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{{Hatnote|For a more general introduction to the topic, see [[Introduction to quantum mechanics#Schrödinger wave equation|Introduction to quantum mechanics]].}}
{{short description|Linear partial differential equation whose solution describes the quantum-mechanical system.}}
{{Use dmy dates|date=June 2016}}
{{Quantum mechanics|cTopic=Equations}}
[[File:Grave Schroedinger (detail).png|alt=|thumb|Schrödinger's equation inscribed on the gravestone of Annemarie and Erwin Schrödinger. (Newton's dot notation for the time derivative is used.)]]
The '''Schrödinger equation''' is a [[linear differential equation|linear]] [[partial differential equation]] that describes the [[wave function]] or state function of a quantum-mechanical system.<ref name="Griffiths2004">{{citation|author=Griffiths, David J.|title=Introduction to Quantum Mechanics (2nd ed.)|year=2004|publisher=Prentice Hall|isbn=978-0-13-111892-8}}</ref>{{rp|1–2}} It is a key result in [[quantum mechanics]], and its discovery was a significant landmark in the development of the subject. The equation is named after [[Erwin Schrödinger]], 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|accessdate=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
| archiveurl = https://web.archive.org/web/20081217040121/http://home.tiscali.nl/physis/HistoricPaper/Schroedinger/Schroedinger1926c.pdf
| archivedate = 17 December 2008
| journal = [[Physical Review]]
| volume = 28 | issue = 6 | pages = 1049–1070
| year = 1926
| doi = 10.1103/PhysRev.28.1049
|bibcode = 1926PhRv...28.1049S }}</ref>

In [[classical mechanics]], [[Newton's second law]] ({{math|'''F''' {{=}} ''m'''''a'''}})<ref group="note">While this is the most famous form of Newton's second law, it is not the most general, being valid only for objects of constant mass. Newton's second law reads <math> \mathbf{F} = \frac{d}{dt}(m \mathbf{v})</math>, the net force acting on a body is equal to the total time derivative of the total momentum of that body—which is equivalent to the given form when mass is constant with time.</ref> is used to make a mathematical prediction as to what path a given physical system will take over time following a set of known initial conditions. Solving this equation gives the position and the momentum of the physical system as a function of the external force <math>\mathbf{F}</math> on the system. Those two parameters are sufficient to describe its state at each time instant. In quantum mechanics, the analogue of Newton's law is Schrödinger's equation.

The concept of a wave function is a fundamental [[Mathematical formulation of quantum mechanics|postulate of quantum mechanics]]; the wave function defines the state of the system at each spatial position, and time. Using these postulates, Schrödinger's equation can be derived from the fact that the time-evolution operator must be [[Unitary operator|unitary]], and must therefore be generated by the exponential of a [[self-adjoint operator]], which is the quantum Hamiltonian. This derivation is explained below.

In the [[Copenhagen interpretation]] of quantum mechanics, the wave function is the most complete description that can be given of a physical system. Solutions to Schrödinger's equation describe not only [[molecule|molecular]], [[atom]]ic, and [[subatomic particle|subatomic]] systems, but also [[macroscopic scale|macroscopic system]]s, possibly even the whole [[universe]].<ref name=Laloe>{{citation|last=Laloe|first=Franck|title=Do We Really Understand Quantum Mechanics| publisher=Cambridge University Press|year=2012| isbn = 978-1-107-02501-1}}</ref>{{rp|292ff}} Schrödinger's equation is central to all applications of quantum mechanics, including [[quantum field theory]], which combines [[special relativity]] with quantum mechanics. Theories of [[quantum gravity]], such as [[string theory]], also do not modify Schrödinger's equation.{{citation needed|date=February 2019}} 
 
The Schrödinger equation is not the only way to study quantum mechanical systems and make predictions. The other formulations of quantum mechanics include [[matrix mechanics]], introduced by [[Werner Heisenberg]], and the [[path integral formulation]], developed chiefly by [[Richard Feynman]]. [[Paul Dirac]] incorporated matrix mechanics and the Schrödinger equation into a single formulation.

{{TOC limit|3}}

==Equation==

===Time-dependent equation===
The form of the Schrödinger equation depends on the physical situation (see below for special cases). The most general form is the time-dependent Schrödinger equation (TDSE), which gives a description of a system evolving with time:<ref name=Shankar1994>
{{cite book
 |last=Shankar |first=R.
 |year=1943
 |title=Principles of Quantum Mechanics
 |edition=2nd
 |publisher=Kluwer Academic/Plenum Publishers
 |isbn=978-0-306-44790-7
}}</ref>{{rp|143}}
[[File:Wave packet (dispersion).gif|thumb|200px|A [[wave function]] that satisfies the nonrelativistic Schrödinger equation with {{math|''V'' {{=}} 0}}. In other words, this corresponds to a particle traveling freely through empty space. The [[real part]] of the [[wave function]] is plotted here.]]
{{Equation box 1
|indent=:
|title='''Time-dependent Schrödinger equation'''&nbsp;''(general)''
|equation=<math>i \hbar \frac{d}{d t}\vert\Psi(t)\rangle = \hat H\vert\Psi(t)\rangle</math>
|cellpadding
|border
|border colour = rgb(80,200,120)
|background colour = rgb(80,200,120,10%)}}

where <math>i</math> is the [[imaginary unit]], <math>\hbar = \frac{h}{2 \pi}</math> is the reduced [[Planck constant]], <math>\Psi</math> (the Greek letter [[psi (letter)|psi]]) is the state vector of the quantum system, <math>t</math> is time, and <math>\hat{H}</math> is the [[Hamiltonian (quantum mechanics)|Hamiltonian]] [[operator (physics)|operator]]. The ''[[wave function|position-space wave function]]'' of the quantum system is nothing but the components in the expansion of the state vector in terms of the position eigenvector <math>\vert\mathbf{r}\rangle</math>. It is a scalar function, expressed as <math>\Psi(\mathbf{r},t) = \langle \mathbf{r}\vert \Psi \rangle </math>. Similarly, the ''[[wave function|momentum-space wave function]]'' can be defined as <math>\tilde\Psi(\mathbf{p},t) = \langle \mathbf{p}\vert \Psi \rangle </math>, where <math>\vert\mathbf{p}\rangle</math> is the momentum eigenvector.

[[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 right column illustrates why stationary states are called "stationary".]]
The most famous example is the [[relativistic quantum mechanics|nonrelativistic]] Schrödinger equation for the wave function in position space <math>\Psi(\mathbf{r},t)</math> of a single particle subject to a potential <math>V(\mathbf{r},t)</math>, such as that due to an [[electric field]].<ref>[http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/scheq.html "Schrodinger equation"]. Hyperphysics. Department of Physics and Astronomy, George State University.</ref><ref group="note">For a charged particle moving under the influence of a magnetic field, see the [[Pauli equation]].</ref>

{{Equation box 1
|indent=:
|title='''Time-dependent Schrödinger equation in position basis'''<br/>''(single [[relativistic quantum mechanics|nonrelativistic]] particle)''
|equation=<math>i\hbar\frac{\partial}{\partial t} \Psi(\mathbf{r},t) = \left [ \frac{-\hbar^2}{2m}\nabla^2 + V(\mathbf{r},t)\right ] \Psi(\mathbf{r},t)</math>
|cellpadding
|border
|border colour = rgb(0,115,207)
|background colour = rgb(0,115,207,10%)}}

where <math>m</math> is the particle's mass, and <math>\nabla^2</math> is the [[Laplacian]].

This is also a [[diffusion equation]], but unlike the [[heat equation]], this one is also a wave equation given the [[imaginary unit]] present in the transient term.

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 a strictly classical approximation to reality and 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 for the system, accounting for the kinetic and 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.

==={{anchor|Time independent equation}}Time-independent equation===
The time-dependent Schrödinger equation described above predicts that wave functions can form [[standing wave]]s, called [[stationary state]]s.<ref group="note">In chemistry, stationary states are atomic and molecular orbitals.</ref> 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'' (TISE).

{{Equation box 1
|indent=:
|title='''Time-independent Schrödinger equation''' (''general'')
|equation=<math>\operatorname{\hat H}|\Psi\rangle = E |\Psi\rangle </math>
|cellpadding
|border
|border colour = rgb(80,200,120)
|background colour = rgb(80,200,120,10%)}}

where <math>E</math> is a constant equal to the energy level of the system. 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 still has a time dependency.

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>.

