Editing Quantum mechanics (section)

== Mathematical formulation ==
{{Main|Mathematical formulation of quantum mechanics}}

In the mathematically rigorous formulation of quantum mechanics, the state of a quantum mechanical system is a vector <math>\psi</math> belonging to a ([[Separable space|separable]]) complex [[Hilbert space]] <math>\mathcal H</math>. This vector is postulated to be normalized under the Hilbert space inner product, that is, it obeys <math>\langle \psi,\psi \rangle = 1</math>, and it is well-defined up to a complex number of modulus 1 (the global phase), that is, <math>\psi</math> and <math>e^{i\alpha}\psi</math> represent the same physical system. In other words, the possible states are points in the [[projective space]] of a Hilbert space, usually called the [[complex projective space]]. The exact nature of this Hilbert space is dependent on the system&nbsp;– for example, for describing position and momentum the Hilbert space is the space of complex [[square-integrable]] functions <math>L^2(\mathbb C)</math>, while the Hilbert space for the [[Spin (physics)|spin]] of a single proton is simply the space of two-dimensional complex vectors <math>\mathbb C^2</math> with the usual inner product.

Physical quantities of interest{{snd}}position, momentum, energy, spin{{snd}}are represented by observables, which are [[Hermitian adjoint#Hermitian operators|Hermitian]] (more precisely, [[self-adjoint operator|self-adjoint]]) linear [[Operator (physics)|operators]] acting on the Hilbert space. A quantum state can be an [[eigenvector]] of an observable, in which case it is called an [[eigenstate]], and the associated [[eigenvalue]] corresponds to the value of the observable in that eigenstate. More generally, a quantum state will be a linear combination of the eigenstates, known as a [[quantum superposition]]. When an observable is measured, the result will be one of its eigenvalues with probability given by the [[Born rule]]: in the simplest case the eigenvalue <math>\lambda</math> is non-degenerate and the probability is given by <math>|\langle \vec\lambda,\psi\rangle|^2</math>, where <math> \vec\lambda</math> is its associated unit-length eigenvector. More generally, the eigenvalue is degenerate and the probability is given by <math>\langle \psi,P_\lambda\psi\rangle</math>, where <math>P_\lambda</math> is the projector onto its associated eigenspace. In the continuous case, these formulas give instead the [[probability density]].

After the measurement, if result <math>\lambda</math> was obtained, the quantum state is postulated to [[Collapse of the wavefunction|collapse]] to <math> \vec\lambda</math>, in the non-degenerate case, or to <math display=inline>P_\lambda\psi\big/\! \sqrt{\langle \psi,P_\lambda\psi\rangle}</math>, in the general case. The [[probabilistic]] nature of quantum mechanics thus stems from the act of measurement. This is one of the most difficult aspects of quantum systems to understand. It was the central topic in the famous [[Bohr–Einstein debates]], in which the two scientists attempted to clarify these fundamental principles by way of [[thought experiment]]s. In the decades after the formulation of quantum mechanics, the question of what constitutes a "measurement" has been extensively studied. Newer [[interpretations of quantum mechanics]] have been formulated that do away with the concept of "[[wave function collapse]]" (see, for example, the [[many-worlds interpretation]]). The basic idea is that when a quantum system interacts with a measuring apparatus, their respective wave functions become [[Quantum entanglement|entangled]] so that the original quantum system ceases to exist as an independent entity (see ''[[Measurement in quantum mechanics]]''<ref name="google215">{{cite book |title=The Quantum Challenge: Modern Research on the Foundations of Quantum Mechanics |edition=2nd |first1=George |last1=Greenstein |first2=Arthur |last2=Zajonc |publisher=Jones and Bartlett |date=2006 |isbn=978-0-7637-2470-2 |page=215 |chapter-url=https://books.google.com/books?id=5t0tm0FB1CsC&pg=PA215 |chapter=8 Measurement |archive-url=https://web.archive.org/web/20230102102134/https://books.google.com/books?id=5t0tm0FB1CsC&pg=PA215 |archive-date=2023-01-02}}</ref>).

