanics]] by Carlo Rovelli 
  * Quantum Mechanics Without State Vectors
    * [[http://www.nybooks.com/articles/2017/01/19/trouble-with-quantum-mechanics/|The Trouble with Quantum Mechanics]] by Steven Weinberg
    * [[https://arxiv.org/abs/1405.3483|Quantum Mechanics Without State Vectors]] by Steven Weinberg
  * The Cellular Automaton Interpretation of Quantum Mechanics
    * [[https://arxiv.org/abs/1405.1548|The Cellular Automaton Interpretation of Quantum Mechanics]] by Gerard 't Hooft
  * Consistent histories interpretation
    * [[http://quantum.phys.cmu.edu/CQT/|Consistent Quantum Theory]] by Robert B. Griffiths
  * Shut up and calculate
    * This is basically the agnostic approach where we say that since the interpretation doesn't matter we shouldn't spent time discussing about it.
    * [[https://arxiv.org/abs/0709.4024|Shut up and calculate]] by Max Tegmark

----

**Recommended Resources:**

* For a quick overview, see http://www.preposterousuniverse.com/blog/2014/05/29/quantum-mechanics-smackdown/
  * For an overview see https://arxiv.org/pdf/1703.08341.pdf
  * A good book on the topic is Roland Omnes, Interpretation of Quantum Mechanics, Princeton U. Press, Princeton, 1994.
  * See also Elegance and Enigma - The Quantum Interviews by Schlosshauer

The [[equations:schroedinger_equation|Schrödinger equation]]

$$i \hbar  \partial_t \Psi(x,t) = H \Psi (x,t) $$

that describes the time-evolution of the wave function. The time-independent version for systems where the Hamiltonian is time-independent is given by

$$H  \psi(x)= E\psi(x),$$
where the complete wave function $\Psi$ is then given by

$${\Psi(x,t) = \phi(t) \psi(x) = e^{-Et/\hbar} \psi(x)}$$
----

The standard Hamiltonian is

$$ H = - \frac{\hbar^2}{2m} \Delta^2 + \hat V \hat{=} \frac{\hat{p}^2}{2m} +\hat{V}. $$
----

The [[equations:heisenberg_equation|Heisenberg equation]]

$$\frac{\mathrm{d}\hat F}{\mathrm{d}t} = -\frac{i}{\hbar}[\hat F,\hat H] + \frac{\partial \hat F}{\partial t},$$

which described the time-evolution of operators and the related time dependence of an expectation value

$$ \langle \frac{\mathrm{d}\hat F}{\mathrm{d}t} \rangle = \langle -\frac{i}{\hbar}[\hat F,\hat H] \rangle + \langle \frac{\partial \hat F}{\partial t} \rangle .$$

----

The [[advanced_notions:uncertainty_principle|uncertainty principle]]

$$ \sigma_x  \sigma_p  \geq \hbar/2,$$

and the generalized version

$$ \sigma_A \sigma_B \geq \big | \frac{1}{2i} \langle [A,B] \rangle \big|^2 .$$

----

The [[equations:canonical_commutation_relations|canonical commutation relations]]

$$ [\hat{p},\hat{x}] =  -i \hbar .$$

-->Recommended Background#

**Physics**

<WRAP group>
<WRAP half column>
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.

</WRAP>

<WRAP half column>
  * [[theories:classical_mechanics|Classical Mechanics]]
  * [[theories:classical_mechanics:hamiltonian|Hamiltonian mechanics]]
</WRAP>
</WRAP>

**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.

</WRAP>
<WRAP half column>

**Essential Math:**
  * [[basic_tools:linear_algebra|Linear Algebra]]
  * [[basic_tools:vector_calculus|Vector Calculus]]
  * [[basic_tools:differential_equations|Differential Equations]]
  * [[basic_tools:probability|Probability Theory]]

</WRAP>
</WRAP>

<--

__The Traditional Roadmap__

**Basics**

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.

</WRAP>

<WRAP half column>
**Essential Concepts:**

* [[theories:quantum_mechanics:canonical|Wave mechanics]]
  * [[equations:schroedinger_equation|Schrödinger equation]]
  * [[advanced_tools:expectation_values|Expectation values]]
  * [[theories:quantum_theory:interpretations:wave_mechanics|Wave packets]]
  * [[advanced_notions:quantum_operators|Quantum operators]]
  * [[advanced_notions:uncertainty_principle|Uncertainty relation]]

</WRAP>
</WRAP>

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."

</WRAP>

<WRAP half column>
**Essential Experiments:**

* [[experiments:double_slit_experiment|Double-slit experiment]]

</WRAP>
</WRAP>

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).

</WRAP>

<WRAP half column>
**Essential Problems:**

* [[theories:quantum_mechanics:canonical#tab__examples|Particle in a box]]
  * [[theories:quantum_mechanics:canonical#tab__examples|Particle in a potential well]]

</WRAP>
</WRAP>

**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.

</WRAP>
<WRAP half column>

**Essential Math:**
  * [[advanced_tools:group_theory|Group theory]]
  * [[advanced_tools:functional_analysis|Functional analysis]]
  * [[basic_tools:complex_analysis|Complex analysis]], especially complex integration

</WRAP>
</WRAP>

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.

