Editing Schrödinger equation

{{Short description|Description of a quantum-mechanical system}}
{{Hatnote|For a more general introduction to the topic, see [[Introduction to quantum mechanics]].}}
{{Use dmy dates|date=June 2016}} 
{{Quantum mechanics|cTopic=Equations}}
{{Modern physics}}
The '''Schrödinger equation''' is a [[partial differential equation]] that governs the [[wave function]] of a non-relativistic quantum-mechanical system.<ref name="Griffiths2004">{{cite book |last=Griffiths| first=David J.|title=Introduction to Quantum Mechanics (2nd ed.)|title-link=Introduction to Quantum Mechanics (book)|publisher=Prentice Hall| year=2004|isbn=978-0-13-111892-8|location=|pages=|author-link=David J. Griffiths}}</ref>{{rp|1–2}} Its discovery was a significant landmark in the development of [[quantum mechanics]]. It is named after [[Erwin Schrödinger]], an Austrian physicist, who postulated the equation in 1925 and published it in 1926, forming the basis for the work that resulted in his [[Nobel Prize in Physics]] in 1933.<ref>{{cite news|title=Physicist Erwin Schrödinger's Google doodle marks quantum mechanics work|url=https://www.theguardian.com/technology/2013/aug/12/erwin-schrodinger-google-doodle|access-date=25 August 2013|newspaper=[[The Guardian]]|date=13 August 2013}}</ref><ref name = sch>
{{cite journal
 | last         = Schrödinger | first = E.
 | title        = An Undulatory Theory of the Mechanics of Atoms and Molecules
 | url          = http://home.tiscali.nl/physis/HistoricPaper/Schroedinger/Schroedinger1926c.pdf
 | archive-url  = https://web.archive.org/web/20081217040121/http://home.tiscali.nl/physis/HistoricPaper/Schroedinger/Schroedinger1926c.pdf
 | archive-date = 17 December 2008
 | journal      = [[Physical Review]]
 | volume       = 28
 | issue        = 6
 | pages        = 1049–70
 | year         = 1926
 | doi          = 10.1103/PhysRev.28.1049
 |bibcode       = 1926PhRv...28.1049S
}}</ref>

Conceptually, the Schrödinger equation is the quantum counterpart of [[Newton's second law]] in [[classical mechanics]]. Given a set of known initial conditions, Newton's second law makes a mathematical prediction as to what path a given physical system will take over time. The Schrödinger equation gives the evolution over time of the [[wave function]], the quantum-mechanical characterization of an isolated physical system. The equation was postulated by Schrödinger based on a postulate of [[Louis de Broglie]] that all matter has an associated [[matter wave]]. The equation predicted bound states of the atom in agreement with experimental observations.<ref>{{Cite book |last=Whittaker |first=Edmund T. |title=A history of the theories of aether & electricity. 2: The modern theories, 1900 – 1926 |date=1989 |publisher=Dover Publ |isbn=978-0-486-26126-3 |edition=Repr |location=New York}}</ref>{{rp|II:268}}

The Schrödinger equation is not the only way to study quantum mechanical systems and make predictions. Other formulations of quantum mechanics include [[matrix mechanics]], introduced by [[Werner Heisenberg]], and the [[path integral formulation]], developed chiefly by [[Richard Feynman]]. When these approaches are compared, the use of the Schrödinger equation is sometimes called "wave mechanics".

The equation given by Schrödinger is nonrelativistic because it contains a first derivative in time and a second derivative in space, and therefore space and time are not on equal footing. [[Paul Dirac]] incorporated [[special relativity]] and quantum mechanics into a [[Dirac equation|single formulation]] that simplifies to the Schrödinger equation in the non-relativistic limit. This is the [[Dirac equation]], which contains a single derivative in both space and time. Another [[partial differential equation]], the [[Klein–Gordon equation]], led to a problem with probability density even though it was a [[Relativistic wave equations|relativistic wave equation]]. The probability density could be negative, which is physically unviable. This was fixed by Dirac by taking the so-called square root of the Klein–Gordon operator and in turn introducing [[Gamma matrices|Dirac matrices]]. In a modern context, the Klein–Gordon equation describes [[Scalar boson|spin-less]] particles, while the Dirac equation describes [[Fermion|spin-1/2]] particles.

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== Definition ==
=== Preliminaries ===
Introductory courses on physics or chemistry typically introduce the Schrödinger equation in a way that can be appreciated knowing only the concepts and notations of basic [[calculus]], particularly [[derivative]]s with respect to space and time. A special case of the Schrödinger equation that admits a statement in those terms is the position-space Schrödinger equation for a single nonrelativistic particle in one dimension:
<math display="block">i\hbar\frac{\partial}{\partial t} \Psi(x,t) = \left [ - \frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} + V(x,t)\right ] \Psi(x,t).</math>
Here, <math>\Psi(x,t)</math> is a wave function, a function that assigns a [[complex number]] to each point <math>x</math> at each time <math>t</math>. The parameter <math>m</math> is the mass of the particle, and <math>V(x,t)</math> is the ''[[Scalar Potential|potential]]'' that represents the environment in which the particle exists.<ref name="Zwiebach2022">{{cite book|first=Barton |last=Zwiebach |author-link=Barton Zwiebach |title=Mastering Quantum Mechanics: Essentials, Theory, and Applications |publisher=MIT Press |year=2022 |isbn=978-0-262-04613-8 |oclc=1347739457}}</ref>{{rp|74}} The constant <math>i</math> is the [[imaginary unit]], and <math>\hbar</math> is the reduced [[Planck constant]], which has units of [[Action (physics)|action]] ([[energy]] multiplied by time).<ref name="Zwiebach2022"/>{{rp|10}}
[[File:Wavepacket-a2k4-en.gif|300px|thumb|Complex plot of a [[wave function]] that satisfies the nonrelativistic [[Free particle#Quantum free particle|free]] Schrödinger equation with {{math|1=''V'' = 0}}.  For more details see [[Wave packet#Gaussian wave packets in quantum mechanics|wave packet]]]]

Broadening beyond this simple case, the [[mathematical formulation of quantum mechanics]] developed by [[Paul Dirac]],<ref>{{cite book|first=Paul Adrien Maurice |last=Dirac |author-link=Paul Dirac |title=The Principles of Quantum Mechanics |title-link=The Principles of Quantum Mechanics |publisher=Clarendon Press |location=Oxford |year=1930}}</ref> [[David Hilbert]],<ref>{{cite book|first=David |last=Hilbert |author-link=David Hilbert |title=Lectures on the Foundations of Physics 1915–1927: Relativity, Quantum Theory and Epistemology |publisher=Springer |doi=10.1007/b12915 |editor-first1=Tilman |editor-last1=Sauer |editor-first2=Ulrich |editor-last2=Majer |year=2009 |isbn=978-3-540-20606-4 |oclc=463777694}}</ref> [[John von Neumann]],<ref>{{cite book|first=John |last=von Neumann |author-link=John von Neumann |title=Mathematische Grundlagen der Quantenmechanik |publisher=Springer |location=Berlin |year=1932}} English translation: {{cite book|title=Mathematical Foundations of Quantum Mechanics |title-link=Mathematical Foundations of Quantum Mechanics |publisher=Princeton University Press |year=1955 |translator-first=Robert T. |translator-last=Beyer |translator-link=Robert T. Beyer}}</ref> and [[Hermann Weyl]]<ref>{{cite book| first=Hermann |last=Weyl |author-link=Hermann Weyl |title=The Theory of Groups and Quantum Mechanics |orig-year=1931 |publisher=Dover |year=1950 |isbn=978-0-486-60269-1 |translator-first=H. P. |translator-last=Robertson |translator-link=Howard P. Robertson}} Translated from the German {{cite book |title=Gruppentheorie und Quantenmechanik |title-link=Gruppentheorie und Quantenmechanik |year=1931 |edition=2nd |publisher={{Interlanguage link|S. Hirzel Verlag|de}}}}</ref> defines the state of a quantum mechanical system to be a vector <math>|\psi\rangle</math> belonging to a [[Separable space|separable]] [[complex number|complex]] [[Hilbert space]] <math>\mathcal H</math>. This vector is postulated to be normalized under the Hilbert space's inner product, that is, in [[Dirac notation]] it obeys <math>\langle \psi | \psi \rangle = 1</math>. 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 [[square-integrable function]]s <math>L^2</math>, while the Hilbert space for the [[Spin (physics)|spin]] of a single proton is the two-dimensional [[complex vector space]] <math>\Complex^2</math> with the usual inner product.<ref name="Zwiebach2022"/>{{rp|322}}

Physical quantities of interest – position, momentum, energy, spin – are represented by [[observable]]s, which are [[self-adjoint operator]]s acting on the Hilbert space. A wave function 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 \lambda | \psi\rangle|^2</math>, where <math> |\lambda\rangle</math> is its associated 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 [[Projection-valued measure#Application in quantum mechanics|projector]] onto its associated eigenspace.{{refn|group=note|This rule for obtaining probabilities from a state vector implies that vectors that only differ by an overall phase are physically equivalent; <math>|\psi\rangle</math> and <math>e^{i\alpha}|\psi\rangle</math> represent the same quantum states. In other words, the possible states are points in the [[projective space]] of a Hilbert space, usually called the [[projective Hilbert space]].}}