=== Time evolution of a quantum state ===

The time evolution of a quantum state is described by the Schrödinger equation:
<math display=block>i\hbar {\frac {\partial}{\partial t}} \psi (t) =H \psi (t). </math>
Here <math>H</math> denotes the [[Hamiltonian (quantum mechanics)|Hamiltonian]], the observable corresponding to the [[total energy]] of the system, and <math>\hbar</math> is the reduced [[Planck constant]]. The constant <math>i\hbar</math> is introduced so that the Hamiltonian is reduced to the [[Hamiltonian mechanics|classical Hamiltonian]] in cases where the quantum system can be approximated by a classical system; the ability to make such an approximation in certain limits is called the [[correspondence principle]].

The solution of this differential equation is given by
<math display=block> \psi(t) = e^{-iHt/\hbar }\psi(0). </math>
The operator <math>U(t) = e^{-iHt/\hbar } </math> is known as the time-evolution operator, and has the crucial property that it is [[Unitarity (physics)|unitary]]. This time evolution is [[deterministic]] in the sense that&nbsp;– given an initial quantum state <math>\psi(0)</math> – it makes a definite prediction of what the quantum state <math>\psi(t)</math> will be at any later time.<ref>{{cite book |title=Dreams Of A Final Theory: The Search for The Fundamental Laws of Nature |first1=Steven |last1=Weinberg |publisher=Random House |year=2010 |isbn=978-1-4070-6396-6 |page=[https://books.google.com/books?id=OLrZkgPsZR0C&pg=PT82 82] |url=https://books.google.com/books?id=OLrZkgPsZR0C}}</ref>
{{anchor|fig1}}
[[File:Atomic-orbital-clouds spd m0.png|thumb|upright=1.25|Fig. 1: [[Probability density function|Probability densities]] corresponding to the wave functions of an electron in a hydrogen atom possessing definite energy levels (increasing from the top of the image to the bottom: ''n'' = 1, 2, 3, ...) and angular momenta (increasing across from left to right: ''s'', ''p'', ''d'', ...). Denser areas correspond to higher probability density in a position measurement.{{pb}}Such wave functions are directly comparable to [[Chladni's figures]] of [[acoustics|acoustic]] modes of vibration in classical physics and are modes of oscillation as well, possessing a sharp energy and thus, a definite frequency. The [[angular momentum]] and energy are [[quantization (physics)|quantized]] and take <em>only</em> discrete values like those shown – as is the case for [[resonant frequencies]] in acoustics.]]

Some wave functions produce probability distributions that are independent of time, such as [[eigenstate]]s of the Hamiltonian.<ref name="Zwiebach2022">{{cite book |first=Barton |last=Zwiebach |title=Mastering Quantum Mechanics: Essentials, Theory, and Applications |author-link=Barton Zwiebach |publisher=MIT Press |year=2022 |isbn=978-0-262-04613-8}}</ref>{{rp|133–137}} Many systems that are treated dynamically in classical mechanics are described by such "static" wave functions. For example, a single electron in an unexcited atom is pictured classically as a particle moving in a circular trajectory around the [[atomic nucleus]], whereas in quantum mechanics, it is described by a static wave function surrounding the nucleus. For example, the electron wave function for an unexcited hydrogen atom is a spherically symmetric function known as an [[atomic orbital|''s'' orbital]] ([[#fig1|Fig. 1]]).