As before, the most common manifestation is the [[relativistic quantum mechanics|nonrelativistic]] Schrödinger equation for a single particle moving in an electric field (but not a magnetic field):

{{Equation box 1
|indent=:
|title='''Time-independent Schrödinger equation''' (''single nonrelativistic particle'')
|equation=<math>\left[ \frac{-\hbar^2}{2m}\nabla^2 + V(\mathbf{r}) \right] \Psi(\mathbf{r}) = E \Psi(\mathbf{r})</math>
|cellpadding
|border
|border colour = rgb(0,115,207)
|background colour = rgb(0,115,207,10%)}}

with definitions as above. Here, the form of the Hamiltonian operator comes from classical mechanics, where the Hamiltonian ''function'' is the sum of the kinetic and potential energies. That is, <math>H = T + V = \frac{\|\mathbf{p}\|^2}{2m} + V(x, y, z)</math> for a single particle in the non-relativistic limit.

The time-independent Schrödinger equation is discussed further below.

==Derivation==
One may derive the Schrodinger equation starting from the [[Dirac-von Neumann axioms]]. Suppose the [[wave function]] <math>\psi(t_0)</math> represents a unit vector defined on a complex [[Hilbert space]] at some initial time <math>t_0</math>. The [[Unitarity (physics)|unitarity principle]] requires that there must exist a linear operator, <math>\hat{U}(t)</math>, such that for any time <math>t > t_0</math>,
{{NumBlk|:|<math>\psi(t) = \hat{U}(t)\psi(t_0).</math>|{{EquationRef|1}}}}
Given that <math>\psi(t)</math> must remain a unit vector, the operator <math>\hat{U}(t)</math> must therefore be a [[Unitary transformation (quantum mechanics)|unitary transformation]]. As such, there exists an [[Exponential map (Lie theory)|exponential map]] such that <math>\hat{U}(t) = e^{-i\hat{\mathcal{H}}t}</math> where <math>\hat{\mathcal{H}}</math> is a [[Hermitian operator]] (given by the fact that the [[Lie algebra]] of the [[unitary group]] is generated by skew-[[Hermitian operators]], and if <math>\hat{\mathcal{H}}</math> is Hermitian, then <math>-i \hat{\mathcal{H}}</math> is skew-Hermitian). Therefore, the first-order Taylor expansion of <math>\hat{U}(t)</math> centered at <math>t_0</math> takes the form
:<math>\hat{U}(t) \approx 1 - i(t-t_0)\hat{\mathcal{H}}.</math>

And so, substituting the above expansion into ({{EquationNote|1}}) thus yields
:<math>\psi(t) = \psi(t_0) - i(t-t_0)\hat{\mathcal{H}}\psi(t_0),</math>

which, when rearranged and taken in the limit <math>t \rightarrow t_0</math>, provides an equation of the same form as the Schr&ouml;dinger equation
:<math>i\frac{d\psi}{dt} = \hat{\mathcal{H}}\psi,</math>

where the ordinary definition for the derivative was used. However, the operator <math>\hat{\mathcal{H}}</math> used here denotes an ''arbitrary'' Hermitian operator. Nonetheless, by using the [[correspondence principle]] it is possible to show that, in the classical limit and using appropriate units, the expectation value of <math>\hat{\mathcal{H}}</math> indeed corresponds to the Hamiltonian of the system.<ref>{{cite book|title=Modern Quantum Mechanics|last1=Sakurai|first1=J. J.|date=1995|publisher=Addison-Wesley|location=Reading, Massachusetts|page=68}}</ref>

==Implications==
===Energy===
The Hamiltonian is constructed in the same manner as in classical mechanics. However, in classical mechanics, the Hamiltonian is a scalar-valued function, whereas in quantum mechanics, it is an operator on a space of functions. It is not surprising that the [[eigenvalues]] of <math>\hat{H}</math> are the energy levels of the system.

===Quantization===
The Schrödinger equation predicts that if certain properties of a system are measured, the result may be ''quantized'', meaning that only specific discrete values can occur. One example is ''energy quantization'': the energy of an electron in an atom is always one of the [[Energy level|quantized energy levels]], a fact discovered via [[atomic spectroscopy]]. (Energy quantization is discussed [[#Time-independent|below]].) Another example is [[angular momentum operator|quantization of angular momentum]]. This was an ''assumption'' in the earlier [[Bohr model|Bohr model of the atom]], but it is a ''prediction'' of the Schrödinger equation.

Another result of the Schrödinger equation is that not every measurement gives a quantized result in quantum mechanics. For example, position, momentum, time, and (in some situations) energy can have any value across a continuous range.<ref name="Zettili2009">{{cite book|author=Nouredine Zettili|title=Quantum Mechanics: Concepts and Applications|date=17 February 2009|publisher=John Wiley & Sons|isbn=978-0-470-02678-6}}</ref>{{rp|165–167}}

===Quantum tunneling===
{{main|Quantum tunneling}}
[[File:TunnelEffektKling1.png|width = 300px|thumb|[[Quantum tunneling]] through a barrier. A particle coming from the left does not have enough energy to climb the barrier. However, it can sometimes "tunnel" to the other side. ]]

In classical physics, when a ball is rolled slowly up a large hill, it will come to a stop and roll back, because it doesn't have enough energy to get over the top of the hill to the other side. However, the Schrödinger equation predicts that there is a small probability that the ball will get to the other side of the hill, even if it has too little energy to reach the top. This is called [[quantum tunneling]]. It is related to the distribution of energy: although the ball's assumed position seems to be on one side of the hill, there is a chance of finding it on the other side.

===Particles as waves===
{{main|Matter wave|Wave–particle duality|Double-slit experiment}}

[[File:Double-slit experiment results Tanamura 2.jpg|175px|thumb|A double slit experiment showing the accumulation of electrons on a screen as time passes.]]

The nonrelativistic Schrödinger equation is a type of [[partial differential equation]] called a [[wave equation]]. Therefore, it is often said particles can exhibit behavior usually attributed to waves. In some modern interpretations this description is reversed – the quantum state, i.e. wave, is the only genuine physical reality, and under the appropriate conditions it can show features of particle-like behavior. However, Ballentine<ref name=Ballentine>{{citation|last=Ballentine|first=Leslie|title=Quantum Mechanics: A Modern Development | publisher=World Scientific Publishing Co.|year=1998| isbn = 978-9810241056}}</ref>{{rp|Chapter 4, p.99}} shows that such an interpretation has problems. Ballentine points out that whilst it is arguable to associate a physical wave with a single particle, there is still only ''one'' Schrödinger wave equation for many particles. He points out:
:''"If a physical wave field were associated with a particle, or if a particle were identified with a wave packet, then corresponding to N interacting particles there should be N interacting waves in ordinary three-dimensional space. But according to (4.6) that is not the case; instead there is one "wave" function in an abstract 3N-dimensional configuration space. The misinterpretation of psi as a physical wave in ordinary space is possible only because the most common applications of quantum mechanics are to one-particle states, for which configuration space and ordinary space are isomorphic."''

[[Double-slit experiment|Two-slit diffraction]] is a famous example of the strange behaviors that waves regularly display, that are not intuitively associated with particles. The overlapping waves from the two slits cancel each other out in some locations, and reinforce each other in other locations, causing a complex pattern to emerge. Intuitively, one would not expect this pattern from firing a single particle at the slits, because the particle should pass through one slit or the other, not a complex overlap of both.

However, since the Schrödinger equation is a [[wave equation]], a single particle fired through a double-slit ''does'' show this same pattern (figure on right). The experiment must be repeated many times for the complex pattern to emerge. Although this is counterintuitive, the prediction is correct; in particular, [[electron diffraction]] and [[neutron diffraction]] are well understood and widely used in science and engineering.

Related to [[diffraction]], particles also display [[Superposition principle#Application to waves|superposition]] and [[Interference (wave propagation)|interference]].

The superposition property allows the particle to be in a [[quantum superposition]] of two or more quantum states at the same time. However, a "quantum state" in quantum mechanics means the ''probability'' that a system will be, for example at a position {{math|''x''}}, not that the system will actually be at position {{math|''x''}}. It does not imply that the particle itself may be in two classical states at once. Indeed, quantum mechanics is generally unable to assign values for properties prior to measurement at all.