A momentum eigenstate would be a perfectly monochromatic wave of infinite extent, which is not square-integrable. Likewise a [[Position operator#Eigenstates|position eigenstate]] would be a [[Dirac delta function|Dirac delta distribution]], not square-integrable and technically not a function at all. Consequently, neither can belong to the particle's Hilbert space. Physicists sometimes regard these eigenstates, composed of elements outside the Hilbert space, as "[[Dirac delta function#Quantum mechanics|generalized eigenvectors]]". These are used for calculational convenience and do not represent physical states.<ref>{{cite book | last=Hall | first=B. C. | title=Quantum Theory for Mathematicians | publisher=Springer |series=Graduate Texts in Mathematics | volume=267 | year=2013 | bibcode=2013qtm..book.....H | isbn=978-1461471158|chapter= Chapter 6: Perspectives on the Spectral Theorem}}</ref><ref name = "Cohen-Tannoudji"/>{{rp|100–105}} Thus, a position-space wave function <math>\Psi(x,t)</math> as used above can be written as the inner product of a time-dependent state vector <math>|\Psi(t)\rangle</math> with unphysical but convenient "position eigenstates" <math>|x\rangle</math>:
<math display="block">\Psi(x,t) = \langle x | \Psi(t) \rangle.</math>

=== Time-dependent equation ===
[[File:StationaryStatesAnimation.gif|300px|thumb|right|Each of these three rows is a wave function which satisfies the time-dependent Schrödinger equation for a [[quantum harmonic oscillator|harmonic oscillator]]. Left: The real part (blue) and imaginary part (red) of the wave function. Right: The [[probability distribution]] of finding the particle with this wave function at a given position. The top two rows are examples of '''[[stationary state]]s''', which correspond to [[standing wave]]s. The bottom row is an example of a state which is ''not'' a stationary state.]]
The form of the Schrödinger equation depends on the physical situation. The most general form is the time-dependent Schrödinger equation, which gives a description of a system evolving with time:<ref name=Shankar1994>{{cite book | last=Shankar | first=R. | author-link=Ramamurti Shankar | year=1994 | title=Principles of Quantum Mechanics | title-link=Principles of Quantum Mechanics | edition=2nd | publisher=Kluwer Academic/Plenum Publishers | isbn=978-0-306-44790-7}}</ref>{{rp|143}}
{{Equation box 1
|indent=:
|title='''Time-dependent Schrödinger equation'''&nbsp;''(general)''
|equation=<math qid=Q165498>i \hbar \frac{d}{d t}\vert\Psi(t)\rangle = \hat H\vert\Psi(t)\rangle</math>
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|border colour = rgb(80,200,120)
|background colour = rgb(80,200,120,10%)}}
where <math>t</math> is time, <math>\vert\Psi(t)\rangle</math> is the state vector of the quantum system (<math>\Psi</math> being the Greek letter [[psi (letter)|psi]]), and <math>\hat{H}</math> is an observable, the [[Hamiltonian (quantum mechanics)|Hamiltonian]] [[operator (physics)|operator]].

The term "Schrödinger equation" can refer to both the general equation, or the specific nonrelativistic version. The general equation is indeed quite general, used throughout quantum mechanics, for everything from the [[Dirac equation]] to [[quantum field theory]], by plugging in diverse expressions for the Hamiltonian. The specific nonrelativistic version is an approximation that yields accurate results in many situations, but only to a certain extent (see [[relativistic quantum mechanics]] and [[relativistic quantum field theory]]).

To apply the Schrödinger equation, write down the [[Hamiltonian (quantum mechanics)|Hamiltonian]] for the system, accounting for the [[Kinetic energy|kinetic]] and [[Potential energy|potential]] energies of the particles constituting the system, then insert it into the Schrödinger equation. The resulting partial [[differential equation]] is solved for the wave function, which contains information about the system. In practice, the square of the absolute value of the wave function at each point is taken to define a [[probability density function]].<ref name="Zwiebach2022"/>{{rp|78}} For example, given a wave function in position space <math>\Psi(x,t)</math> as above, we have
<math display="block">\Pr(x,t) = |\Psi(x,t)|^2.</math>

=== <span class="anchor" id="Time independent equation"></span> Time-independent equation ===
The time-dependent Schrödinger equation described above predicts that wave functions can form [[standing wave]]s, called [[stationary state]]s. These states are particularly important as their individual study later simplifies the task of solving the time-dependent Schrödinger equation for ''any'' state. Stationary states can also be described by a simpler form of the Schrödinger equation, the time-independent Schrödinger equation.

{{Equation box 1
 |indent=:
 |title='''Time-independent Schrödinger equation''' (''general'')
 |equation=<math>\operatorname{\hat H}|\Psi\rangle = E |\Psi\rangle </math>
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 |border colour = rgb(80,200,120)
 |background colour = rgb(80,200,120,10%)
}}
where <math>E</math> is the energy of the system.<ref name="Zwiebach2022"/>{{rp|134}} This is only used when the [[Hamiltonian (quantum mechanics)|Hamiltonian]] itself is not dependent on time explicitly. However, even in this case the total wave function is dependent on time as explained in the section on [[#Properties|linearity]] below. In the language of [[linear algebra]], this equation is an [[Eigenvalues and eigenvectors|eigenvalue equation]]. Therefore, the wave function is an [[eigenfunction]] of the Hamiltonian operator with corresponding eigenvalue(s) <math>E</math>.

== Properties ==
=== Linearity ===
The Schrödinger equation is a [[linear differential equation]], meaning that if two state vectors <math>|\psi_1\rangle</math> and <math>|\psi_2\rangle</math> are solutions, then so is any [[linear combination]]
<math display="block"> |\psi\rangle = a|\psi_1\rangle + b |\psi_2\rangle </math>
of the two state vectors where {{mvar|a}} and {{mvar|b}} are any complex numbers.<ref name="rieffel"/>{{rp|25}} Moreover, the sum can be extended for any number of state vectors. This property allows [[Quantum superposition|superpositions of quantum states]] to be solutions of the Schrödinger equation. Even more generally, it holds that a general solution to the Schrödinger equation can be found by taking a weighted sum over a basis of states. A choice often employed is the basis of [[energy operator|energy]] eigenstates, which are solutions of the time-independent Schrödinger equation. In this basis, a time-dependent state vector <math>|\Psi(t)\rangle</math> can be written as the linear combination
<math display="block">|\Psi(t)\rangle = \sum_{n} A_n e^{ {-iE_n t}/\hbar} |\psi_{E_n}\rangle , </math>
where <math>A_n</math> are complex numbers and the vectors <math>|\psi_{E_n}\rangle</math> are solutions of the time-independent equation <math>\hat H |\psi_{E_n}\rangle = E_n |\psi_{E_n}\rangle</math>.

=== Unitarity ===
{{Further|Wigner's theorem|Stone's theorem on one-parameter unitary groups{{!}}Stone's theorem}}

Holding the Hamiltonian <math>\hat{H}</math> constant, the Schrödinger equation has the solution<ref name="Shankar1994" />
<math display="block"> |\Psi(t)\rangle = e^{-i\hat{H}t/\hbar }|\Psi(0)\rangle.</math>
The operator <math>\hat{U}(t) = e^{-i\hat{H}t/\hbar}</math> is known as the time-evolution operator, and it is [[unitarity (physics)|unitary]]: it preserves the inner product between vectors in the Hilbert space.<ref name="rieffel">{{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.|date=2011-03-04|publisher=MIT Press|isbn=978-0-262-01506-6|language=en|author-link=Eleanor Rieffel}}</ref> Unitarity is a general feature of time evolution under the Schrödinger equation. If the initial state is <math>|\Psi(0)\rangle</math>, then the state at a later time <math>t</math> will be given by
<math display="block"> |\Psi(t)\rangle = \hat{U}(t) |\Psi(0)\rangle </math>
for some unitary operator <math>\hat{U}(t)</math>. Conversely, suppose that <math>\hat{U}(t)</math> is a continuous family of unitary operators parameterized by <math>t</math>. [[Without loss of generality]],<ref>{{cite web | last=Yaffe | first=Laurence G. | url=https://courses.washington.edu/partsym/12aut/ch06.pdf |title=Chapter 6: Symmetries | website=Physics 226: Particles and Symmetries | year=2015 |access-date=2021-01-01}}</ref> the parameterization can be chosen so that <math>\hat{U}(0)</math> is the identity operator and that <math>\hat{U}(t/N)^N = \hat{U}(t)</math> for any <math>N > 0</math>. Then <math>\hat{U}(t)</math> depends upon the parameter <math>t</math> in such a way that
<math display="block" id="unitary operator given self-adjoint operator">\hat{U}(t) = e^{-i\hat{G}t} </math>
for some self-adjoint operator <math>\hat{G}</math>, called the ''generator'' of the family <math>\hat{U}(t)</math>. A Hamiltonian is just such a generator (up to the factor of the Planck constant that would be set to 1 in [[natural units]]).
To see that the generator is Hermitian, note that with <math>\hat{U}(\delta t) \approx \hat{U}(0)-i\hat{G} \delta t</math>, we have
<math display="block">\hat{U}(\delta t)^\dagger \hat{U}(\delta t)\approx(\hat{U}(0)^\dagger+i\hat{G}^\dagger \delta t)(\hat{U}(0)-i\hat{G}\delta t)=I+i\delta t(\hat{G}^\dagger-\hat{G})+O(\delta t^2),</math> so <math>\hat{U}(t)</math> is unitary only if, to first order, its derivative is Hermitian.<ref>{{cite book |last1=Sakurai |first1=J. J. |last2=Napolitano |first2=J. |author-link1=J. J. Sakurai |title=Modern Quantum Mechanics |title-link=Modern Quantum Mechanics |date=2017 |publisher=Cambridge University Press |location=Cambridge |page=68 |edition=Second |isbn=978-1-108-49999-6 |oclc=1105708539}}</ref>