Analytic solutions of the Schrödinger equation are known for [[List of quantum-mechanical systems with analytical solutions|very few relatively simple model Hamiltonians]] including the [[quantum harmonic oscillator]], the [[particle in a box]], the [[dihydrogen cation]], and the [[hydrogen atom]]. Even the [[helium]] atom&nbsp;– which contains just two electrons&nbsp;– has defied all attempts at a fully analytic treatment, admitting no solution in [[Closed-form expression|closed form]].<ref>{{Cite journal |last1=Zhang |first1=Ruiqin |last2=Deng |first2=Conghao |date=1993 |title=Exact solutions of the Schrödinger equation for some quantum-mechanical many-body systems  |journal=Physical Review A |volume=47 |issue=1 |pages=71–77 |doi=10.1103/PhysRevA.47.71 |pmid=9908895 |bibcode=1993PhRvA..47...71Z |issn=1050-2947}}</ref><ref>{{Cite journal |last1=Li |first1=Jing |last2=Drummond |first2=N. D. |last3=Schuck |first3=Peter |last4=Olevano |first4=Valerio |date=2019-04-01 |title=Comparing many-body approaches against the helium atom exact solution |journal=SciPost Physics |volume=6 |issue=4 |page=40 |doi=10.21468/SciPostPhys.6.4.040 |doi-access=free |arxiv=1801.09977 |bibcode=2019ScPP....6...40L |issn=2542-4653}}</ref><ref>{{cite book |last=Drake |first=Gordon W. F. |chapter=High Precision Calculations for Helium |date=2023 |title=Springer Handbook of Atomic, Molecular, and Optical Physics |series=Springer Handbooks |pages=199–216 |editor-last=Drake |editor-first=Gordon W. F. |place=Cham |publisher=Springer International Publishing |doi=10.1007/978-3-030-73893-8_12 |isbn=978-3-030-73892-1}}</ref>

However, there are techniques for finding approximate solutions. One method, called [[perturbation theory (quantum mechanics)|perturbation theory]], uses the analytic result for a simple quantum mechanical model to create a result for a related but more complicated model by (for example) the addition of a weak [[potential energy]].<ref name="Zwiebach2022" />{{rp|793}} Another approximation method applies to systems for which quantum mechanics produces only small deviations from classical behavior. These deviations can then be computed based on the classical motion.<ref name="Zwiebach2022" />{{rp|849}}

=== Uncertainty principle ===

One consequence of the basic quantum formalism is the uncertainty principle. In its most familiar form, this states that no preparation of a quantum particle can imply simultaneously precise predictions both for a measurement of its position and for a measurement of its momentum.<ref name="Cohen-Tannoudji">{{cite book |last1=Cohen-Tannoudji |first1=Claude |last2=Diu |first2=Bernard |last3=Laloë |first3=Franck |title=Quantum Mechanics |author-link1=Claude Cohen-Tannoudji |publisher=John Wiley & Sons |year=2005 |isbn=0-471-16433-X |translator-first1=Susan Reid |translator-last1=Hemley |translator-first2=Nicole |translator-last2=Ostrowsky |translator-first3=Dan |translator-last3=Ostrowsky}}</ref><ref name="L&L">{{cite book |last1=Landau |first1=Lev D. |author-link1=Lev Landau |url=https://archive.org/details/QuantumMechanics_104 |title=Quantum Mechanics: Non-Relativistic Theory |last2=Lifschitz |first2=Evgeny M. |author-link2=Evgeny Lifshitz |publisher=[[Pergamon Press]] |year=1977 |isbn=978-0-08-020940-1 |edition=3rd |volume=3 |oclc=2284121}}</ref> Both position and momentum are observables, meaning that they are represented by [[Hermitian operators]]. The position operator <math>\hat{X}</math> and momentum operator <math>\hat{P}</math> do not commute, but rather satisfy the [[canonical commutation relation]]:
<math display=block>[\hat{X}, \hat{P}] = i\hbar.</math>
Given a quantum state, the Born rule lets us compute expectation values for both <math>X</math> and <math>P</math>, and moreover for powers of them. Defining the uncertainty for an observable by a [[standard deviation]], we have
<math display=block>\sigma_X={\textstyle \sqrt{\left\langle X^2 \right\rangle - \left\langle X \right\rangle^2}},</math>
and likewise for the momentum:
<math display=block>\sigma_P=\sqrt{\left\langle P^2 \right\rangle - \left\langle P \right\rangle^2}.</math>
The uncertainty principle states that
<math display=block>\sigma_X \sigma_P \geq \frac{\hbar}{2}.</math>
Either standard deviation can in principle be made arbitrarily small, but not both simultaneously.<ref name="ballentine1970">Section 3.2 of {{Citation |last=Ballentine |first=Leslie E. |title=The Statistical Interpretation of Quantum Mechanics |journal=Reviews of Modern Physics |volume=42 |issue=4 |pages=358–381 |year=1970 |bibcode=1970RvMP...42..358B |doi=10.1103/RevModPhys.42.358 |s2cid=120024263}}. This fact is experimentally well-known for example in quantum optics; see e.g. chap. 2 and Fig. 2.1 {{Citation |last=Leonhardt |first=Ulf |title=Measuring the Quantum State of Light |year=1997 |url=https://archive.org/details/measuringquantum0000leon |location=Cambridge |publisher=Cambridge University Press |bibcode=1997mqsl.book.....L |isbn=0-521-49730-2}}.</ref> This inequality generalizes to arbitrary pairs of self-adjoint operators <math>A</math> and <math>B</math>. The [[commutator]] of these two operators is
<math display=block>[A,B]=AB-BA,</math>
and this provides the lower bound on the product of standard deviations:
<math display=block>\sigma_A \sigma_B \geq \tfrac12 \left|\bigl\langle[A,B]\bigr\rangle \right|.</math>

