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{{Short description|Equation governing the wave function of a quantum-mechanical system}}
{{Quantum book backlink|Quantum dynamics and evolution}}

The '''Schrödinger equation''' is the fundamental equation of nonrelativistic [[Physics:Quantum mechanics|quantum mechanics]]. It governs the time evolution of the [[Wave function|wave function]] of a quantum-mechanical system and provides the mathematical basis for predicting measurable quantities such as probability densities, energies, and stationary states.<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}}</ref>{{rp|1–2}} It was introduced by [[Biography:Erwin Schrödinger|Erwin Schrödinger]] in 1925–1926 and became one of the central achievements in the development of modern physics.<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={{wipe|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 = [[Physics:Physical Review|Physical Review]]
| volume = 28
| issue = 6
| pages = 1049–1070
| year = 1926
| doi = 10.1103/PhysRev.28.1049
| bibcode = 1926PhRv...28.1049S
}}</ref>

[[File:Schrodinger equation light.jpg|thumb|400px|The Schrödinger equation is the central equation of wave mechanics and describes the time evolution of a quantum state.]]

== Overview ==
Conceptually, the Schrödinger equation plays a role in quantum mechanics analogous to that of Newton's second law in classical mechanics. Given an initial quantum state, it determines how that state changes over time.<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 |location=New York}}</ref>{{rp|II:268}} In Schrödinger’s formulation, matter is associated with a wave, following the earlier proposal of [[Biography:Louis de Broglie|Louis de Broglie]] that particles possess wave-like properties.

The Schrödinger equation is one of several equivalent formulations of quantum mechanics. Other approaches include [[Physics:Matrix mechanics|matrix mechanics]], developed by [[Biography:Werner Heisenberg|Werner Heisenberg]], and the [[Physics:Path integral formulation|path integral formulation]], associated chiefly with [[Biography:Richard Feynman|Richard Feynman]]. The Schrödinger approach is often called '''wave mechanics'''.<ref>{{cite book|first=Paul Adrien Maurice |last=Dirac |title=The Principles of Quantum Mechanics |title-link=The Principles of Quantum Mechanics |publisher=Clarendon Press |location=Oxford |year=1930}}</ref>

== Definition ==
=== Preliminaries ===
In introductory treatments, the Schrödinger equation is often presented in position space 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 the wave function, <math>m</math> is the particle mass, and <math>V(x,t)</math> is the potential energy function describing the particle’s environment.<ref name="Zwiebach2022">{{cite book|first=Barton |last=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 constants <math>i</math> and <math>\hbar</math> denote the imaginary unit and the reduced [[Physics:Planck constant|Planck constant]], respectively.<ref name="Zwiebach2022"/>{{rp|10}}

In the more general mathematical formulation of quantum mechanics, a system is described by a state vector <math>|\psi\rangle</math> in a [[Hilbert space]] <math>\mathcal H</math>.<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><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> Physical observables such as position, momentum, and energy are represented by self-adjoint operators acting on this space.<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}}</ref>

A position-space wave function is obtained from the state vector through
<math display="block">\Psi(x,t)=\langle x|\Psi(t)\rangle.</math>

=== Time-dependent Schrödinger equation ===
The most general form used in ordinary quantum mechanics is the time-dependent Schrödinger equation:
<math display="block">i \hbar \frac{d}{d t}\vert\Psi(t)\rangle = \hat H\vert\Psi(t)\rangle</math>
where <math>\hat H</math> is the [[Physics:Hamiltonian (quantum mechanics)|Hamiltonian]] operator of the system.<ref name=Shankar1994>{{cite book | last=Shankar | first=R. | 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}}

To apply the equation, one specifies the Hamiltonian of the system, including kinetic and potential energy terms, and solves for the wave function. The square modulus of the wave function gives the probability density:
<math display="block">\Pr(x,t)=|\Psi(x,t)|^2.</math><ref name="Zwiebach2022"/>{{rp|78}}