=== Changes of basis ===
The Schrödinger equation is often presented using quantities varying as functions of position, but as a vector-operator equation it has a valid representation in any arbitrary complete basis of [[Bra–ket notation|kets]] in [[Hilbert space]]. As mentioned above, "bases" that lie outside the physical Hilbert space are also employed for calculational purposes. This is illustrated by the ''position-space'' and ''momentum-space'' Schrödinger equations for a nonrelativistic, spinless particle.<ref name="Cohen-Tannoudji" />{{rp|182}} The Hilbert space for such a particle is the space of complex square-integrable functions on three-dimensional Euclidean space, and its Hamiltonian is the sum of a kinetic-energy term that is quadratic in the momentum operator and a potential-energy term:
<math display="block">i\hbar \frac{d}{dt}|\Psi(t)\rangle = \left(\frac{1}{2m}\hat{p}^2 + \hat{V}\right)|\Psi(t)\rangle.</math>
Writing <math>\mathbf{r}</math> for a three-dimensional position vector and <math>\mathbf{p}</math> for a three-dimensional momentum vector, the position-space Schrödinger equation is
<math display="block">i\hbar\frac{\partial}{\partial t} \Psi(\mathbf{r},t) = - \frac{\hbar^2}{2m} \nabla^2 \Psi(\mathbf{r},t) + V(\mathbf{r}) \Psi(\mathbf{r},t).</math>
The momentum-space counterpart involves the [[Fourier transform]]s of the wave function and the potential:
<math display="block"> i\hbar \frac{\partial}{\partial t} \tilde{\Psi}(\mathbf{p}, t) = \frac{\mathbf{p}^2}{2m} \tilde{\Psi}(\mathbf{p},t) + (2\pi\hbar)^{-3/2} \int d^3 \mathbf{p}' \, \tilde{V}(\mathbf{p} - \mathbf{p}') \tilde{\Psi}(\mathbf{p}',t).</math>
The functions <math>\Psi(\mathbf{r},t)</math> and <math>\tilde{\Psi}(\mathbf{p},t)</math> are derived from <math>|\Psi(t)\rangle</math> by
<math display="block">\Psi(\mathbf{r},t) = \langle \mathbf{r} | \Psi(t)\rangle,</math>
<math display="block">\tilde{\Psi}(\mathbf{p},t) = \langle \mathbf{p} | \Psi(t)\rangle,</math>
where <math>|\mathbf{r}\rangle</math> and <math>|\mathbf{p}\rangle</math> do not belong to the Hilbert space itself, but have well-defined inner products with all elements of that space.

When restricted from three dimensions to one, the position-space equation is just the first form of the Schrödinger equation given [[#Preliminaries|above]]. The relation between position and momentum in quantum mechanics can be appreciated in a single dimension. In [[canonical quantization]], the classical variables <math>x</math> and <math>p</math> are promoted to self-adjoint operators <math>\hat{x}</math> and <math>\hat{p}</math> that satisfy the [[canonical commutation relation]]
<math display="block">[\hat{x}, \hat{p}] = i\hbar.</math>
This implies that<ref name="Cohen-Tannoudji" />{{rp|190}}
<math display="block">\langle x | \hat{p} | \Psi \rangle = -i\hbar \frac{d}{dx} \Psi(x),</math>
so the action of the momentum operator <math>\hat{p}</math> in the position-space representation is <math display="inline">-i\hbar \frac{d}{dx}</math>. Thus, <math>\hat{p}^2</math> becomes a [[second derivative]], and in three dimensions, the second derivative becomes the [[Laplace operator|Laplacian]] <math>\nabla^2</math>.

The canonical commutation relation also implies that the position and momentum operators are Fourier conjugates of each other. Consequently, functions originally defined in terms of their position dependence can be converted to functions of momentum using the Fourier transform.<ref name="Zwiebach2022"/>{{rp|103–104}} In [[solid-state physics]], the Schrödinger equation is often written for functions of momentum, as [[Bloch's theorem]] ensures the periodic crystal lattice potential couples <math>\tilde{\Psi}(p) </math> with <math>\tilde{\Psi}(p + \hbar K) </math> for only discrete [[reciprocal lattice]] vectors <math>K </math>. This makes it convenient to solve the momentum-space Schrödinger equation at each [[Crystal momentum|point]] in the [[Brillouin zone]] independently of the other points in the Brillouin zone.<ref name="Ashcroft1976">{{cite book|first1=Neil W. |last1=Ashcroft |author-link1=Neil Ashcroft |first2=N. David |last2=Mermin |author-link2=N. David Mermin |title=Solid State Physics |title-link=Ashcroft and Mermin |year=1976 |publisher=Harcourt College Publishers |isbn=0-03-083993-9}}</ref>{{rp|138}}

=== Probability current ===
{{Main|Probability current|Continuity equation}}

The Schrödinger equation is consistent with [[conservation of probability|local probability conservation]].<ref name = "Cohen-Tannoudji"/>{{rp|238}} It also ensures that a normalized wavefunction remains normalized after time evolution. In matrix mechanics, this means that the [[Time evolution|time evolution operator]] is a [[unitary operator]].<ref name=":1">{{Cite book |last1=Sakurai |first1=Jun John |title=Modern quantum mechanics |last2=Napolitano |first2=Jim |date=2021 |publisher=Cambridge University Press |isbn=978-1-108-47322-4 |edition=3rd |location=Cambridge}}</ref> In contrast to, for example, the Klein Gordon equation, although a redefined inner product of a wavefunction can be time independent, the total volume integral of modulus square of the wavefunction need not be time independent.<ref>{{Cite journal |last=Mostafazadeh |first=Ali |date=2003-01-07 |title=Hilbert Space Structures on the Solution Space of Klein-Gordon Type Evolution Equations |journal=Classical and Quantum Gravity |volume=20 |issue=1 |pages=155–171 |doi=10.1088/0264-9381/20/1/312 |arxiv=math-ph/0209014 |bibcode=2003CQGra..20..155M |issn=0264-9381}}</ref>

The continuity equation for probability in non relativistic quantum mechanics is stated as: 
<math display="block">\frac{\partial}{\partial t} \rho\left(\mathbf{r},t\right) + \nabla \cdot \mathbf{j} = 0, </math>where
<math display="block"> \mathbf{j} = \frac{1}{2m} \left( \Psi^*\hat{\mathbf{p}}\Psi - \Psi\hat{\mathbf{p}}\Psi^* \right) = -\frac{i\hbar}{2m}(\psi^*\nabla\psi-\psi\nabla\psi^*) = \frac \hbar m \operatorname{Im} (\psi^*\nabla \psi)     </math>
is the [[probability current]] or probability flux (flow per unit area).

If the wavefunction is represented as <math display="inline">\psi( {\bf x},t)=\sqrt{\rho({\bf x},t)}\exp\left(\frac{i S({\bf x},t)}{\hbar}\right), </math> where <math>S(\mathbf x,t) </math> is a real function which represents the complex phase of the wavefunction, then the probability flux is calculated as:<math display="block"> \mathbf{j} = \frac{\rho \nabla S}  {m} </math>Hence, the spatial variation of the phase of a wavefunction is said to characterize the probability flux of the wavefunction. Although the <math display="inline"> \frac{ \nabla S}  {m} </math> term appears to play the role of velocity, it does not represent velocity at a point since simultaneous measurement of position and velocity violates [[uncertainty principle]].<ref name=":1" />

=== Separation of variables ===
If the Hamiltonian is not an explicit function of time, Schrödinger's equation reads: 
<math display="block">i\hbar\frac{\partial}{\partial t} \Psi(\mathbf{r},t) = \left [ - \frac{\hbar^2}{2m}\nabla^2 + V(\mathbf{r})\right ] \Psi(\mathbf{r},t).</math> The operator on the left side depends only on time; the one on the right side depends only on space.
Solving the equation by [[separation of variables]] means seeking a solution of the form of a product of spatial and temporal parts<ref>{{Cite journal|last=Singh|first=Chandralekha|author-link=Chandralekha Singh|date=March 2008|title=Student understanding of quantum mechanics at the beginning of graduate instruction|url=http://aapt.scitation.org/doi/10.1119/1.2825387|journal=American Journal of Physics|language=en|volume=76|issue=3|pages=277–287|doi=10.1119/1.2825387|arxiv=1602.06660 |bibcode=2008AmJPh..76..277S |s2cid=118493003 |issn=0002-9505}}</ref>
<math display="block">\Psi(\mathbf{r},t)=\psi(\mathbf{r})\tau(t),</math>
where <math>\psi(\mathbf{r})</math> is a function of all the spatial coordinate(s) of the particle(s) constituting the system only, and <math>\tau(t)</math> is a function of time only. Substituting this expression for <math>\Psi</math> into the time dependent left hand side shows that <math>\tau(t)</math> is a phase factor:
<math display="block"> \Psi(\mathbf{r},t) = \psi(\mathbf{r}) e^{-i{E t/\hbar}}.</math>
A solution of this type is called ''stationary,'' since the only time dependence is a phase factor that cancels when the probability density is calculated via the Born rule.<ref name=Shankar1994/>{{rp|143ff}}

The spatial part of the full wave function solves the equation<ref name="Adams Sigel Mlynek 1994 pp. 143–210">{{cite journal | last1=Adams | first1=C.S | last2=Sigel | first2=M | last3=Mlynek | first3=J | title=Atom optics | journal=Physics Reports | publisher=Elsevier BV | volume=240 | issue=3 | year=1994 | issn=0370-1573 | doi=10.1016/0370-1573(94)90066-3 | pages=143–210| bibcode=1994PhR...240..143A | doi-access=free }}</ref>
<math display="block"> \nabla^2\psi(\mathbf{r}) + \frac{2m}{\hbar^2} \left [E - V(\mathbf{r})\right ] \psi(\mathbf{r}) = 0,</math>
where the energy <math>E</math> appears in the phase factor.