Another consequence of the canonical commutation relation is that the position and momentum operators are [[Fourier transform#Uncertainty principle|Fourier transforms]] of each other, so that a description of an object according to its momentum is the Fourier transform of its description according to its position. The fact that dependence in momentum is the Fourier transform of the dependence in position means that the momentum operator is equivalent (up to an <math>i/\hbar</math> factor) to taking the derivative according to the position, since in Fourier analysis [[Fourier transform#Differentiation|differentiation corresponds to multiplication in the dual space]]. This is why in quantum equations in position space, the momentum <math> p_i</math> is replaced by <math>-i \hbar \frac {\partial}{\partial x}</math>, and in particular in the [[Schrödinger equation#Equation|non-relativistic Schrödinger equation in position space]] the momentum-squared term is replaced with a Laplacian times <math>-\hbar^2</math>.<ref name="Cohen-Tannoudji" />

=== Composite systems and entanglement ===
When two different quantum systems are considered together, the Hilbert space of the combined system is the [[tensor product]] of the Hilbert spaces of the two components. For example, let {{mvar|A}} and {{mvar|B}} be two quantum systems, with Hilbert spaces <math> \mathcal H_A </math> and <math> \mathcal H_B </math>, respectively. The Hilbert space of the composite system is then
<math display=block> \mathcal H_{AB} = \mathcal H_A \otimes \mathcal H_B.</math>
If the state for the first system is the vector <math>\psi_A</math> and the state for the second system is <math>\psi_B</math>, then the state of the composite system is
<math display=block>\psi_A \otimes \psi_B.</math>
Not all states in the joint Hilbert space <math>\mathcal H_{AB}</math> can be written in this form, however, because the superposition principle implies that linear combinations of these "separable" or "product states" are also valid. For example, if <math>\psi_A</math> and <math>\phi_A</math> are both possible states for system <math>A</math>, and likewise <math>\psi_B</math> and <math>\phi_B</math> are both possible states for system <math>B</math>, then
<math display=block>\tfrac{1}{\sqrt{2}} \left ( \psi_A \otimes \psi_B + \phi_A \otimes \phi_B \right )</math>
is a valid joint state that is not separable. States that are not separable are called [[quantum entanglement|entangled]].<ref name=":0">{{Cite book |last1=Nielsen |first1=Michael A. |last2=Chuang |first2=Isaac L. |title=Quantum Computation and Quantum Information |publisher=Cambridge University Press |year=2010 |edition=2nd |oclc=844974180 |isbn=978-1-107-00217-3 |author-link1=Michael Nielsen |author-link2=Isaac Chuang}}</ref><ref name=":1">{{Cite book |title-link=Quantum Computing: A Gentle Introduction |title=Quantum Computing: A Gentle Introduction |last1=Rieffel |first1=Eleanor G. |last2=Polak |first2=Wolfgang H. |year=2011 |publisher=MIT Press |isbn=978-0-262-01506-6 |author-link=Eleanor Rieffel}}</ref>