For a nonrelativistic spinless particle in three dimensions, the position-space form 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>

=== Time-independent Schrödinger equation ===
When the Hamiltonian does not depend explicitly on time, solutions of definite energy can be found using the time-independent Schrödinger equation:
<math display="block">\hat H|\Psi\rangle = E |\Psi\rangle</math>
where <math>E</math> is the energy eigenvalue.<ref name="Zwiebach2022"/>{{rp|134}}

This is an eigenvalue equation: the wave function is an eigenfunction of the Hamiltonian, and the corresponding eigenvalue is the measurable energy. Such solutions describe [[Physics:Stationary state|stationary states]], whose probability density does not change with time.

== Properties ==
=== Linearity ===
The Schrödinger equation is linear. If <math>|\psi_1\rangle</math> and <math>|\psi_2\rangle</math> are solutions, then any linear combination
<math display="block">|\psi\rangle = a|\psi_1\rangle + b|\psi_2\rangle</math>
is also a solution, where <math>a</math> and <math>b</math> are complex numbers.<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|author-link=Eleanor Rieffel}}</ref>{{rp|25}} This allows for [[Physics:Quantum superposition|quantum superposition]], one of the defining features of quantum theory.

When expressed in an energy eigenbasis, a general time-dependent state may be written as
<math display="block">|\Psi(t)\rangle = \sum_{n} A_n e^{-iE_n t/\hbar} |\psi_{E_n}\rangle.</math>

=== Unitarity ===
Time evolution under the Schrödinger equation is unitary. If the Hamiltonian is time-independent, the formal solution is
<math display="block">|\Psi(t)\rangle = e^{-i\hat{H}t/\hbar }|\Psi(0)\rangle.</math>
The operator
<math display="block">\hat{U}(t)=e^{-i\hat{H}t/\hbar}</math>
is called the time-evolution operator and preserves the inner product in Hilbert space.<ref name="rieffel"/>

As a result, if a state is normalized at one time, it remains normalized at all later times.

=== Probability current ===
The Schrödinger equation is consistent with local conservation of probability.<ref name="Cohen-Tannoudji">{{cite book|last1=Cohen-Tannoudji |first1=Claude |last2=Diu |first2=Bernard |last3=Laloë |first3=Franck |title=Quantum Mechanics |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|238}} In nonrelativistic quantum mechanics, the probability density <math>\rho(\mathbf{r},t)</math> and probability current <math>\mathbf{j}</math> satisfy the continuity equation
<math display="block">\frac{\partial}{\partial t} \rho\left(\mathbf{r},t\right) + \nabla \cdot \mathbf{j} = 0.</math>

For a wave function <math>\psi</math>, the probability current is
<math display="block">\mathbf{j} = \frac \hbar m \operatorname{Im} (\psi^*\nabla \psi).</math>

=== Separation of variables ===
If the potential does not depend explicitly on time, one may seek solutions of the form
<math display="block">\Psi(\mathbf{r},t)=\psi(\mathbf{r})\tau(t).</math>
This leads to stationary-state solutions
<math display="block">\Psi(\mathbf{r},t) = \psi(\mathbf{r}) e^{-iEt/\hbar}.</math>

The spatial part then satisfies
<math display="block">\nabla^2\psi(\mathbf{r}) + \frac{2m}{\hbar^2} \left [E - V(\mathbf{r})\right ] \psi(\mathbf{r}) = 0.</math>

This method is especially useful in systems with symmetry, where Cartesian or spherical coordinates allow further separation.