This generalizes to any number of particles in any number of dimensions (in a time-independent potential): the [[standing wave]] solutions of the time-independent equation are the states with definite energy, instead of a probability distribution of different energies. In physics, these standing waves are called "[[stationary state]]s" or "[[energy eigenstate]]s"; in chemistry they are called "[[atomic orbital]]s" or "[[molecular orbital]]s". Superpositions of energy eigenstates change their properties according to the relative phases between the energy levels. The energy eigenstates form a basis: any wave function may be written as a sum over the discrete energy states or an integral over continuous energy states, or more generally as an integral over a measure. This is an example of the [[spectral theorem]], and in a finite-dimensional state space it is just a statement of the completeness of the eigenvectors of a [[Hermitian matrix]].

Separation of variables can also be a useful method for the time-independent Schrödinger equation. For example, depending on the symmetry of the problem, the [[Cartesian coordinates|Cartesian axes]] might be separated, as in
<math display="block">\psi(\mathbf{r}) = \psi_x(x)\psi_y(y)\psi_z(z),</math>
or [[spherical coordinates|radial and angular coordinates]] might be separated:
<math display="block">\psi(\mathbf{r}) = \psi_r(r)\psi_\theta(\theta)\psi_\phi(\phi).</math>

== Examples ==
{{See also|List of quantum-mechanical systems with analytical solutions}}

=== Particle in a box ===
[[File:Infinite potential well.svg|thumb|1-dimensional potential energy box (or infinite potential well)]]
{{Main|Particle in a box}}
The particle in a one-dimensional potential energy box is the most mathematically simple example where restraints lead to the quantization of energy levels. The box is defined as having zero potential energy ''inside'' a certain region and infinite potential energy ''outside''.<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>{{rp|77–78}} For the one-dimensional case in the <math>x</math> direction, the time-independent Schrödinger equation may be written
<math display="block"> - \frac {\hbar ^2}{2m} \frac {d ^2 \psi}{dx^2} = E \psi.</math>

With the differential operator defined by
<math display="block"> \hat{p}_x = -i\hbar\frac{d}{dx} </math>
the previous equation is evocative of the [[Kinetic energy#Kinetic energy of rigid bodies|classic kinetic energy analogue]],
<math display="block"> \frac{1}{2m} \hat{p}_x^2 = E,</math>
with state <math>\psi</math> in this case having energy <math>E</math> coincident with the kinetic energy of the particle.

The general solutions of the Schrödinger equation for the particle in a box are
<math display="block"> \psi(x) = A e^{ikx} + B e ^{-ikx} \qquad\qquad E =  \frac{\hbar^2 k^2}{2m}</math>
or, from [[Euler's formula]],
<math display="block"> \psi(x) = C \sin(kx) + D \cos(kx).</math>

The infinite potential walls of the box determine the values of <math>C, D, </math> and <math>k</math> at <math>x=0</math> and <math>x=L</math> where <math>\psi</math> must be zero. Thus, at <math>x=0</math>,
<math display="block">\psi(0) = 0 = C\sin(0) + D\cos(0) = D</math>
and <math>D=0</math>. At <math>x=L</math>,
<math display="block"> \psi(L) = 0 = C\sin(kL),</math>
in which <math>C</math> cannot be zero as this would conflict with the postulate that <math>\psi</math> has norm 1. Therefore, since <math>\sin(kL)=0</math>, <math>kL</math> must be an integer multiple of <math>\pi</math>,
<math display="block">k = \frac{n\pi}{L}\qquad\qquad n=1,2,3,\ldots.</math>

This constraint on <math>k</math> implies a constraint on the energy levels, yielding
<math display="block">E_n = \frac{\hbar^2 \pi^2 n^2}{2mL^2} = \frac{n^2h^2}{8mL^2}.</math>

A [[finite potential well]] is the generalization of the infinite potential well problem to potential wells having finite depth. The finite potential well problem is mathematically more complicated than the infinite particle-in-a-box problem as the wave function is not pinned to zero at the walls of the well. Instead, the wave function must satisfy more complicated mathematical boundary conditions as it is nonzero in regions outside the well. Another related problem is that of the [[rectangular potential barrier]], which furnishes a model for the [[quantum tunneling]] effect that plays an important role in the performance of modern technologies such as [[flash memory]] and [[scanning tunneling microscope|scanning tunneling microscopy]].

=== Harmonic oscillator ===
[[File:QuantumHarmonicOscillatorAnimation.gif|300px|thumb|right|A [[harmonic oscillator]] in classical mechanics (A–B) and quantum mechanics (C–H). In (A–B), a ball, attached to a [[Hooke's law|spring]], oscillates back and forth. (C–H) are six solutions to the Schrödinger Equation for this situation. The horizontal axis is position, the vertical axis is the real part (blue) or imaginary part (red) of the [[wave function]]. [[Stationary state]]s, or energy eigenstates, which are solutions to the time-independent Schrödinger equation, are shown in C, D, E, F, but not G or H.]]

{{Main|Quantum harmonic oscillator}}

The Schrödinger equation for this situation is
<math display="block"> E\psi = -\frac{\hbar^2}{2m}\frac{d^2}{d x^2}\psi + \frac{1}{2} m\omega^2 x^2\psi, </math>
where <math> x </math> is the displacement and <math> \omega </math> the angular frequency. Furthermore, it can be used to describe approximately a wide variety of other systems, including [[Molecular vibration|vibrating atoms, molecules]],<ref>{{cite book|title=Physical Chemistry |first=P. W. |last=Atkins |publisher=Oxford University Press |year=1978 |isbn=0-19-855148-7}}</ref> and atoms or ions in lattices,<ref>{{cite book|title=Solid State Physics |edition=2nd |first1=J. R. |last1=Hook |first2=H. E. |last2=Hall |series=Manchester Physics Series |publisher=John Wiley & Sons |year=2010 |isbn=978-0-471-92804-1}}</ref> and approximating other potentials near equilibrium points. It is also the [[Perturbation theory (quantum mechanics)#Applying perturbation theory|basis of perturbation methods]] in quantum mechanics.

The solutions in position space are
<math display="block"> \psi_n(x) = \sqrt{\frac{1}{2^n\,n!}} \ \left(\frac{m\omega}{\pi \hbar}\right)^{1/4} \ e^{
- \frac{m\omega x^2}{2 \hbar}} \ \mathcal{H}_n\left(\sqrt{\frac{m\omega}{\hbar}} x \right), </math>
where <math>n \in  \{0, 1, 2, \ldots \}</math>, and the functions <math> \mathcal{H}_n </math> are the [[Hermite polynomials]] of order <math> n </math>. The solution set may be generated by
<math display="block">\psi_n(x) = \frac{1}{\sqrt{n!}} \left( \sqrt{\frac{m \omega}{2 \hbar}} \right)^{n} \left( x - \frac{\hbar}{m \omega} \frac{d}{dx}\right)^n \left( \frac{m \omega}{\pi \hbar} \right)^{\frac{1}{4}} e^{\frac{-m \omega x^2}{2\hbar}}.</math>

The eigenvalues are
<math display="block">  E_n = \left(n + \frac{1}{2} \right) \hbar \omega. </math>

The case <math>  n = 0 </math> is called the [[ground state]], its energy is called the [[zero-point energy]], and the wave function is a [[Normal distribution|Gaussian]].<ref>{{Cite book|title=A Modern Approach to Quantum Mechanics |last=Townsend |first=John S. |publisher=University Science Books|year=2012|isbn=978-1-891389-78-8|pages=247–250, 254–5, 257, 272 |chapter=Chapter 7: The One-Dimensional Harmonic Oscillator}}</ref>

The harmonic oscillator, like the particle in a box, illustrates the generic feature of the Schrödinger equation that the energies of bound eigenstates are discretized.<ref name="Cohen-Tannoudji" />{{rp|352}}

=== Hydrogen atom ===
[[File:Hydrogen Density Plots.png|thumb|[[Wave function]]s of the [[electron]] in a hydrogen atom at different [[energy level]]s. They are plotted according to solutions of the Schrödinger equation.]]
The Schrödinger equation for the electron in a [[hydrogen atom]] (or a hydrogen-like atom) is
<math display="block"> E \psi = -\frac{\hbar^2}{2\mu}\nabla^2\psi - \frac{q^2}{4\pi\varepsilon_0 r}\psi </math>
where <math> q </math> is the electron charge, <math> \mathbf{r} </math> is the position of the electron relative to the nucleus, <math> r = |\mathbf{r}| </math> is the magnitude of the relative position, the potential term is due to the [[Coulomb's law|Coulomb interaction]], wherein <math> \varepsilon_0 </math> is the [[permittivity of free space]] and
<math display="block"> \mu = \frac{m_q m_p}{m_q+m_p} </math>
is the 2-body [[reduced mass]] of the hydrogen [[Nucleus (atomic structure)|nucleus]] (just a [[proton]]) of mass <math> m_p </math> and the electron of mass <math> m_q </math>. The negative sign arises in the potential term since the proton and electron are oppositely charged. The reduced mass in place of the electron mass is used since the electron and proton together orbit each other about a common center of mass, and constitute a two-body problem to solve. The motion of the electron is of principal interest here, so the equivalent one-body problem is the motion of the electron using the reduced mass.