====Measurement and uncertainty====
{{main|Measurement in quantum mechanics|Heisenberg uncertainty principle|Interpretations of quantum mechanics}}
In classical mechanics, a particle has, at every moment, an exact position and an exact momentum. These values change [[determinism|deterministically]] as the particle moves according to [[Newton's laws]]. Under the [[Copenhagen interpretation]] of quantum mechanics, particles do not have exactly determined properties, and when they are measured, the result is randomly drawn from a [[probability distribution]]. The Schrödinger equation predicts what the probability distributions are, but fundamentally cannot predict the exact result of each measurement.

The [[Heisenberg uncertainty principle]] is one statement of the inherent measurement uncertainty in quantum mechanics. It states that the more precisely a particle's position is known, the less precisely its momentum is known, and vice versa.

The Schrödinger equation describes the (deterministic) evolution of the [[wave function]] of a particle. However, even if the wave function is known exactly, the result of a specific measurement on the wave function is uncertain.

==Interpretation of the wave function==
{{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. [[Interpretations of quantum mechanics]] address questions such as what the relation is between the wave function, the underlying reality, and the results of experimental measurements.

An important aspect is the relationship between the Schrödinger equation and [[wave function collapse]]. In the oldest [[Copenhagen interpretation]], particles follow the Schrödinger equation ''except'' during wave function collapse, during which they behave entirely differently. The advent of [[quantum decoherence|quantum decoherence theory]] allowed alternative approaches (such as the [[Everett many-worlds interpretation]] and [[consistent histories]]), wherein the Schrödinger equation is ''always'' satisfied, and wave function collapse should be explained as a consequence of the Schrödinger equation.

In 1952, [[Erwin Schrödinger]] gave a lecture during which he commented,
:Nearly every result [a quantum theorist] pronounces is about the probability of this ''or'' that or that ... happening—with usually a great many alternatives. The idea that they be not alternatives but ''all'' really happen simultaneously seems lunatic to him, just ''impossible.''<ref>{{cite book|last=Schrödinger |first=Erwin |authorlink=Erwin Schrödinger|title=The interpretation of quantum mechanics: Dublin seminars (1949–1955) and other unpublished essays |year=1995 |publisher=Ox Bow Press |isbn=9781881987086}}</ref>

[[David Deutsch]] regarded this as the earliest known reference to a many-worlds interpretation of quantum mechanics, an interpretation generally credited to [[Hugh Everett III]],<ref>David Deutsch, ''The Beginning of Infinity'', page 310</ref> while [[Jeffrey A. Barrett]] took the more modest position that it indicates a "similarity in ... general views" between Schrödinger and Everett.<ref>{{cite book|last=Barrett |first=Jeffrey A. |authorlink=Jeffrey A. Barrett |title=The Quantum Mechanics of Minds and Worlds |pages=63 |publisher=[[Oxford University Press]] |year=1999 |isbn=9780191583254}}</ref>

==Historical background and development==
[[File:Erwin Schrodinger2.jpg|right|thumb|[[Erwin Schrödinger]]]]
{{Main|Theoretical and experimental justification for the Schrödinger equation}}

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>p = \frac{h}{\lambda} = \hbar k,</math>

where <math>h</math> is [[Planck's constant]] and <math>\hbar = {h}/{2\pi}</math> is the reduced Planck constant (or the Dirac 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 waves 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.
 |year        = 1925
 |title       = Recherches sur la théorie des quanta
 |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
 |archiveurl  = https://web.archive.org/web/20090509012910/http://www.ensmp.fr/aflb/LDB-oeuvres/De_Broglie_Kracklauer.pdf
 |archivedate = 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> L = n{h \over 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>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}}</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. Unfortunately the paper was rejected by the ''Physical Review'', as recounted by 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=29–32|url=https://archive.org/details/radiantscienceda00kame}}</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|William R. Hamilton]]'s 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=Schrodinger |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><!--
--- Is the non-formulation by Hamilton really relevant?

Hamilton believed that mechanics was the zero-wavelength limit of wave propagation, but did not formulate an equation for those waves.<ref>
{{cite web
 |last=Michon |first=G.P.
 |year=2009
 |title=Hamilton's Analogy: Paths to the Schrödinger Equation
 |url=http://home.att.net/~numericana/answer/schrodinger.htm#1928
 |work=Final Answers: The Schrödinger Equation
 |accessdate=28 February 2010
}}</ref>

----> A modern version of his reasoning is reproduced below. The equation he found is:<ref name="verlagsgesellschaft1991">''Encyclopaedia of Physics'' (2nd Edition), R. G. Lerner, G. L. Trigg, VHC publishers, 1991, (Verlagsgesellschaft) 3-527-26954-1, (VHC Inc.) {{isbn|0-89573-752-3}}</ref>

:<math>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>

However, 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.
 |year=1919
 |title=Atombau und Spektrallinien
 |publisher=Friedrich Vieweg und Sohn
 |location=Braunschweig
 |isbn=978-3-87144-484-5
}}</ref><ref>For an English source, see {{Cite journal
 |last=Haar |first=T.
 |title=The Old Quantum Theory
 |url=
 |pages=
 |isbn=
}}</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>\left(E + {e^2\over r} \right)^2 \psi(x) = - \nabla^2\psi(x) + m^2 \psi(x).</math>
<!-- <math>\frac{1}{c^2}\left(E + {e^2\over 4 \pi \varepsilon_0 r} \right)^2 \psi(x) = -\hbar^2 \nabla^2\psi(x) + \frac{m^2c^2}{\hbar^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 in an isolated mountain cabin in December 1925.<ref>
{{cite book
 |last=Rhodes |first=R.
 |year=1986
 |title=Making of the Atomic Bomb
 |publisher=Touchstone
 |pages=
 |isbn=978-0-671-44133-3
}}</ref>{{failed verification|date=September 2018}}

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|author=Erwin Schrödinger|title=Collected Papers on Wave Mechanics: Third Edition|year=1982|publisher=American Mathematical Soc.|isbn=978-0-8218-3524-1}}</ref>{{rp|1}}<ref>{{cite journal
 |last=Schrödinger |first=E.
 |year=1926
 |title=Quantisierung als Eigenwertproblem; von Erwin Schrödinger
 |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
|bibcode = 1926AnP...384..361S }}</ref> In the equation, 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]]. In a paper, Schrödinger himself explained this equation as follows:

{{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. |20px|20px|Erwin Schrödinger<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, 323–38. 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.</ref>}}

This 1926 paper was enthusiastically endorsed by Einstein, who saw the matter-waves as an intuitive depiction of nature, as opposed to Heisenberg's [[matrix mechanics]], which he considered overly formal.<ref>
{{Cite journal
 |last=Einstein |first=A.
 |display-authors=etal 
 |title=Letters on Wave Mechanics: Schrodinger–Planck–Einstein–Lorentz
 |url=
}}</ref>

The Schrödinger equation details the behavior of <math>\Psi</math> but says nothing of its ''nature''. Schrödinger tried to interpret it as a charge density in his fourth paper, but he was unsuccessful.<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}} In 1926, just a few days after Schrödinger's fourth and final 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}} Schrödinger, though, always opposed a statistical or probabilistic approach, with its associated [[wave function collapse|discontinuities]]—much like Einstein, who believed that quantum mechanics was a statistical approximation to an underlying [[determinism|deterministic theory]]—and never reconciled with the [[Copenhagen interpretation]].{{#tag:ref |It is clear that even in his last year of life, as shown in a letter to Max Born, that Schrödinger never accepted the Copenhagen interpretation.<ref name=Moore1992/>{{rp|220}}}}

Louis de Broglie in his later years proposed a real valued [[wave function]] connected to the complex wave function by a proportionality constant and developed the [[De Broglie–Bohm theory]].