== Examples ==
=== Particle in a box ===
The [[Physics:Particle in a box|particle in a box]] is one of the simplest exactly solvable quantum systems. A particle confined to a region of zero potential with infinite walls outside satisfies
<math display="block"> - \frac {\hbar ^2}{2m} \frac {d ^2 \psi}{dx^2} = E \psi.</math>

The boundary conditions force the wave function to vanish at the walls, so only certain wavelengths are allowed. This yields quantized energies
<math display="block">E_n = \frac{\hbar^2 \pi^2 n^2}{2mL^2} = \frac{n^2h^2}{8mL^2}, \qquad n=1,2,3,\ldots</math>
demonstrating that bound states in quantum mechanics have discrete energy levels.<ref name="Cohen-Tannoudji"/>{{rp|77–78}}

=== Harmonic oscillator ===
For the [[Physics:Quantum harmonic oscillator|quantum harmonic oscillator]], the time-independent Schrödinger equation 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>

Its energy eigenvalues are
<math display="block">E_n = \left(n + \frac{1}{2} \right) \hbar \omega.</math>
The lowest state, <math>n=0</math>, has nonzero energy, called the [[Physics:Zero-point energy|zero-point energy]].<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>

=== Hydrogen atom ===
The Schrödinger equation for the electron in a [[Physics:Hydrogen atom|hydrogen 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>

It can be solved exactly by separation of variables in spherical coordinates.<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> The resulting solutions explain the discrete spectral lines of hydrogen and were among the earliest major successes of wave mechanics.<ref name="Schrödinger1982">{{cite book |first=Erwin |last=Schrödinger |title=Collected Papers on Wave Mechanics |edition=3rd |year=1982 |publisher=American Mathematical Society |isbn=978-0-8218-3524-1}}</ref>{{rp|1}}

=== Approximate solutions ===
Most quantum systems cannot be solved exactly. Approximate methods such as [[Chemistry:Variational method (quantum mechanics)|variational methods]], [[Physics:WKB approximation|WKB approximation]], and [[Perturbation theory (quantum mechanics)|perturbation theory]] are therefore widely used.

== Semiclassical limit ==
The connection between quantum and classical mechanics can be explored through the [[Ehrenfest theorem]], which relates the time evolution of expectation values to classical equations of motion.<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|oclc=28854083|author-link=Asher Peres}}</ref>{{rp|302}} In the limit where the wave function remains highly localized, quantum expectation values approximately follow classical trajectories.

Writing the wave function as
<math display="block">\Psi = \sqrt{\rho(\mathbf{r},t)} e^{iS(\mathbf{r},t)/\hbar}</math>
and taking the formal limit <math>\hbar \to 0</math> leads to the [[Physics:Hamilton–Jacobi equation|Hamilton–Jacobi equation]], illustrating the correspondence between quantum and classical mechanics.<ref name=":0"/>{{rp|308}}

== Density matrix formulation ==
Wave functions are not always the most convenient way to describe a system. When the preparation is not perfectly known, or when a system is part of a larger one, [[Density matrix|density matrices]] are often used instead.<ref name=":0"/>{{rp|74}} The density-matrix analogue of the Schrödinger equation is
<math display="block"> i \hbar \frac{\partial \hat{\rho}}{\partial t} = [\hat{H}, \hat{\rho}],</math>
where the brackets denote the [[Commutator|commutator]].<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}}</ref><ref>{{cite book|title=Statistical mechanics|last=Schwabl|first=Franz|page=16|isbn=978-3-540-43163-3|year=2002|publisher=Springer }}</ref>

For time-independent Hamiltonians, the solution is
<math display="block">\hat{\rho}(t) = e^{-i \hat{H} t/\hbar} \hat{\rho}(0) e^{i \hat{H} t/\hbar}.</math>