The Schrödinger equation for a hydrogen atom can be solved by separation of variables.<ref>{{cite book|title=Physics for Scientists and Engineers – with Modern Physics |edition=6th |first1=P. A. |last1=Tipler |first2=G. |last2=Mosca |publisher=Freeman |year=2008 |isbn=978-0-7167-8964-2}}</ref> In this case, [[spherical polar coordinates]] are the most convenient. Thus,
<math display="block"> \psi(r,\theta,\varphi) = R(r)Y_\ell^m(\theta, \varphi) = R(r)\Theta(\theta)\Phi(\varphi),</math>
where {{math|''R''}} are radial functions and <math> Y^m_l (\theta, \varphi) </math> are [[spherical harmonic]]s of degree <math> \ell </math> and order <math> m </math>. This is the only atom for which the Schrödinger equation has been solved for exactly. Multi-electron atoms require approximate methods. The family of solutions are:<ref>{{cite book|first=David J. |last=Griffiths |author-link=David J. Griffiths |title=Introduction to Elementary Particles|url=https://books.google.com/books?id=w9Dz56myXm8C&pg=PA162 | access-date=27 June 2011|year=2008|publisher=Wiley-VCH|isbn=978-3-527-40601-2|pages=162–}}</ref>
<math display="block"> \psi_{n\ell m}(r,\theta,\varphi) = \sqrt {\left (  \frac{2}{n a_0} \right )^3\frac{(n-\ell-1)!}{2n[(n+\ell)!]} } e^{- r/na_0} \left(\frac{2r}{na_0}\right)^\ell L_{n-\ell-1}^{2\ell+1}\left(\frac{2r}{na_0}\right) \cdot Y_{\ell}^m(\theta, \varphi ) </math>
where
* <math> a_0 = \frac{4 \pi \varepsilon_0 \hbar^2}{m_q q^2} </math> is the [[Bohr radius]],
* <math> L_{n-\ell-1}^{2\ell+1}(\cdots) </math> are the [[Laguerre polynomial#Generalized Laguerre polynomials|generalized Laguerre polynomials]] of degree <math> n - \ell - 1 </math>,
* <math> n, \ell, m </math> are the [[principal quantum number|principal]], [[azimuthal quantum number|azimuthal]], and [[magnetic quantum number|magnetic]] [[quantum numbers]] respectively, which take the values <math>n = 1, 2, 3, \dots,</math> <math>\ell = 0, 1, 2, \dots, n - 1,</math> <math>m = -\ell, \dots, \ell.</math>

=== Approximate solutions ===
It is typically not possible to solve the Schrödinger equation exactly for situations of physical interest. Accordingly, approximate solutions are obtained using techniques like [[Variational method (quantum mechanics)|variational methods]] and [[WKB approximation]]. It is also common to treat a problem of interest as a small modification to a problem that can be solved exactly, a method known as [[perturbation theory (quantum mechanics)|perturbation theory]].

== Semiclassical limit ==
One simple way to compare classical to quantum mechanics is to consider the time-evolution of the ''expected'' position and ''expected'' momentum, which can then be compared to the time-evolution of the ordinary position and momentum in classical mechanics.<ref name=":0" />{{rp|302}} The quantum expectation values satisfy the [[Ehrenfest theorem]]. For a one-dimensional quantum particle moving in a potential <math>V</math>, the Ehrenfest theorem says
<math display="block">m\frac{d}{dt}\langle x\rangle = \langle p\rangle;\quad \frac{d}{dt}\langle p\rangle =  -\left\langle V'(X)\right\rangle.</math>
Although the first of these equations is consistent with the classical behavior, the second is not: If the pair <math>(\langle X\rangle, \langle P\rangle)</math> were to satisfy Newton's second law, the right-hand side of the second equation would have to be
<math display="block">-V'\left(\left\langle X\right\rangle\right)</math>
which is typically not the same as <math>-\left\langle V'(X)\right\rangle</math>. For a general <math>V'</math>, therefore, quantum mechanics can lead to predictions where expectation values do not mimic the classical behavior. In the case of the quantum harmonic oscillator, however, <math>V'</math> is linear and this distinction disappears, so that in this very special case, the expected position and expected momentum do exactly follow the classical trajectories.

For general systems, the best we can hope for is that the expected position and momentum will ''approximately'' follow the classical trajectories. If the wave function is highly concentrated around a point <math>x_0</math>, then <math>V'\left(\left\langle X\right\rangle\right)</math> and <math>\left\langle V'(X)\right\rangle</math> will be ''almost'' the same, since both will be approximately equal to <math>V'(x_0)</math>. In that case, the expected position and expected momentum will remain very close to the classical trajectories, at least for as long as the wave function remains highly localized in position.

The Schrödinger equation in its general form
<math display="block"> i\hbar \frac{\partial}{\partial t} \Psi\left(\mathbf{r},t\right) = \hat{H} \Psi\left(\mathbf{r},t\right)</math>
is closely related to the [[Hamilton–Jacobi equation]] (HJE)
<math display="block"> -\frac{\partial}{\partial t} S(q_i,t) = H\left(q_i,\frac{\partial S}{\partial q_i},t \right) </math>
where <math>S</math> is the classical [[action (physics)|action]] and <math>H</math> is the [[Hamiltonian mechanics|Hamiltonian function]] (not operator).<ref name=":0" />{{rp|308}} Here the [[generalized coordinates]] <math>q_i</math> for <math>i = 1, 2, 3</math> (used in the context of the HJE) can be set to the position in Cartesian coordinates as <math>\mathbf{r} = (q_1, q_2, q_3) = (x, y, z)</math>.

Substituting
<math display="block"> \Psi = \sqrt{\rho(\mathbf{r},t)} e^{iS(\mathbf{r},t)/\hbar}</math>
where <math>\rho</math> is the probability density, into the Schrödinger equation and then taking the limit <math>\hbar \to 0</math> in the resulting equation yield the [[Hamilton–Jacobi equation]].

== Density matrices==
{{main|Density matrix }}
Wave functions are not always the most convenient way to describe quantum systems and their behavior. When the preparation of a system is only imperfectly known, or when the system under investigation is a part of a larger whole, [[density matrix|density matrices]] may be used instead.<ref name=":0" />{{rp|74}} A density matrix is a [[Positive-semidefinite matrix|positive semi-definite operator]] whose [[Trace class|trace]] is equal to 1. (The term "density operator" is also used, particularly when the underlying Hilbert space is infinite-dimensional.) The set of all density matrices is [[convex set|convex]], and the extreme points are the operators that project onto vectors in the Hilbert space. These are the density-matrix representations of wave functions; in Dirac notation, they are written <math display="block"> \hat{\rho} = |\Psi\rangle\langle \Psi|.</math>

The density-matrix analogue of the Schrödinger equation for wave functions is<ref>{{cite book |title=The theory of open quantum systems| last1= Breuer |first1=Heinz|last2= Petruccione|first2=Francesco|page=110|isbn=978-0-19-852063-4 |year=2002 | publisher= Oxford University Press | url=https://books.google.com/books?id=0Yx5VzaMYm8C&pg=PA110}}</ref><ref>{{cite book|url=https://books.google.com/books?id=o-HyHvRZ4VcC&pg=PA16 |title=Statistical mechanics|last=Schwabl|first=Franz|page=16|isbn=978-3-540-43163-3|year=2002|publisher=Springer }}</ref>
<math display="block"> i \hbar \frac{\partial \hat{\rho}}{\partial t} = [\hat{H}, \hat{\rho}],</math>
where the brackets denote a [[commutator]]. This is variously known as the von Neumann equation, the Liouville–von Neumann equation, or just the Schrödinger equation for density matrices.<ref name=":0" />{{rp|312}} If the Hamiltonian is time-independent, this equation can be easily solved to yield
<math display="block">\hat{\rho}(t) = e^{-i \hat{H} t/\hbar} \hat{\rho}(0) e^{i \hat{H} t/\hbar}.</math>

More generally, if the unitary operator <math>\hat{U}(t)</math> describes wave function evolution over some time interval, then the time evolution of a density matrix over that same interval is given by
<math display="block"> \hat{\rho}(t) = \hat{U}(t) \hat{\rho}(0) \hat{U}(t)^\dagger.</math>

Unitary evolution of a density matrix conserves its [[von Neumann entropy]].<ref name=":0" />{{rp|267}}

== Relativistic quantum physics and quantum field theory ==
The one-particle Schrödinger equation described  above is valid essentially in the nonrelativistic domain. For one reason, it is essentially invariant under [[Galilean transformation]]s, which form the symmetry group of [[Newtonian dynamics]].{{refn|group=note|More precisely, the effect of a Galilean transformation upon the Schrödinger equation can be canceled by a phase transformation of the wave function that leaves the probabilities, as calculated via the Born rule, unchanged.<ref>{{cite book|last=Home|first=Dipankar|title=Conceptual Foundations of Quantum Physics |publisher=Springer US|year=2013|isbn=9781475798081|pages=4–5 |oclc=1157340444}}</ref>}} Moreover, processes that change particle number are natural in relativity, and so an equation for one particle (or any fixed number thereof) can only be of limited use.<ref name="Coleman"/> A more general form of the Schrödinger equation that also applies in relativistic situations can be formulated within [[quantum field theory]] (QFT), a framework that allows the combination of quantum mechanics with special relativity. The region in which both simultaneously apply may be described by [[relativistic quantum mechanics]]. Such descriptions may use time evolution generated by a Hamiltonian operator, as in the [[Schrödinger functional]] method.<ref>{{Cite journal|last=Symanzik|first=K.|author-link=Kurt Symanzik|date=1981-07-06|title=Schrödinger representation and Casimir effect in renormalizable quantum field theory|url=https://dx.doi.org/10.1016/0550-3213%2881%2990482-X|journal=Nuclear Physics B|language=en|volume=190|issue=1|pages=1–44|doi=10.1016/0550-3213(81)90482-X|bibcode=1981NuPhB.190....1S |issn=0550-3213|url-access=subscription}}</ref><ref>{{Cite journal|last=Kiefer|first=Claus|date=1992-03-15|title=Functional Schrödinger equation for scalar QED|url=https://link.aps.org/doi/10.1103/PhysRevD.45.2044|journal=Physical Review D|language=en|volume=45|issue=6|pages=2044–2056|doi=10.1103/PhysRevD.45.2044|pmid=10014577 |bibcode=1992PhRvD..45.2044K |issn=0556-2821|url-access=subscription}}</ref><ref>{{Cite book|last=Hatfield|first=Brian|url=https://www.worldcat.org/oclc/170230278|title=Quantum Field Theory of Point Particles and Strings|date=1992|publisher=Perseus Books|isbn=978-1-4294-8516-6|location=Cambridge, Mass.|oclc=170230278}}</ref><ref>{{Cite journal|last=Islam|first=Jamal Nazrul|date=May 1994|title=The Schrödinger equation in quantum field theory|url=http://link.springer.com/10.1007/BF02054667|journal=Foundations of Physics|language=en|volume=24|issue=5|pages=593–630|doi=10.1007/BF02054667|bibcode=1994FoPh...24..593I |s2cid=120883802 |issn=0015-9018|url-access=subscription}}</ref>