==The wave equation for particles==
{{main|Wave–particle duality}}

The Schrödinger equation is a variation on the [[diffusion equation]] where the diffusion constant is imaginary. A spike of heat will decay in amplitude and spread out; however, because the imaginary i is the generator of rotations in the complex plane, a spike in the amplitude of a matter wave will also rotate in the complex plane over time. The solutions are therefore functions which describe wave-like motions. Wave equations in physics can normally be derived from other physical laws – the wave equation for [[Vibration|mechanical vibrations]] on strings and in matter can be derived from [[Newton's laws]], where the wave function represents the [[Displacement (vector)|displacement]] of matter, and [[electromagnetic waves]] from [[Maxwell's equations]], where the wave functions are [[electric field|electric]] and [[magnetic field|magnetic]] fields. The basis for Schrödinger's equation, on the other hand, is the energy of the system and a separate [[Mathematical formulation of quantum mechanics#Postulates of quantum mechanics|postulate of quantum mechanics]]: the wave function is a description of the system.<ref name="Quantum Chemistry 1977">''Molecular Quantum Mechanics Parts I and II: An Introduction to Quantum Chemistry'' (Volume 1), P. W. Atkins, Oxford University Press, 1977, {{isbn|0-19-855129-0}}</ref> The Schrödinger equation is therefore a new concept in itself; as Feynman put it:

{{cquote|Where did we get that (equation) from? Nowhere. It is not possible to derive it from anything you know. It came out of the mind of Schrödinger. |20px|20px|Richard Feynman<ref>''The New Quantum Universe'', T. Hey, P. Walters, Cambridge University Press, 2009, {{isbn|978-0-521-56457-1}}</ref>}}

The foundation of the equation is structured to be a linear differential equation based on classical energy conservation, and consistent with the De Broglie relations. The solution is the wave function {{math|''ψ''}}, which contains all the information that can be known about the system. In the [[Copenhagen interpretation]], the modulus of {{math|''ψ''}} is related to the [[probability]] the particles are in some spatial configuration at some instant of time. Solving the equation for {{math|''ψ''}} can be used to predict how the particles will behave under the influence of the specified potential and with each other.

The Schrödinger equation was developed principally from the [[De Broglie hypothesis]], a wave equation that would describe particles,<ref name="Quanta 1974">''Quanta: A handbook of concepts'', P. W. Atkins, Oxford University Press, 1974, {{isbn|0-19-855493-1}}</ref> and can be constructed as shown informally in the following sections.<ref name="Molecules, B.H. Bransden 1983">''Physics of Atoms and Molecules'', B. H. Bransden, C. J. Joachain, Longman, 1983, {{isbn|0-582-44401-2}}</ref> For a more rigorous description of Schrödinger's equation, see also Resnick ''et al''.<ref name="Atoms, Molecules 1985"/>

=== Consistency with energy conservation ===
The total energy {{math|''E''}} of a particle is the sum of kinetic energy <math>T</math> and potential energy <math>V</math>, this sum is also the frequent expression for the [[Hamiltonian mechanics#Basic physical interpretation|Hamiltonian]] <math>H</math> in classical mechanics: 
:<math>E = T + V =H \,\!</math>
Explicitly, for a particle in one dimension with position <math>x</math>, [[mass]] <math>m</math> and [[momentum]] <math>p</math>, and potential energy <math>V</math> which generally [[Harmonic function|varies with position]] and time <math>t</math>:

:<math> E = \frac{p^2}{2m}+V(x,t)=H.</math>

For three dimensions, the [[position vector]] {{math|'''r'''}} and momentum vector {{math|'''p'''}} must be used:
:<math>E = \frac{\mathbf{p}\cdot\mathbf{p}}{2m}+V(\mathbf{r},t)=H</math>

This formalism can be extended to any fixed number of particles: the total energy of the system is then the total kinetic energies of the particles, plus the total potential energy, again the Hamiltonian. However, there can be [[interactions#Physics|interactions]] between the particles (an [[many body problem|{{math|''N''}}-body problem]]), so the potential energy {{math|''V''}} can change as the spatial configuration of particles changes, and possibly with time. The potential energy, in general, is ''not'' the sum of the separate potential energies for each particle, it is a function of all the spatial positions of the particles. Explicitly:

:<math>E=\sum_{n=1}^N \frac{\mathbf{p}_n\cdot\mathbf{p}_n}{2m_n} + V(\mathbf{r}_1,\mathbf{r}_2,\ldots,\mathbf{r}_N,t) = H \,\!</math>

=== Linearity ===
The simplest wave function is a [[plane wave]] of the form:

:<math> \Psi(\mathbf{r},t) = A e^{i(\mathbf{k}\cdot\mathbf{r}-\omega t)}</math>

where the {{math|''A''}} is the amplitude, {{math|'''k'''}} the wave vector, and <math>\omega</math> the angular frequency, of the plane wave. In general, physical situations are not purely described by plane waves, so for generality the [[superposition principle]] is required; any wave can be made by superposition of sinusoidal plane waves. So if the equation is linear, a [[linear combination]] of plane waves is also an allowed solution. Hence a necessary and separate requirement is that the Schrödinger equation is a [[linear differential equation]].

For discrete <math>\mathbf{k}</math> the sum is a [[superposition principle#Application to waves|superposition]] of plane waves:

:<math> \Psi(\mathbf{r},t) = \sum_{n=1}^\infty A_n e^{i(\mathbf{k}_n\cdot\mathbf{r}-\omega_n t)} \,\!</math>

for some real amplitude coefficients <math>A_n</math>, and for continuous <math>\mathbf{k}</math> the sum becomes an integral, the [[Fourier transform]] of a momentum space wave function:<ref name="Quantum Mechanics Demystified 2006">''Quantum Mechanics Demystified'', D. McMahon, McGraw Hill (USA), 2006, {{isbn|0-07-145546-9}}</ref>

:<math> \Psi(\mathbf{r},t) = \frac{1}{\left(\sqrt{2\pi}\,\right)^3}\int\Phi(\mathbf{k})e^{i(\mathbf{k}\cdot\mathbf{r}-\omega t)}d^3\mathbf{k} \,\!</math>

where <math>d^3 \mathbf{k} = dk_x \,dk_x \,dk_z</math>is the differential volume element in [[momentum space|{{math|'''k'''}}-space]], and the integrals are taken over all <math>\mathbf{k}</math>-space. The momentum wave function <math>\Phi (\mathbf{k})</math> arises in the integrand since the position and momentum space wave functions are Fourier transforms of each other.

=== Consistency with the de Broglie relations ===
[[File:Wavefunction values.svg|300px|thumb|Diagrammatic summary of the quantities related to the wave function, as used in De broglie's hypothesis and development of the Schrödinger equation.<ref name="Quanta 1974"/>]]

[[Photoelectric effect|Einstein's light quanta hypothesis]] (1905) states that the energy {{math|''E''}} of a quantum of light or photon is proportional to its [[frequency]] <math>\nu</math> (or [[angular frequency]], <math>\omega = 2\pi \nu</math>)
:<math>E = h\nu = \hbar \omega \,\!</math>
Likewise [[Matter wave|De Broglie's hypothesis]] (1924) states that any particle can be associated with a wave, and that the momentum <math>p</math> of the particle is inversely proportional to the [[wavelength]] <math>\lambda</math> of such a wave (or proportional to the [[wavenumber]], <math>k = \frac{2\pi}{\lambda}</math>), in one dimension, by:
:<math>p = \frac{h}{\lambda} =  \hbar k\;,</math>
while in three dimensions, wavelength {{math|''λ''}} is related to the magnitude of the [[wavevector]] {{math|'''k'''}}:
:<math>\mathbf{p} = \hbar \mathbf{k}\,,\quad |\mathbf{k}| = \frac{2\pi}{\lambda} \,. </math>

The Planck–Einstein and de Broglie relations illuminate the deep connections between energy with time, and space with momentum, and express [[wave–particle duality]]. In practice, [[natural units]] comprising <math>\hbar = 1</math> are used, as the De Broglie ''equations'' reduce to ''identities'': allowing momentum, wave number, energy and frequency to be used interchangeably, to prevent duplication of quantities, and reduce the number of dimensions of related quantities. For familiarity SI units are still used in this article.