</WRAP>

<WRAP half column>
**Essential Concepts:**

* [[advanced_tools:quantum_mechanics:dirac_notation|Dirac notation]]
  * [[basic_tools:hilbert_space|Hilbert space]]
  * [[basic_notions:angular_momentum|Angular momentum theory]]
  * [[basic_notions:spin|Spin]]
  * [[advanced_notions:quantum_field_theory:perturbation_theory|Perturbation theory]]
  * [[advanced_tools:scattering_theory|Scattering theory]]
  * [[theories:quantum_mechanics:path_integral|Path integral formulation]]

</WRAP>
</WRAP>

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.

</WRAP>

<WRAP half column>
**Essential Experiments:**

* [[experiments:stern-gerlach|Stern-Gerlach experiment]]
  * [[experiments:aharonov-bohm|Aharonov-Bohm experiment]]

</WRAP>
</WRAP>

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.

</WRAP>

<WRAP half column>
**Essential Problems:**

* [[models:basic_models:harmonic_oscillator|Harmonic oscillator]]
  * [[models:hydrogen_atom|Hydrogen atom]]

</WRAP>

Quantum mechanics is technically difficult. Only a few extremely artificial textbook examples can be solved exactly. For everything else, we need to use approximation techniques to tackle realistic systems.

The most important approximation schemes in quantum mechanics are

* time-independent perturbation theory,
  * WKB approximation
  * time-dependent perturbation theory,
  * adiabatic approximation,
  * semi-classical approximation.

To describe scattering processes in quantum mechanics additional tools are needed, especially

* partial wave analysis and
  * the Born approximation,
  * Fermi's golden rule.
</WRAP>

<tabbox History> 
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"((see [[https://en.wikipedia.org/wiki/Old_quantum_theory]] and [[https://arxiv.org/abs/0802.1366|Rise and fall of the old quantum theory by Manfred Bucher]])).

----

Historically, Heisenberg's matrix mechanics((Heisenberg, W.: Uber die quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen, Z. Phys. 33 (1925) 879-893. English translation in Sources of quantum mechanics, ed. B.L. van der Waerden)) 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((Schrödinger, E.: Quantisierung als Eiegnwertproblem. Ann. d. Physik 79 (1926), 361–376, 489–527)) whose main focus is the classical geometric structure of configuration space and all about wave functions.

Only one year afterward, Neumann((Neumann, J. von: Mathematische Grundlagen der Quantenmechanik. Springer, Heidelberg, 1932)) unified the two approaches by introducing the abstract concept of a [[basic_tools:hilbert_space|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 ((M. Born, P. Jordan, and W. Heisenberg, “Zur Quantenmechanik II,” Zeitschrift f¨ur Physik 35 (1926) 557–615.))

Around the same time, Dirac discovered independently the same structure((P. A. M. Dirac, “The fundamental equations of quantum mechanics,” c. London, Ser. A 645-653 (1925))).

The spectrum of the Hydrogen atom was first calculated by Pauli((W. Pauli, “Über das Wasserstoffspektrum vom Standpunkt der neuen Quantenmechanik [On the hydrogen spectrum from the standpoint of the new quantum mechanics],” Zeitschrift f¨ur Physik 36 (1926) 336–363)).

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?//"

<blockquote>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.<cite>"[[https://arxiv.org/abs/1712.02894|Space is blue and birds fly through it]]" by Carlo Rovelli</cite></blockquote>

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".

<blockquote>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”.  <cite>"[[https://arxiv.org/abs/1712.02894|Space is blue and birds fly through it]]" by Carlo Rovelli</cite></blockquote>

----

**Recommended Resources**

* Jim Baggott; The Quantum Story
  * Robert P. Crease and Charles C. Mann, The Second Creation: Makers of the Revolution in Twentieth-Century Physics
  * Abraham Pais, Inward Bound: of Matter and Forces in the Physical World
  * Uncertainty by David Lindley
  * See also Ron Maimon's answer, [[https://physics.stackexchange.com/q/18643|Good book on the history of Quantum Mechanics?]], URL (version: 2011-12-23): 
  * W. a. Fedak and J. J. Prentis, “The 1925 Born and Jordan paper ’On quantum mechanics’,” American Journal of Physics 77 (2009) 128.
  * B. van der Waerden, Sources of quantum mechanics. North Holland, 1967.

<tabbox FAQ> 
For many more questions and answers see: http://www.mat.univie.ac.at/~neum/physfaq/physics-faq.html

-->What makes a theory Quantum?#
see https://physics.stackexchange.com/questions/397694/what-makes-a-theory-quantum

<--

-->What is coherence in quantum mechanics?#

https://physics.stackexchange.com/questions/77495/what-is-coherence-in-quantum-mechanics/77515#77515
<--

-->What is the relation between phase space formulation with Wigner quasi-probability distributions and path integral formulation of quantum mechanics?#
see https://physics.stackexchange.com/questions/177800/what-is-the-relation-between-phase-space-formulation-with-wigner-quasi-probabili
<--

</tabbox>

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