=== Klein–Gordon and Dirac equations ===
Attempts to combine quantum physics with special relativity began with building [[relativistic wave equations]] from the relativistic [[energy–momentum relation]]
<math display="block">E^2 = (pc)^2 + \left(m_0 c^2\right)^2,</math>
instead of nonrelativistic energy equations. The [[Klein–Gordon equation]] and the [[Dirac equation]] are two such equations. The Klein–Gordon equation,
<math display="block"> -\frac {1}{c^2} \frac{\partial^2}{\partial t^2} \psi + \nabla^2 \psi = \frac {m^2 c^2}{\hbar^2} \psi,</math>
was the first such equation to be obtained, even before the nonrelativistic one-particle Schrödinger equation, and applies to massive spinless particles. Historically, Dirac obtained the Dirac equation by seeking a differential equation that would be first-order in both time and space, a desirable property for a relativistic theory. Taking the "square root" of the left-hand side of the Klein–Gordon equation in this way required factorizing it into a product of two operators, which Dirac wrote using 4 × 4 matrices <math>\alpha_1,\alpha_2,\alpha_3,\beta</math>. Consequently, the wave function also became a four-component function, governed by the Dirac equation that, in free space, read
<math display="block">\left(\beta mc^2 + c\left(\sum_{n \mathop = 1}^{3}\alpha_n p_n\right)\right) \psi = i \hbar \frac{\partial\psi }{\partial t}. </math>

This has again the form of the Schrödinger equation, with the time derivative of the wave function being given by a Hamiltonian operator acting upon the wave function. Including influences upon the particle requires modifying the Hamiltonian operator. For example, the Dirac Hamiltonian for a particle of mass {{math|''m''}} and electric charge {{math|''q''}} in an electromagnetic field (described by the [[electromagnetic potential]]s {{math|''φ''}} and {{math|'''A'''}}) is:
<math display="block">\hat{H}_{\text{Dirac}}= \gamma^0 \left[c  \boldsymbol{\gamma}\cdot\left(\hat{\mathbf{p}} - q \mathbf{A}\right) + mc^2 + \gamma^0q \varphi \right],</math>
in which the {{math|1='''γ''' = (''γ''<sup>1</sup>, ''γ''<sup>2</sup>, ''γ''<sup>3</sup>)}} and {{math|''γ''<sup>0</sup>}} are the Dirac [[gamma matrices]] related to the spin of the particle. The Dirac equation is true for all {{nowrap|[[spin-1/2|spin-{{frac|2}}]]}} particles, and the solutions to the equation are {{nowrap|4-component}} [[spinor field]]s with two components corresponding to the particle and the other two for the [[antiparticle]].

For the Klein–Gordon equation, the general form of the Schrödinger equation is inconvenient to use, and in practice the Hamiltonian is not expressed in an analogous way to the Dirac Hamiltonian. The equations for relativistic quantum fields, of which the Klein–Gordon and Dirac equations are two examples, can be obtained in other ways, such as starting from a [[Lagrangian (field theory)|Lagrangian density]] and using the [[Euler–Lagrange equation]]s for fields, or using the [[representation theory of the Lorentz group]] in which certain representations can be used to fix the equation for a [[free particle]] of given spin (and mass).

In general, the Hamiltonian to be substituted in the general Schrödinger equation is not just a function of the position and momentum operators (and possibly time), but also of spin matrices. Also, the solutions to a relativistic wave equation, for a massive particle of spin {{math|''s''}}, are complex-valued {{nowrap|{{math|2(2''s'' + 1)}}-component}} [[spinor field]]s.

=== Fock space ===
As originally formulated, the Dirac equation is an equation for a single quantum particle, just like the single-particle Schrödinger equation with wave function {{nowrap|<math>\Psi(x,t)</math>.}} This is of limited use in relativistic quantum mechanics, where particle number is not fixed. Heuristically, this complication can be motivated by noting that mass–energy equivalence implies material particles can be created from energy. A common way to address this in QFT is to introduce a Hilbert space where the basis states are labeled by particle number, a so-called [[Fock space]]. The Schrödinger equation can then be formulated for quantum states on this Hilbert space.<ref name="Coleman">{{Cite book|editor-last1=Derbes |editor-first1=David  |title=Lectures Of Sidney Coleman On Quantum Field Theory |editor-last2=Ting |editor-first2=Yuan-sen |editor-last3=Chen |editor-first3=Bryan Gin-ge |editor-last4=Sohn |editor-first4=Richard |editor-last5=Griffiths |editor-first5=David |editor-last6=Hill |editor-first6=Brian |date=2018-11-08 |publisher=World Scientific Publishing |isbn=978-9-814-63253-9 |oclc=1057736838 |language=en |first=Sidney |last=Coleman |author-link=Sidney Coleman}}</ref> However, because the Schrödinger equation picks out a preferred time axis, the Lorentz invariance of the theory is no longer manifest, and accordingly, the theory is often formulated in other ways.<ref>{{Cite book|last=Srednicki|first=Mark Allen|url=https://www.worldcat.org/oclc/71808151|title=Quantum Field Theory|publisher=Cambridge University Press|year=2012|isbn=978-0-521-86449-7|location=Cambridge|oclc=71808151}}</ref>

== History ==
[[File:Erwin Schrödinger (1933).jpg|thumb|upright|[[Erwin Schrödinger]]]]
Following [[Max Planck]]'s quantization of light (see [[black-body radiation]]), [[Albert Einstein]] interpreted Planck's [[quantum|quanta]] to be [[photon]]s, [[corpuscular theory of light|particles of light]], and proposed that the [[Planck relation|energy of a photon is proportional to its frequency]], one of the first signs of [[wave–particle duality]]. Since energy and [[momentum]] are related in the same way as [[frequency]] and [[wavenumber|wave number]] in [[special relativity]], it followed that the momentum <math>p</math> of a photon is inversely proportional to its [[wavelength]] <math>\lambda</math>, or proportional to its wave number <math>k</math>:
<math display="block">p = \frac{h}{\lambda} = \hbar k,</math>
where <math>h</math> is the [[Planck constant]] and <math>\hbar = {h}/{2\pi}</math> is the reduced Planck constant. [[Louis de Broglie]] hypothesized that this is true for all particles, even particles which have mass such as electrons. He showed that, assuming that the [[matter wave]]s propagate along with their particle counterparts, electrons form [[standing wave]]s, meaning that only certain discrete rotational frequencies about the nucleus of an atom are allowed.<ref>
{{cite journal
 |last        = de Broglie
 |first       = L.
 |author-link = Louis de Broglie
 |year        = 1925
 |title       = Recherches sur la théorie des quanta
 |language    = fr
 |trans-title = On the Theory of Quanta
 |url         = http://tel.archives-ouvertes.fr/docs/00/04/70/78/PDF/tel-00006807.pdf
 |journal     = [[Annales de Physique]]
 |volume      = 10
 |issue       = 3
 |pages       = 22–128
 |doi         = 10.1051/anphys/192510030022
 |url-status  = dead
 |archive-url  = https://web.archive.org/web/20090509012910/http://www.ensmp.fr/aflb/LDB-oeuvres/De_Broglie_Kracklauer.pdf
 |archive-date = 9 May 2009
 |df           = dmy-all
 |bibcode      = 1925AnPh...10...22D
}}</ref>
These quantized orbits correspond to discrete [[energy level]]s, and de Broglie reproduced the [[Bohr model]] formula for the energy levels. The Bohr model was based on the assumed quantization of angular momentum <math>L</math> according to
<math display="block"> L = n \frac{h}{2\pi} = n\hbar.</math>
According to de Broglie, the electron is described by a wave, and a whole number of wavelengths must fit along the circumference of the electron's orbit:
<math display="block">n \lambda = 2 \pi r.</math>

This approach essentially confined the electron wave in one dimension, along a circular orbit of radius <math>r</math>.