Schrödinger's insight,{{citation needed|date=January 2014}} late in 1925, was to express the [[Phase (waves)|phase]] of a [[plane wave]] as a [[complex number|complex]] [[phase factor]] using these relations:

:<math>\Psi = Ae^{i(\mathbf{k}\cdot\mathbf{r}-\omega t)} = Ae^{i(\mathbf{p}\cdot\mathbf{r}-Et)/\hbar} </math>

and to realize that the first order [[partial derivatives]] with respect to space were

:<math> \nabla\Psi = \dfrac{i}{\hbar}\mathbf{p}Ae^{i(\mathbf{p}\cdot\mathbf{r}-Et)/\hbar} = \dfrac{i}{\hbar}\mathbf{p}\Psi. </math>

Taking partial derivatives with respect to time gives

:<math> \dfrac{\partial \Psi}{\partial t} = -\dfrac{i E}{\hbar} Ae^{i(\mathbf{p}\cdot\mathbf{r}-Et)/\hbar} = -\dfrac{i E}{\hbar} \Psi. </math>

Another postulate of quantum mechanics is that all observables are represented by [[linear operator|linear]] [[Self-adjoint operator|Hermitian operators]] which act on the wave function, and the eigenvalues of the operator are the values the observable takes. The previous derivatives are consistent with the [[energy operator]] (or Hamiltonian operator), corresponding to the time derivative,

:<math>\hat{E} \Psi = i\hbar\dfrac{\partial}{\partial t}\Psi = E\Psi  </math>

where {{math|''E''}} are the energy [[eigenvalue]]s, and the [[momentum operator]], corresponding to the spatial derivatives (the [[del|gradient]] <math>\nabla </math>),

:<math>\hat{\mathbf{p}} \Psi = -i\hbar\nabla \Psi = \mathbf{p} \Psi </math>

where {{math|'''p'''}} is a vector of the momentum eigenvalues. In the above, the "[[circumflex|hats]]" ( {{math|ˆ}} ) indicate these observables are operators, not simply ordinary numbers or vectors. The energy and momentum operators are ''[[differential operators]]'', while the potential energy operator <math>V</math> is just a multiplicative factor.

Substituting the energy and momentum operators into the classical energy conservation equation obtains the operator:

:<math>E= \dfrac{\mathbf{p}\cdot\mathbf{p}}{2m}+V \quad \rightarrow \quad \hat{E} = \dfrac{\hat{\mathbf{p}}\cdot\hat{\mathbf{p}}}{2m}  + V </math>

so in terms of derivatives with respect to time and space, acting this operator on the wave function {{math|''Ψ''}} immediately led Schrödinger to his equation:{{citation needed|date=January 2014}}

:<math>i\hbar\dfrac{\partial \Psi}{\partial t}= -\dfrac{\hbar^2}{2m}\nabla^2\Psi +V\Psi</math>

Wave–particle duality can be assessed from these equations as follows. The kinetic energy {{math|''T''}} is related to the square of momentum {{math|'''p'''}}. As the particle's momentum increases, the kinetic energy increases more rapidly, but since the wave number {{math|{{!}}'''k'''{{!}}}} increases the wavelength {{math|''λ''}} decreases. In terms of ordinary scalar and vector quantities (not operators):

:<math> \mathbf{p}\cdot\mathbf{p} \propto \mathbf{k}\cdot\mathbf{k} \propto T \propto \dfrac{1}{\lambda^2}</math>

The kinetic energy is also proportional to the second spatial derivatives, so it is also proportional to the magnitude of the ''[[curvature]]'' of the wave, in terms of operators:

:<math> \hat{T} \Psi = \frac{-\hbar^2}{2m}\nabla\cdot\nabla  \Psi \, \propto \, \nabla^2 \Psi \,.</math>

As the curvature increases, the amplitude of the wave alternates between positive and negative more rapidly, and also shortens the wavelength. So the inverse relation between momentum and wavelength is consistent with the energy the particle has, and so the energy of the particle has a connection to a wave, all in the same mathematical formulation.<ref name="Quanta 1974"/>

===Wave and particle motion===
 {{multiple image
   | align = right
   | direction = horizontal
   | image1    = Quantum mechanics travelling wavefunctions.svg
   | caption1  = Increasing levels of [[wavepacket]] localization, meaning the particle has a more localized position.
   | width1     = 400
   | image2    = Perfect localization.svg
   | caption2  = In the limit ''ħ'' → 0, the particle's position and momentum become known exactly. This is equivalent to the classical particle.
   | width2     = 150
  }}

Schrödinger required that a [[wave packet]] solution near position <math>\mathbf{r}</math> with wave vector near <math>\mathbf{k}</math> will move along the trajectory determined by classical mechanics for times short enough for the spread in <math>\mathbf{k}</math> (and hence in velocity) not to substantially increase the spread in {{math|'''r'''}}. Since, for a given spread in {{math|'''k'''}}, the spread in velocity is proportional to Planck's constant <math>\hbar</math>, it is sometimes said that in the limit as <math>\hbar</math> approaches zero, the equations of classical mechanics are restored from quantum mechanics.<ref name="Analytical Mechanics 2008">''Analytical Mechanics'', L. N. Hand, J. D. Finch, Cambridge University Press, 2008, {{isbn|978-0-521-57572-0}}</ref> Great care is required in how that limit is taken, and in what cases.

The limiting short-wavelength is equivalent to <math>\hbar</math> tending to zero because this is limiting case of increasing the wave packet localization to the definite position of the particle (see images right). Using the [[Heisenberg uncertainty principle]] for position and momentum, the products of uncertainty in position and momentum become zero as <math>\hbar \longrightarrow 0</math>:

:<math> \sigma(x) \sigma(p_x) \geqslant \frac{\hbar}{2} \quad \rightarrow \quad \sigma(x) \sigma(p_x) \geqslant 0 \,\!</math>

where {{math|''σ''}} denotes the (root mean square) [[measurement uncertainty]] in {{math|''x''}} and {{math|''p<sub>x</sub>''}} (and similarly for the {{math|''y''}} and {{math|''z''}} directions) which implies the position and momentum can only be known to arbitrary precision in this 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. 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<ref>{{harvnb|Hall|2013}} Section 3.7.5</ref>
:<math>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>-V'\left(\left\langle X\right\rangle\right)</math>,
which is typically not the same as <math>-\left\langle V'(X)\right\rangle</math>. 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.<ref>{{harvnb|Hall|2013}} p. 78</ref> When Planck's constant is small, it is possible to have a state that is well localized in ''both'' position and momentum. The small uncertainty in momentum ensures that the particle ''remains'' well localized in position for a long time, so that expected position and momentum continue to closely track the classical trajectories.

The Schrödinger equation in its general form

:<math> 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> -\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). 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>.<ref name="Analytical Mechanics 2008"/>

Substituting

:<math> \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 \longrightarrow 0</math> in the resulting equation yield the Hamilton–Jacobi equation.

The implications are as follows:

* The motion of a particle, described by a (short-wavelength) wave packet solution to the Schrödinger equation, is also described by the Hamilton–Jacobi equation of motion.
* The Schrödinger equation includes the wave function, so its wave packet solution implies the position of a (quantum) particle is fuzzily spread out in wave fronts. On the contrary, the Hamilton–Jacobi equation applies to a (classical) particle of definite position and momentum, instead the position and momentum at all times (the trajectory) are deterministic and can be simultaneously known.

==Nonrelativistic quantum mechanics==
The quantum mechanics of particles without accounting for the effects of [[special relativity]], for example particles propagating at speeds much less than [[speed of light|light]], is known as '''nonrelativistic quantum mechanics'''. Following are several forms of Schrödinger's equation in this context for different situations: time independence and dependence, one and three spatial dimensions, and one and {{math|''N''}} particles.

In actuality, the particles constituting the system do not have the numerical labels used in theory. The language of mathematics forces us to label the positions of particles one way or another, otherwise there would be confusion between symbols representing which variables are for which particle.<ref name="Atoms, Molecules 1985">''Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles'' (2nd Edition), R. Resnick, R. Eisberg, John Wiley & Sons, 1985, {{isbn|978-0-471-87373-0}}</ref>

===Time-independent===
If the Hamiltonian is not an explicit function of time, the equation is [[Separation of variables|separable]] into a product of spatial and temporal parts. In general, the wave function takes the form:

:<math>\Psi(\text{space coords},t)=\psi(\text{space coords})\tau(t)\,.</math>

where {{math|''ψ''(space coords)}} is a function of all the spatial coordinate(s) of the particle(s) constituting the system only, and {{math|''τ''(''t'')}} is a function of time only.

Substituting for {{math|''&psi;''}} into the Schrödinger equation for the relevant number of particles in the relevant number of dimensions, solving by [[separation of variables]] implies the general solution of the time-dependent equation has the form:<ref name="verlagsgesellschaft1991"/>

:<math> \Psi(\text{space coords},t) = \psi(\text{space coords}) e^{-i{E t/\hbar}} \,.</math>

Since the time dependent phase factor is always the same, only the spatial part needs to be solved for in time independent problems. Additionally, the energy operator {{math|''Ê'' {{=}} ''iħ''{{sfrac|∂|∂''t''}}}} can always be replaced by the energy eigenvalue {{math|''E''}}, thus the time independent Schrödinger equation is an [[eigenvalue]] equation for the Hamiltonian operator:<ref name=Shankar1994/>{{rp|143ff}}
:<math>\hat{H} \psi = E \psi </math>

This is true for any number of particles in any number of dimensions (in a time independent potential). This case describes the [[standing wave]] solutions of the time-dependent equation, which 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 eigenvalues from this equation form a discrete [[Spectrum (functional analysis)|spectrum]] of values, so mathematically energy must be quantized. More specifically, 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 the [[spectral theorem]] in mathematics, and in a finite state space it is just a statement of the completeness of the eigenvectors of a [[Hermitian matrix]].