== Relativistic generalizations ==
The ordinary one-particle Schrödinger equation is fundamentally nonrelativistic. In relativistic settings, one instead uses equations such as the [[Physics:Klein–Gordon equation|Klein–Gordon equation]] for spin-0 particles and the [[Physics:Dirac equation|Dirac equation]] for spin-{{frac|1|2}} particles.<ref>{{Cite journal|last=Symanzik|first=K.|date=1981-07-06|title=Schrödinger representation and Casimir effect in renormalizable quantum field theory|journal=Nuclear Physics B|volume=190|issue=1|pages=1–44|doi=10.1016/0550-3213(81)90482-X|bibcode=1981NuPhB.190....1S}}</ref><ref>{{Cite journal|last=Kiefer|first=Claus|date=1992-03-15|title=Functional Schrödinger equation for scalar QED|journal=Physical Review D|volume=45|issue=6|pages=2044–2056|doi=10.1103/PhysRevD.45.2044|pmid=10014577 |bibcode=1992PhRvD..45.2044K}}</ref>

In [[Physics:Quantum field theory|quantum field theory]], the Schrödinger picture can still be formulated, but the Hilbert space is generally a [[Physics:Fock space|Fock space]] in which particle number is not fixed.<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>

== History ==
The origins of the Schrödinger equation lie in the development of wave–particle duality. Following [[Biography:Max Planck|Max Planck]]’s quantization of radiation and [[Biography:Albert Einstein|Albert Einstein]]’s light quantum hypothesis, [[Biography:Louis de Broglie|Louis de Broglie]] proposed that matter particles also have an associated wavelength:
<math display="block">p = \frac{h}{\lambda} = \hbar k.</math>

Building on this idea, Schrödinger sought a wave equation for electrons. His nonrelativistic equation,
<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>
reproduced the observed spectral energies of the hydrogen atom and provided a compelling alternative to earlier quantum models.<ref name="Schrödinger1982"/><ref>{{cite journal
|last=Schrödinger |first=E.
|year=1926
|title=Quantisierung als Eigenwertproblem; von Erwin Schrödinger
|language=de
|journal=[[Physics:Annalen der Physik|Annalen der Physik]]
|volume=384
|issue=4
|pages=361–377
|doi=10.1002/andp.19263840404
|doi-access=free
|bibcode = 1926AnP...384..361S
}}</ref>

Soon afterward, [[Biography:Max Born|Max Born]] interpreted the wave function probabilistically, showing that <math>|\Psi|^2</math> gives a probability density rather than a literal charge distribution.<ref name=Moore1992>{{cite book | last=Moore | first=W. J. | year=1992 | title=Schrödinger: Life and Thought | publisher={{wipe|Cambridge University Press}} | isbn=978-0-521-43767-7}}</ref>{{rp|220}}

== Interpretation ==
The Schrödinger equation determines how the wave function changes in time, but it does not by itself specify what the wave function ''is''. Different [[Physics:Interpretation of quantum mechanics|interpretations of quantum mechanics]] assign different meanings to the wave function and to the measurement process.

In the [[Physics:Copenhagen interpretation|Copenhagen interpretation]], the wave function represents information or probability amplitudes, and measurement involves probabilistic outcomes governed by the [[Physics:Born rule|Born rule]].<ref name=":0"/><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> Other interpretations, including [[Physics:Many-worlds interpretation|many-worlds]] and [[Physics:Bohmian mechanics|Bohmian mechanics]], retain the same Schrödinger equation while differing in how they understand physical reality and measurement.<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}}</ref><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 ==

* [[Physics:Dirac equation|Dirac equation]]
* [[Physics:Klein–Gordon equation|Klein–Gordon equation]]
* [[Physics:Particle in a box|Particle in a box]]
* [[Physics:Quantum harmonic oscillator|Quantum harmonic oscillator]]
* [[Density matrix]]
* [[Physics:Interpretations of quantum mechanics|Interpretations of quantum mechanics]]
* [[Physics:Probability current|Probability current]]
* [[Physics:Schrödinger picture|Schrödinger picture]]
== Notes ==
{{reflist|group=note}}

= References =
{{reflist|3}}

{{Author|Harold Foppele}}

{{Sourceattribution|Physics:Quantum Schrödinger equation|1}}

Physics:Quantum Schrödinger equation - Revision history

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