In 1921, prior to de Broglie, Arthur C. Lunn at the University of Chicago had used the same argument based on the completion of the relativistic energy–momentum [[Four-vector|4-vector]] to derive what we now call the de Broglie relation.<ref>{{cite journal |last=Weissman |first=M. B. |author2=V. V. Iliev |author3=I. Gutman |title=A pioneer remembered: biographical notes about Arthur Constant Lunn |journal=Communications in Mathematical and in Computer Chemistry |year=2008 |volume=59 |issue=3 |pages=687–708 |url=https://match.pmf.kg.ac.rs/electronic_versions/Match59/n3/match59n3_687-708.pdf}}</ref><ref>{{cite journal |title=Alan Sokal's Hoax and A. Lunn's Theory of Quantum Mechanics |author=Samuel I. Weissman |author2=Michael Weissman |year=1997 |journal=Physics Today |volume=50 |issue=6 |page=15 |doi=10.1063/1.881789 |bibcode=1997PhT....50f..15W}}</ref> Unlike de Broglie, Lunn went on to formulate the differential equation now known as the Schrödinger equation and solve for its energy eigenvalues for the hydrogen atom; the paper was rejected by the ''Physical Review'', according to Kamen.<ref>{{cite book |last=Kamen |first=Martin D. |title=Radiant Science, Dark Politics |year=1985 |publisher=University of California Press |location=Berkeley and Los Angeles, California |isbn=978-0-520-04929-1 |pages=[https://archive.org/details/radiantscienceda00kame/page/29 29–32] |url=https://archive.org/details/radiantscienceda00kame/page/29}}</ref>

Following up on de Broglie's ideas, physicist [[Peter Debye]] made an offhand comment that if particles behaved as waves, they should satisfy some sort of wave equation. Inspired by Debye's remark, Schrödinger decided to find a proper 3-dimensional wave equation for the electron. He was guided by [[William Rowan Hamilton]]'s [[Hamilton's optico-mechanical analogy|analogy between mechanics and optics]], encoded in the observation that the zero-wavelength limit of optics resembles a mechanical system—the trajectories of [[light rays]] become sharp tracks that obey [[Fermat's principle]], an analog of the [[principle of least action]].<ref>{{Cite book | last=Schrödinger | first=E. | year=1984 | title=Collected papers | publisher=Friedrich Vieweg und Sohn | isbn=978-3-7001-0573-2}} See introduction to first 1926 paper.</ref>

[[File:Grave Schroedinger (detail).png|alt=|thumb|Schrödinger's equation inscribed on the gravestone of Annemarie and Erwin Schrödinger. ([[Notation for differentiation#Newton's notation|Newton's dot notation]] for the time derivative is used.)]]
The equation he found is<ref name="verlagsgesellschaft1991">{{cite book |title=Encyclopaedia of Physics |edition=2nd |first1=R. G. |last1=Lerner |author1-link=Rita G. Lerner |first2=G. L. |last2=Trigg |publisher=VHC publishers |year=1991 | isbn=0-89573-752-3}}</ref>
<math display="block">i\hbar \frac{\partial}{\partial t} \Psi(\mathbf{r}, t) = -\frac{\hbar^2}{2m} \nabla^2 \Psi(\mathbf{r}, t) + V(\mathbf{r})\Psi(\mathbf{r}, t).</math>

By that time [[Arnold Sommerfeld]] had [[Sommerfeld–Wilson quantization|refined the Bohr model]] with [[fine structure|relativistic corrections]].<ref>{{Cite book |last=Sommerfeld |first=A. |author-link=Arnold Sommerfeld |year=1919 |title=Atombau und Spektrallinien |language=de |publisher=Friedrich Vieweg und Sohn |location=Braunschweig |isbn=978-3-87144-484-5}}</ref><ref>For an English source, see {{Cite book |last=Haar |first=T. |title=The Old Quantum Theory |year=1967 |location=Oxford, New York |publisher=Pergamon Press |url=https://archive.org/details/oldquantumtheory00haar |url-access=registration }}</ref> Schrödinger used the relativistic energy–momentum relation to find what is now known as the [[Klein–Gordon equation]] in a [[Coulomb potential]] (in [[natural units]]):
<math display="block">\left(E + \frac{e^2}{r}\right)^2 \psi(x) = - \nabla^2 \psi(x) + m^2 \psi(x).</math>

He found the standing waves of this relativistic equation, but the relativistic corrections disagreed with Sommerfeld's formula. Discouraged, he put away his calculations and secluded himself with a mistress in a mountain cabin in December 1925.<ref>{{Cite news |last=Teresi |first=Dick |date=1990-01-07 |title=The Lone Ranger of Quantum Mechanics |language=en-US |work=[[The New York Times]] |url=https://www.nytimes.com/1990/01/07/books/the-lone-ranger-of-quantum-mechanics.html |access-date=2020-10-13 |issn=0362-4331}}</ref>

While at the cabin, Schrödinger decided that his earlier nonrelativistic calculations were novel enough to publish and decided to leave off the problem of relativistic corrections for the future. Despite the difficulties in solving the differential equation for hydrogen (he had sought help from his friend the mathematician [[Hermann Weyl]]<ref name="Schrödinger1982"/>{{rp|3}}) Schrödinger showed that his nonrelativistic version of the wave equation produced the correct spectral energies of hydrogen in a paper published in 1926.<ref name="Schrödinger1982">{{cite book |first=Erwin |last=Schrödinger |author-link=Erwin Schrödinger |title=Collected Papers on Wave Mechanics |edition=3rd |year=1982 |publisher=[[American Mathematical Society]] |isbn=978-0-8218-3524-1}}</ref>{{rp|1}}<ref>
{{cite journal
 |last=Schrödinger |first=E.
 |author-link=Erwin Schrödinger
 |year=1926
 |title=Quantisierung als Eigenwertproblem; von Erwin Schrödinger
 |language=de
 |url=http://gallica.bnf.fr/ark:/12148/bpt6k153811.image.langFR.f373.pagination
 |journal=[[Annalen der Physik]]
 |volume= 384 |issue=4 |pages=361–377
 |doi=10.1002/andp.19263840404 |doi-access=
 |bibcode = 1926AnP...384..361S
|url-access=subscription
 }}</ref> Schrödinger computed the [[hydrogen spectral series]] by treating a [[hydrogen atom]]'s [[electron]] as a wave <math>\Psi(\mathbf{x}, t)</math>, moving in a [[potential well]] <math>V</math>, created by the [[proton]]. This computation accurately reproduced the energy levels of the [[Bohr model]].

The Schrödinger equation details the behavior of <math>\Psi</math> but says nothing of its ''nature''. Schrödinger tried to interpret the real part of <math>\Psi \frac{\partial \Psi^*}{\partial t}</math> as a charge density, and then revised this proposal, saying in his next paper that the [[modulus squared]] of <math>\Psi</math> is a charge density. This approach was, however, unsuccessful.{{refn|group=note|For details, see Moore,<ref name=Moore1992>{{cite book | last=Moore | first=W. J. | year=1992 | title=Schrödinger: Life and Thought | publisher=[[Cambridge University Press]] | isbn=978-0-521-43767-7}}</ref>{{rp|219}} Jammer,<ref name="jammer1974">{{cite book | last=Jammer | first=Max | author-link=Max Jammer | title=Philosophy of Quantum Mechanics: The interpretations of quantum mechanics in historical perspective | url=https://archive.org/details/philosophyofquan0000jamm | url-access=registration | year=1974 | publisher=Wiley-Interscience| isbn=9780471439585 }}</ref>{{rp|24–25}} and Karam.<ref>{{Cite journal|last=Karam|first=Ricardo|date=June 2020|title=Schrödinger's original struggles with a complex wave function|url=http://aapt.scitation.org/doi/10.1119/10.0000852|journal=[[American Journal of Physics]]|language=en|volume=88|issue=6|pages=433–438|doi=10.1119/10.0000852|bibcode=2020AmJPh..88..433K |s2cid=219513834 |issn=0002-9505|url-access=subscription}}</ref>}} In 1926, just a few days after this paper was published, [[Max Born]] successfully interpreted <math>\Psi</math> as the [[probability amplitude]], whose modulus squared is equal to [[Probability density function|probability density]].<ref name=Moore1992/>{{rp|220}} Later, Schrödinger himself explained this interpretation as follows:<ref>Erwin Schrödinger, "The Present situation in Quantum Mechanics", p. 9 of 22. The English version was translated by John D. Trimmer. The translation first appeared first in ''Proceedings of the American Philosophical Society'', 124, [https://www.jstor.org/stable/986572 323–338]. It later appeared as Section I.11 of Part I of ''Quantum Theory and Measurement'' by J. A. Wheeler and W. H. Zurek, eds., Princeton University Press, New Jersey 1983, {{ISBN|0691083169}}.</ref>
{{cquote|
The already ... mentioned psi-function.... is now the means for predicting probability of measurement results. In it is embodied the momentarily attained sum of theoretically based future expectation, somewhat as laid down in a catalog.
|author=Erwin Schrödinger
}}

== Interpretation ==
{{main|Interpretations of quantum mechanics}}
The Schrödinger equation provides a way to calculate the wave function of a system and how it changes dynamically in time. However, the Schrödinger equation does not directly say ''what,'' exactly, the wave function is. The meaning of the Schrödinger equation and how the mathematical entities in it relate to physical reality depends upon the [[interpretation of quantum mechanics]] that one adopts.