===One-dimensional examples===
For a particle in one dimension, the Hamiltonian is:
 
:<math> \hat{H} = \frac{\hat{p}^2}{2m} + V(x) \,, \quad \hat{p} = -i\hbar \frac{d}{d x}  </math>

and substituting this into the general Schrödinger equation gives:

:<math> \left[-\frac{\hbar^2}{2m}\frac{d^2}{d x^2} + V(x) \right]\psi(x) = E\psi(x) </math>

This is the only case the Schrödinger equation is an [[ordinary derivative|ordinary]] differential equation, rather than a [[partial derivative|partial]] differential equation. The general solutions are always of the form:

:<math> \Psi(x,t)=\psi(x) e^{-iEt/\hbar} \, . </math>

For {{math|''N''}} particles in one dimension, the Hamiltonian is:

:<math> \hat{H} = \sum_{n=1}^{N}\frac{\hat{p}_n^2}{2m_n} + V(x_1,x_2,\ldots, x_N) \,,\quad \hat{p}_n = -i\hbar \frac{\partial}{\partial x_n}</math>

where the position of particle {{math|''n''}} is {{math|''x<sub>n</sub>''}}. The corresponding Schrödinger equation is:

:<math> -\frac{\hbar^2}{2}\sum_{n=1}^N \frac{1}{m_n}\frac{\partial^2}{\partial x_n^2}\psi(x_1,x_2,\ldots, x_N) + V(x_1,x_2,\ldots, x_N)\psi(x_1,x_2,\ldots, x_N) = E\psi(x_1,x_2,\ldots, x_N) \, .</math>

so the general solutions have the form:

:<math> \Psi(x_1,x_2,\ldots, x_N,t) = e^{-iEt/\hbar}\psi(x_1,x_2\ldots, x_N) </math>

For non-interacting distinguishable particles,<ref>{{cite book|author=N. Zettili|title=Quantum Mechanics: Concepts and Applications|edition=2nd|page=458|isbn=978-0-470-02679-3|date=2009-02-24}}</ref> the potential of the system only influences each particle separately, so the total potential energy is the sum of potential energies for each particle:

:<math> V(x_1,x_2,\ldots, x_N) = \sum_{n=1}^N V(x_n) \, .</math>

and the wave function can be written as a product of the wave functions for each particle:

:<math> \Psi(x_1,x_2,\ldots, x_N,t) = e^{-i{E t/\hbar}}\prod_{n=1}^N\psi(x_n) \, ,</math>

For non-interacting [[identical particles]], the potential is still a sum, but wave function is a bit more complicated – it is a sum over the permutations of products of the separate wave functions to account for particle exchange. In general for interacting particles, the above decompositions are ''not'' possible.

====Free particle====
For no potential, {{math|''V'' {{=}} 0}}, so the particle is free and the equation reads:<ref name=Shankar1994/>{{rp|151ff}}
:<math> - E \psi = \frac{\hbar^2}{2m}{d^2 \psi \over d x^2}\,</math>

which has oscillatory solutions for {{math|''E'' > 0}} (the {{math|''C''<sub>n</sub>}} are arbitrary constants):

:<math>\psi_E(x) = C_1 e^{i\sqrt{2mE/\hbar^2}\,x} + C_2 e^{-i\sqrt{2mE/\hbar^2}\,x}\,</math>

and exponential solutions for {{math|''E'' < 0}}

:<math>\psi_{-|E|}(x) = C_1 e^{\sqrt{2m|E|/\hbar^2}\,x} + C_2 e^{-\sqrt{2m|E|/\hbar^2}\,x}.\,</math>

The exponentially growing solutions have an infinite norm, and are not physical. They are not allowed in a finite volume with periodic or fixed boundary conditions.

See also [[free particle]] and [[wavepacket]] for more discussion on the free particle.

====Constant potential====
[[File:EffetTunnel.gif|left|Animation of a de Broglie wave incident on a barrier.]]

For a constant potential, {{math|''V'' {{=}} ''V''<sub>0</sub>}}, the solution is oscillatory for {{math|''E'' > ''V''<sub>0</sub>}} and exponential for {{math|''E'' < ''V''<sub>0</sub>}}, corresponding to energies that are allowed or disallowed in classical mechanics. Oscillatory solutions have a classically allowed energy and correspond to actual classical motions, while the exponential solutions have a disallowed energy and describe a small amount of quantum bleeding into the classically disallowed region, due to [[quantum tunneling]]. If the potential {{math|''V''<sub>0</sub>}} grows to infinity, the motion is classically confined to a finite region. Viewed far enough away, every solution is reduced to an exponential; the condition that the exponential is decreasing restricts the energy levels to a discrete set, called the allowed energies.<ref name="Quantum Mechanics Demystified 2006"/>
{{Clear}}

====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> E\psi = -\frac{\hbar^2}{2m}\frac{d^2}{d x^2}\psi + \frac{1}{2}m\omega^2x^2\psi, </math>

where <math> x </math> is the displacement and <math> \omega </math> the angular frequency. This is an example of a quantum-mechanical system whose wave function can be solved for exactly. Furthermore, it can be used to describe approximately a wide variety of other systems, including [[Molecular vibration|vibrating atoms, molecules]],<ref>''Physical Chemistry'', P. W. Atkins, Oxford University Press, 1978, {{isbn|0-19-855148-7}}</ref> and atoms or ions in lattices,<ref>''Solid State Physics'' (2nd Edition), J. R. Hook, H. E. Hall, Manchester Physics Series, John Wiley & Sons, 2010, {{isbn|978-0-471-92804-1}}</ref> and approximating other potentials near equilibrium points. It is also the [[Perturbation theory (quantum mechanics)#Applications of perturbation theory|basis of perturbation methods]] in quantum mechanics.

The solutions in position space are

:<math>  \psi_n(x) = \sqrt{\frac{1}{2^n\,n!}} \cdot \left(\frac{m\omega}{\pi \hbar}\right)^{1/4} \cdot e^{
- \frac{m\omega x^2}{2 \hbar}} \cdot \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>\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>  E_n = \left(n + \frac{1}{2} \right) \hbar \omega. </math>

The case <math>  n = 0 </math> 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|location=|pages=247–250, 254–5, 257, 272|chapter=Chapter 7: The One-Dimensional Harmonic Oscillator}}</ref>

===Three-dimensional examples===
The extension from one dimension to three dimensions is straightforward, all position and momentum operators are replaced by their three-dimensional expressions and the partial derivative with respect to space is replaced by the [[gradient]] operator.

The Hamiltonian for one particle in three dimensions is:

:<math> \hat{H} = \frac{\hat{\mathbf{p}}\cdot\hat{\mathbf{p}}}{2m} + V(\mathbf{r}) \,, \quad \hat{\mathbf{p}} = -i\hbar \nabla  </math>

generating the equation

:<math> \left[-\frac{\hbar^2}{2m}\nabla^2 + V(\mathbf{r}) \right]\psi(\mathbf{r}) = E\psi(\mathbf{r}) </math>

with stationary state solutions of the form

:<math> \Psi(\mathbf{r},t) = \psi(\mathbf{r}) e^{-iEt/\hbar}, </math>

where the position of the particle is <math> \mathbf{r} </math>.