If the state for a composite system is entangled, it is impossible to describe either component system {{mvar|A}} or system {{mvar|B}} by a state vector. One can instead define [[reduced density matrix|reduced density matrices]] that describe the statistics that can be obtained by making measurements on either component system alone. This necessarily causes a loss of information, though: knowing the reduced density matrices of the individual systems is not enough to reconstruct the state of the composite system.<ref name=":0" /><ref name=":1" /> Just as density matrices specify the state of a subsystem of a larger system, analogously, [[POVM|positive operator-valued measures]] (POVMs) describe the effect on a subsystem of a measurement performed on a larger system. POVMs are extensively used in quantum information theory.<ref name=":0" /><ref name="wilde">{{Cite book |last=Wilde |first=Mark M. |title=Quantum Information Theory |publisher=Cambridge University Press |year=2017 |isbn=978-1-107-17616-4 |edition=2nd |doi=10.1017/9781316809976.001 |arxiv=1106.1445 |s2cid=2515538 |oclc=973404322}}</ref>

As described above, entanglement is a key feature of models of measurement processes in which an apparatus becomes entangled with the system being measured. Systems interacting with the environment in which they reside generally become entangled with that environment, a phenomenon known as [[quantum decoherence]]. This can explain why, in practice, quantum effects are difficult to observe in systems larger than microscopic.<ref>{{Cite journal |last=Schlosshauer |first=Maximilian |date=October 2019 |title=Quantum decoherence |journal=Physics Reports |volume=831 |pages=1–57 |arxiv=1911.06282 |bibcode=2019PhR...831....1S |doi=10.1016/j.physrep.2019.10.001 |s2cid=208006050}}</ref>

=== Equivalence between formulations ===
There are many mathematically equivalent formulations of quantum mechanics. One of the oldest and most common is the "[[transformation theory (quantum mechanics)|transformation theory]]" proposed by [[Paul Dirac]], which unifies and generalizes the two earliest formulations of quantum mechanics&nbsp;– [[matrix mechanics]] (invented by [[Werner Heisenberg]]) and wave mechanics (invented by [[Erwin Schrödinger]]).<ref>{{cite journal |last=Rechenberg |first=Helmut |author-link=Helmut Rechenberg |year=1987 |title=Erwin Schrödinger and the creation of wave mechanics |url=http://www.actaphys.uj.edu.pl/fulltext?series=Reg&vol=19&page=683 |format=PDF |journal=[[Acta Physica Polonica B]] |volume=19 |issue=8 |pages=683–695 |access-date=13 June 2016}}</ref> An alternative formulation of quantum mechanics is [[Feynman]]'s [[path integral formulation]], in which a quantum-mechanical amplitude is considered as a sum over all possible classical and non-classical paths between the initial and final states. This is the quantum-mechanical counterpart of the [[action principle]] in classical mechanics.<ref>{{cite book |first1=Richard P. |last1=Feynman |first2=Albert R. |last2=Hibbs |title=Quantum Mechanics and Path Integrals |edition=Emended |editor-first=Daniel F. |editor-last=Steyer |year=2005 |publisher=McGraw-Hill |isbn=978-0-486-47722-0 |pages=v–vii}}</ref>

=== Symmetries and conservation laws ===
{{Main|Noether's theorem}}

The Hamiltonian <math>H</math> is known as the ''generator'' of time evolution, since it defines a unitary time-evolution operator <math>U(t) = e^{-iHt/\hbar}</math> for each value of <math>t</math>. From this relation between <math>U(t)</math> and <math>H</math>, it follows that any observable <math>A</math> that commutes with <math>H</math> will be <em>conserved</em>: its expectation value will not change over time.<ref name="Zwiebach2022" />{{rp|471}} This statement generalizes, as mathematically, any Hermitian operator <math>A</math> can generate a family of unitary operators parameterized by a variable <math>t</math>. Under the evolution generated by <math>A</math>, any observable <math>B</math> that commutes with <math>A</math> will be conserved. Moreover, if <math>B</math> is conserved by evolution under <math>A</math>, then <math>A</math> is conserved under the evolution generated by <math>B</math>. This implies a quantum version of the result proven by [[Emmy Noether]] in classical ([[Lagrangian mechanics|Lagrangian]]) mechanics: for every [[differentiable]] [[Symmetry (physics)|symmetry]] of a Hamiltonian, there exists a corresponding [[conservation law]].