In the views often grouped together as the [[Copenhagen interpretation]], a system's wave function is a collection of statistical information about that system. The Schrödinger equation relates information about the system at one time to information about it at another. While the time-evolution process represented by the Schrödinger equation is continuous and deterministic, in that knowing the wave function at one instant is in principle sufficient to calculate it for all future times, wave functions can also change discontinuously and stochastically during a [[measurement in quantum mechanics|measurement]]. The wave function changes, according to this school of thought, because new information is available. The post-measurement wave function generally cannot be known prior to the measurement, but the probabilities for the different possibilities can be calculated using the [[Born rule]].<ref name=":0">{{cite book|last=Peres|first=Asher|title=Quantum Theory: Concepts and Methods|title-link=Quantum Theory: Concepts and Methods|publisher=[[Kluwer]]|year=1993|isbn=0-7923-2549-4|location=|oclc=28854083|author-link=Asher Peres}}</ref><ref name="omnes">{{cite book|first=R. |last=Omnès |author-link=Roland Omnès |title=The Interpretation of Quantum Mechanics |publisher=Princeton University Press |year=1994 |isbn=978-0-691-03669-4 |oclc=439453957}}</ref>{{refn|group=note|One difficulty in discussing the philosophical position of "the Copenhagen interpretation" is that there is no single, authoritative source that establishes what the interpretation is. Another complication is that the philosophical background familiar to Einstein, Bohr, Heisenberg, and contemporaries is much less so to physicists and even philosophers of physics in more recent times.<ref name="Faye-Stanford">{{Cite book|last=Faye|first=Jan|title=[[Stanford Encyclopedia of Philosophy]]|publisher=Metaphysics Research Lab, Stanford University|year=2019|editor-last=Zalta|editor-first=Edward N.|chapter=Copenhagen Interpretation of Quantum Mechanics| author-link=Jan Faye|chapter-url=https://plato.stanford.edu/entries/qm-copenhagen/}}</ref><ref name="chevalley1999">{{cite book|first=Catherine |last=Chevalley |chapter=Why Do We Find Bohr Obscure? |title=Epistemological and Experimental Perspectives on Quantum Physics |editor-first1=Daniel |editor-last1=Greenberger |editor-first2=Wolfgang L. |editor-last2=Reiter |editor-first3=Anton |editor-last3=Zeilinger |publisher=Springer Science+Business Media |doi=10.1007/978-94-017-1454-9 |isbn=978-9-04815-354-1 |year=1999 |pages=59–74}}</ref>}} Other, more recent interpretations of quantum mechanics, such as [[relational quantum mechanics]] and [[QBism]] also give the Schrödinger equation a status of this sort.<ref>{{Cite journal| last=van Fraassen|first=Bas C.|author-link=Bas van Fraassen|date=April 2010|title=Rovelli's World|url=http://link.springer.com/10.1007/s10701-009-9326-5|journal=[[Foundations of Physics]]|language=en|volume=40|issue=4|pages=390–417| doi=10.1007/s10701-009-9326-5| bibcode=2010FoPh...40..390V|s2cid=17217776|issn=0015-9018|url-access=subscription}}</ref><ref>{{Cite book| last=Healey|first=Richard|title=[[Stanford Encyclopedia of Philosophy]]|publisher=Metaphysics Research Lab, Stanford University| year=2016|editor-last=Zalta|editor-first=Edward N.|chapter=Quantum-Bayesian and Pragmatist Views of Quantum Theory|chapter-url=https://plato.stanford.edu/entries/quantum-bayesian/}}</ref>

Schrödinger himself suggested in 1952 that the different terms of a superposition evolving under the Schrödinger equation are "not alternatives but all really happen simultaneously". This has been interpreted as an early version of Everett's [[many-worlds interpretation]].<ref>{{Cite book| last1=Deutsch |first1=David | chapter=Apart from Universes|title=Many Worlds? Everett, Quantum Theory and Reality |editor=S. Saunders |editor2=J. Barrett |editor3=A. Kent |editor4=D. Wallace |publisher=Oxford University Press|year=2010}}</ref><ref>{{cite book |last1=Schrödinger |first1=Erwin |editor1-last=Bitbol |editor1-first=Michel |title=The Interpretation of Quantum Mechanics: Dublin Seminars (1949–1955) and other unpublished essays |date=1996 |publisher=OxBow Press}}</ref>{{refn|group=note|Schrödinger's later writings also contain elements resembling the [[modal interpretation]] originated by [[Bas van Fraassen]]. Because Schrödinger subscribed to a kind of post-[[Ernst Mach|Machian]] [[neutral monism]], in which "matter" and "mind" are only different aspects or arrangements of the same common elements, treating the wavefunction as physical and treating it as information became interchangeable.<ref>{{Cite book|last=Bitbol|first=Michel|author-link=Michel Bitbol |url=https://www.worldcat.org/oclc/851376153|title=Schrödinger's Philosophy of Quantum Mechanics|date=1996|publisher=Springer Netherlands| isbn=978-94-009-1772-9|location=Dordrecht|oclc=851376153}}</ref>}} This interpretation, formulated independently in 1956, holds that ''all'' the possibilities described by quantum theory ''simultaneously'' occur in a multiverse composed of mostly independent parallel universes.<ref>{{Cite book|first=Jeffrey |last=Barrett|title=[[Stanford Encyclopedia of Philosophy]]|publisher=Metaphysics Research Lab, Stanford University|year=2018|editor-last=Zalta|editor-first=Edward N.|chapter=Everett's Relative-State Formulation of Quantum Mechanics|chapter-url=https://plato.stanford.edu/entries/qm-everett/}}</ref> This interpretation removes the axiom of wave function collapse, leaving only continuous evolution under the Schrödinger equation, and so all possible states of the measured system and the measuring apparatus, together with the observer, are present in a real physical [[quantum superposition]]. While the multiverse is deterministic, we perceive non-deterministic behavior governed by probabilities, because we do not observe the multiverse as a whole, but only one parallel universe at a time. Exactly how this is supposed to work has been the subject of much debate. Why should we assign probabilities at all to outcomes that are certain to occur in some worlds, and why should the probabilities be given by the Born rule?<ref name="wallace2003">{{cite journal|last1=Wallace|first1=David|year=2003|title=Everettian Rationality: defending Deutsch's approach to probability in the Everett interpretation|journal=Stud. Hist. Phil. Mod. Phys.|volume=34|issue=3|pages=415–438|arxiv=quant-ph/0303050|bibcode=2003SHPMP..34..415W|doi=10.1016/S1355-2198(03)00036-4|s2cid=1921913}}</ref> Several ways to answer these questions in the many-worlds framework have been proposed, but there is no consensus on whether they are successful.<ref name="ballentine1973">{{cite journal|first1=L. E. |last1=Ballentine|date=1973|title=Can the statistical postulate of quantum theory be derived?—A critique of the many-universes interpretation|journal=Foundations of Physics| volume=3| issue=2| pages=229–240| doi=10.1007/BF00708440|bibcode=1973FoPh....3..229B|s2cid=121747282}}</ref><ref>{{cite book|first=N. P. |last=Landsman |chapter=The Born rule and its interpretation |chapter-url=http://www.math.ru.nl/~landsman/Born.pdf |quote=The conclusion seems to be that no generally accepted derivation of the Born rule has been given to date, but this does not imply that such a derivation is impossible in principle. |title=Compendium of Quantum Physics |editor-first1=F. |editor-last1=Weinert |editor-first2=K. |editor-last2=Hentschel |editor-first3=D. |editor-last3=Greenberger  |editor-first4=B. |editor-last4=Falkenburg |publisher=Springer |year=2008 |isbn=978-3-540-70622-9}}</ref><ref name="kent2009">{{Cite book|last1=Kent|first1=Adrian|author-link=Adrian Kent|title=Many Worlds? Everett, Quantum Theory and Reality|publisher=Oxford University Press|year=2010|editor=S. Saunders|chapter=One world versus many: The inadequacy of Everettian accounts of evolution, probability, and scientific confirmation|arxiv=0905.0624|bibcode=2009arXiv0905.0624K|editor2=J. Barrett|editor3=A. Kent|editor4=D. Wallace}}</ref>

[[Bohmian mechanics]] reformulates quantum mechanics to make it deterministic, at the price of adding a force due to a "quantum potential". It attributes to each physical system not only a wave function but in addition a real position that evolves deterministically under a nonlocal guiding equation. The evolution of a physical system is given at all times by the Schrödinger equation together with the guiding equation.<ref>{{cite book|chapter-url=https://plato.stanford.edu/entries/qm-bohm/ |last=Goldstein |first=Sheldon |chapter=Bohmian Mechanics |title=[[Stanford Encyclopedia of Philosophy]] |year=2017 |editor-first1=Edward N. |editor-last=Zalta |publisher=Metaphysics Research Lab, Stanford University}}</ref>

== See also ==
{{div col}}
* [[Eckhaus equation]]
* [[Fokker–Planck equation]]
* [[Interpretations of quantum mechanics]]
* [[List of things named after Erwin Schrödinger]]
* [[Logarithmic Schrödinger equation]]
* [[Nonlinear Schrödinger equation]]
* [[Pauli equation]]
* [[Quantum channel]]
* [[Relation between Schrödinger's equation and the path integral formulation of quantum mechanics]]
* [[Schrödinger picture]]
* [[Wigner quasiprobability distribution]]
{{div col end}}

== Notes ==
{{reflist|group=note}}

== References ==
{{reflist}}

== External links ==
{{wikiquote}}
* {{springer|title=Schrödinger equation|id=p/s083410|mode=cs1}}
* [http://oyc.yale.edu/sites/default/files/notes_quantum_cookbook.pdf Quantum Cook Book] (PDF) and [http://oyc.yale.edu/physics/phys-201#sessions PHYS 201: Fundamentals of Physics II] by [[Ramamurti Shankar]], Yale OpenCourseware
* [http://www.lightandmatter.com/lm/ The Modern Revolution in Physics] – an online textbook.
* [https://ocw.mit.edu/courses/physics/8-04-quantum-physics-i-spring-2016/ Quantum Physics I] at [[MIT OpenCourseWare]]
{{QED}}
{{Quantum information}}
{{Quantum mechanics topics}}
{{Quantum field theories}}
{{Quantum gravity}}
{{Authority control}}

{{DEFAULTSORT:Schrodinger Equation}}
[[Category:Schrödinger equation| ]]
[[Category:Partial differential equations]]
[[Category:Wave mechanics]]
[[Category:Functions of space and time]]