For <math> N </math> particles in three dimensions, the Hamiltonian is

:<math> \hat{H} = \sum_{n=1}^{N}\frac{\hat{\mathbf{p}}_n\cdot\hat{\mathbf{p}}_n}{2m_n} + V(\mathbf{r}_1,\mathbf{r}_2,\ldots,\mathbf{r}_N) \,,\quad \hat{\mathbf{p}}_n = -i\hbar \nabla_n </math>

where the position of particle {{math|''n''}} is {{math|'''r'''<sub>''n''</sub>}} and the gradient operators are partial derivatives with respect to the particle's position coordinates. In Cartesian coordinates, for particle {{math|''n''}}, the position vector is {{math|'''r'''''<sub>n</sub>'' {{=}} (''x<sub>n</sub>'', ''y<sub>n</sub>'', ''z<sub>n</sub>'')}} while the gradient and [[Laplacian operator]] are respectively:

:<math>\nabla_n = \mathbf{e}_x \frac{\partial}{\partial x_n} + \mathbf{e}_y\frac{\partial}{\partial y_n} + \mathbf{e}_z\frac{\partial}{\partial z_n}\,,\quad \nabla_n^2 = \nabla_n\cdot\nabla_n = \frac{\partial^2}{{\partial x_n}^2} + \frac{\partial^2}{{\partial y_n}^2} + \frac{\partial^2}{{\partial z_n}^2}</math>

The Schrödinger equation is:

:<math> -\frac{\hbar^2}{2}\sum_{n=1}^{N}\frac{1}{m_n}\nabla_n^2\Psi(\mathbf{r}_1,\mathbf{r}_2,\ldots,\mathbf{r}_N) + V(\mathbf{r}_1,\mathbf{r}_2,\ldots,\mathbf{r}_N)\Psi(\mathbf{r}_1,\mathbf{r}_2,\ldots,\mathbf{r}_N) = E\Psi(\mathbf{r}_1,\mathbf{r}_2,\ldots,\mathbf{r}_N) </math>

with stationary state solutions:

:<math> \Psi(\mathbf{r}_1,\mathbf{r}_2,\ldots, \mathbf{r}_N,t) = e^{-iEt/\hbar}\psi(\mathbf{r}_1,\mathbf{r}_2,\ldots, \mathbf{r}_N) </math>

Again, for non-interacting distinguishable particles the potential is the sum of particle potentials

:<math> V(\mathbf{r}_1,\mathbf{r}_2,\ldots, \mathbf{r}_N) = \sum_{n=1}^N V(\mathbf{r}_n) </math>

and the wave function is a product of the particle wave functions

:<math> \Psi(\mathbf{r}_1,\mathbf{r}_2 ,\ldots, \mathbf{r}_N,t) = e^{-i{E t/\hbar}} \prod_{n=1}^N\psi(\mathbf{r}_n) \, . </math>

For non-interacting identical particles, the potential is a sum but the wave function is a sum over permutations of products. The previous two equations do not apply to interacting particles.

Following are examples where exact solutions are known. See the main articles for further details.

====Hydrogen atom====
The Schrödinger equation for the [[hydrogen atom]] (or a hydrogen-like atom) is<ref name="Quantum Chemistry 1977"/><ref name="Quanta 1974"/>

:<math> 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> \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 centre of mass, and constitute a two-body problem to solve. The motion of the electron is of principle interest here, so the equivalent one-body problem is the motion of the electron using the reduced mass.

The Schrödinger for a hydrogen atom can be solved by separation of variables.<ref>''Physics for Scientists and Engineers – with Modern Physics'' (6th Edition), P. A. Tipler, G. Mosca, Freeman, 2008, {{isbn|0-7167-8964-7}}</ref> In this case, [[spherical polar coordinates]] are the most convenient. Thus,

:<math> \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|author=David Griffiths|title=Introduction to elementary particles|url=https://books.google.com/books?id=w9Dz56myXm8C&pg=PA162|accessdate=27 June 2011|year=2008|publisher=Wiley-VCH|isbn=978-3-527-40601-2|pages=162–}}</ref>

:<math> \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> 
\begin{align} 
n & = 1,2,3, \dots \\
\ell & = 0,1,2, \dots, n-1 \\
m & = -\ell,\dots,\ell \\
\end{align}
</math>

The [[Laguerre polynomial#Generalized Laguerre polynomials|generalized Laguerre polynomials]] are defined differently by different authors. See main article on them and the hydrogen atom.

====Two-electron atoms or ions====
The equation for any two-electron system, such as the neutral [[helium atom]] (He, <math> Z = 2 </math>), the negative [[hydrogen]] [[ion]] (H<sup>−</sup>, <math> Z = 1 </math>), or the positive [[lithium]] ion (Li<sup>+</sup>, <math> Z = 3 </math>) is:<ref name="Molecules, B.H. Bransden 1983"/>

:<math> E\psi = -\hbar^2\left[\frac{1}{2\mu}\left(\nabla_1^2 +\nabla_2^2 \right) + \frac{1}{M}\nabla_1\cdot\nabla_2\right] \psi + \frac{e^2}{4\pi\varepsilon_0}\left[ \frac{1}{r_{12}} -Z\left( \frac{1}{r_1}+\frac{1}{r_2} \right) \right] \psi </math>

where {{math|'''r'''<sub>1</sub>}} is the relative position of one electron ({{math|''r''<sub>1</sub> {{=}} {{abs|'''r'''<sub>1</sub>}}}} is its relative magnitude), {{math|'''r'''<sub>2</sub>}} is the relative position of the other electron ({{math|''r''<sub>2</sub> {{=}} {{!}}'''r'''<sub>2</sub>{{!}}}} is the magnitude), {{math|''r''<sub>12</sub> {{=}} {{!}}'''r'''<sub>12</sub>{{!}}}} is the magnitude of the separation between them given by

:<math> |\mathbf{r}_{12}| = |\mathbf{r}_2 - \mathbf{r}_1 | \,\!</math>

{{math|''μ''}} is again the two-body reduced mass of an electron with respect to the nucleus of mass {{math|''M''}}, so this time

:<math> \mu = \frac{m_e M}{m_e+M} \,\!</math>

and {{math|''Z''}} is the [[atomic number]] for the element (not a [[quantum number]]).

The cross-term of two Laplacians

:<math>\frac{1}{M}\nabla_1\cdot\nabla_2\,\!</math>

is known as the ''mass polarization term'', which arises due to the motion of [[Atomic nucleus|atomic nuclei]]. The wave function is a function of the two electron's positions:

:<math> \psi = \psi(\mathbf{r}_1,\mathbf{r}_2). </math>

There is no closed form solution for this equation.

===Time-dependent===
This is the equation of motion for the quantum state. In the most general form, it is written:<ref name=Shankar1994/>{{rp|143ff}}

:<math>i \hbar \frac{\partial}{\partial t}\Psi = \hat H \Psi.</math>

and the solution, the wave function, is a function of all the particle coordinates of the system and time. Following are specific cases.

For one particle in one dimension, the Hamiltonian

:<math> \hat{H} = \frac{\hat{p}^2}{2m} + V(x,t) \,,\quad \hat{p} = -i\hbar \frac{\partial}{\partial x} </math>

generates the equation:

:<math> i\hbar\frac{\partial}{\partial t}\Psi(x,t) = -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}\Psi(x,t) + V(x,t)\Psi(x,t) </math>

For {{math|''N''}} particles in one dimension, the Hamiltonian is:

:<math> \hat{H} = \sum_{n=1}^{N}\frac{\hat{p}_n^2}{2m_n} + V(x_1,x_2,\ldots, x_N,t) \,,\quad \hat{p}_n = -i\hbar \frac{\partial}{\partial x_n} </math>

where the position of particle {{math|''n''}} is {{math|''x<sub>n</sub>''}}, generating the equation:

:<math> i\hbar\frac{\partial}{\partial t}\Psi(x_1,x_2,\ldots, x_N,t) = -\frac{\hbar^2}{2}\sum_{n=1}^{N}\frac{1}{m_n}\frac{\partial^2}{\partial x_n^2}\Psi(x_1,x_2,\ldots, x_N,t) + V(x_1,x_2,\ldots, x_N,t)\Psi(x_1,x_2,\ldots, x_N,t) \, .</math>

For one particle in three dimensions, the Hamiltonian is:

:<math> \hat{H} = \frac{\hat{\mathbf{p}}\cdot\hat{\mathbf{p}}}{2m} + V(\mathbf{r},t) \,,\quad \hat{\mathbf{p}} = -i\hbar \nabla </math>

generating the equation:

:<math> i\hbar\frac{\partial}{\partial t}\Psi(\mathbf{r},t) = -\frac{\hbar^2}{2m}\nabla^2\Psi(\mathbf{r},t) + V(\mathbf{r},t)\Psi(\mathbf{r},t) </math>

For {{math|''N''}} particles in three dimensions, the Hamiltonian is:

:<math>  \hat{H} = \sum_{n=1}^{N}\frac{\hat{\mathbf{p}}_n\cdot\hat{\mathbf{p}}_n}{2m_n} + V(\mathbf{r}_1,\mathbf{r}_2,\ldots,\mathbf{r}_N,t)\,,\quad \hat{\mathbf{p}}_n = -i\hbar \nabla_n </math>

where the position of particle {{math|''n''}} is {{math|'''r'''<sub>''n''</sub>}}, generating the equation:<ref name=Shankar1994/>{{rp|141}}
:<math> i\hbar\frac{\partial}{\partial t}\Psi(\mathbf{r}_1,\mathbf{r}_2,\ldots,\mathbf{r}_N,t) = -\frac{\hbar^2}{2} \sum_{n=1}^N \frac{1}{m_n}\nabla_n^2\Psi(\mathbf{r}_1,\mathbf{r}_2,\ldots,\mathbf{r}_N,t) + V(\mathbf{r}_1, \mathbf{r}_2, \ldots, \mathbf{r}_N,t)\Psi(\mathbf{r}_1,\mathbf{r}_2,\ldots,\mathbf{r}_N,t) </math>

This last equation is in a very high dimension, so the solutions are not easy to visualize.

== Solution methods ==
{{prose|date=October 2016}}
{{Col-begin}}
{{Col-break}}
General techniques:
* Separation of variables
* [[Perturbation theory (quantum mechanics)|Perturbation theory]]
* The [[variational method]]
* [[Quantum Monte Carlo]] methods
* [[Density functional theory]]
* The [[WKB approximation]] and semi-classical expansion
{{Col-break}}

Methods for special cases:
* [[List of quantum-mechanical systems with analytical solutions]]
* [[Hartree–Fock]] method and [[post Hartree–Fock]] methods
{{col-end}}

== Properties ==
The Schrödinger equation has the following properties: some are useful, but there are shortcomings. Ultimately, these properties arise from the Hamiltonian used, and the solutions to the equation.

===Linearity===
{{See also|Linear differential equation}}

In the development above, the Schrödinger equation was made to be linear for generality, though this has other implications. If two wave functions {{math|''ψ''<sub>1</sub>}} and {{math|''ψ''<sub>2</sub>}} are solutions, then so is any [[linear combination]] of the two:

:<math>\displaystyle \psi = a\psi_1 + b \psi_2 </math>

where {{math|''a''}} and {{math|''b''}} are any complex numbers (the sum can be extended for any number of wave functions). 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 all single state solutions achievable. For example, consider a wave function {{math|''Ψ''(''x'', ''t'')}} such that the wave function is a product of two functions: one time independent, and one time dependent. If states of definite energy found using the time independent Schrödinger equation are given by {{math|''ψ<sub>E</sub>''(''x'')}} with amplitude {{math|''A<sub>n</sub>''}} and time dependent phase factor is given by

:<math>e^{{-iE_n t}/\hbar}, </math>

then a valid general solution is

:<math>\displaystyle \Psi(x,t) = \sum\limits_{n} A_n \psi_{E_n}(x) e^{{-iE_n t}/\hbar}. </math>

Additionally, the ability to scale solutions allows one to solve for a wave function without normalizing it first. If one has a set of normalized solutions {{math|''ψ<sub>n</sub>''}}, then

:<math>\displaystyle \Psi = \sum\limits_{n} A_n \psi_n </math>

can be normalized by ensuring that

:<math>\displaystyle \sum\limits_{n}|A_n|^2 = 1. </math>

This is much more convenient than having to verify that

:<math>\displaystyle \int\limits_{-\infty}^{\infty}|\Psi(x)|^2\,dx = \int\limits_{-\infty}^{\infty}\Psi(x)\Psi^{*}(x)\,dx = 1. </math>

=== Momentum space Schrödinger equation ===
The Schrödinger equation <math>i\hbar \partial_t |\psi\rangle = \hat{H}|\psi \rangle </math> is often presented in the position basis form <math>i\hbar \partial_t \psi(x) = -\frac{\hbar^2}{2m}\nabla^2\psi(x) + V(x)\psi(x) </math> (with <math>\langle x |\psi\rangle \equiv \psi(x) </math>). But as a vector operator equation it has a valid representation in any arbitrary complete basis of kets in [[Hilbert space]]. For example, in the momentum space basis the equation reads

<math>\displaystyle i\hbar \partial_t f(p) = \frac{p^2}{2m}f(p) + (\tilde{V}*f) (p) </math>

where <math>|p\rangle </math> is the plane wave state of definite momentum <math>p </math>, <math>\langle p |\psi\rangle \equiv f(p) = \int \psi(x)e^{-ipx}dx </math>, <math>\tilde{V} </math> is the Fourier transform of <math>V </math>, and <math>* </math> denotes [[Convolution theorem|Fourier convolution]].

In the 1D example with absence of a potential, <math>\tilde V = 0 </math> (or similarly <math>\tilde V(k) \propto \delta(k) </math> in the case of a background potential constant throughout space), each stationary state of energy <math> \hbar \omega = q^2/2m </math> is of the form

<math> f_q(p) = (c_{+}\delta(p-q)+c_{-}\delta(p+q))e^{-i\omega t} </math>

for arbitrary complex coefficients <math> c_{\pm} </math>. Such a wave function, as expected in free space, is a superposition of plane waves moving right and left with momenta <math> \pm q </math>; upon momentum measurement the state would collapse to one of definite momentum <math> \pm q </math> with probability <math> \propto |c_{\pm}|^2 </math>.

A version of the momentum space Schrödinger equation is often used in [[solid-state physics]], as [[Bloch's Theorem|Bloch's theorem]] ensures the periodic crystal lattice potential couples <math>f(p) </math> with <math>f(p+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.

=== Real energy eigenstates ===
For the time-independent equation, an additional feature of linearity follows: if two wave functions {{math|''ψ''<sub>1</sub>}} and {{math|''&psi;''<sub>2</sub>}} are solutions to the time-independent equation with the same energy {{math|''E''}}, then so is any linear combination:

:<math> \hat H (a\psi_1 + b \psi_2 ) = a \hat H \psi_1 + b \hat H \psi_2 = E (a \psi_1 + b\psi_2). </math>

Two different solutions with the same energy are called ''degenerate''.<ref name="Quantum Mechanics Demystified 2006"/>

In an arbitrary potential, if a wave function {{math|''ψ''}} solves the time-independent equation, so does its [[complex conjugate]], denoted {{math|''ψ''*}}. By taking linear combinations, the real and imaginary parts of {{math|''ψ''}} are each solutions. If there is no degeneracy they can only differ by a factor.

In the time-dependent equation, complex conjugate waves move in opposite directions. If {{math|''Ψ''(''x'', ''t'')}} is one solution, then so is {{math|''Ψ''*(''x'', –''t'')}}. The symmetry of complex conjugation is called [[T-symmetry|time-reversal symmetry]].

===Space and time derivatives===
[[File:Wavefunction continuity space.svg|250px|thumb|Continuity of the wave function and its first spatial derivative (in the {{math|''x''}} direction, {{math|''y''}} and {{math|''z''}} coordinates not shown), at some time {{math|''t''}}.]]

The Schrödinger equation is first order in time and second in space, which describes the time evolution of a quantum state (meaning it determines the future amplitude from the present).

Explicitly for one particle in 3-dimensional Cartesian coordinates – the equation is

:<math>i\hbar{\partial \Psi \over \partial t} = - {\hbar^2\over 2m} \left ( {\partial^2 \Psi \over \partial x^2} + {\partial^2 \Psi \over \partial y^2} + {\partial^2 \Psi \over \partial z^2} \right ) + V(x,y,z,t)\Psi.\,\!</math>

The first time partial derivative implies the initial value (at {{math|''t'' {{=}} 0}}) of the wave function

:<math> \Psi(x,y,z,0) \,\!</math>

is an arbitrary constant. Likewise – the second order derivatives with respect to space implies the wave function ''and'' its first order spatial derivatives

:<math> \begin{align} 
& \Psi(x_b,y_b,z_b,t) \\
& \frac{\partial}{\partial x}\Psi(x_b,y_b,z_b,t) \quad \frac{\partial}{\partial y}\Psi(x_b,y_b,z_b,t) \quad \frac{\partial}{\partial z}\Psi(x_b,y_b,z_b,t) 
\end{align} \,\!</math>

are all arbitrary constants at a given set of points, where {{math|''x<sub>b</sub>''}}, {{math|''y<sub>b</sub>''}}, {{math|''z<sub>b</sub>''}} are a set of points describing boundary {{math|''b''}} (derivatives are evaluated at the boundaries). Typically there are one or two boundaries, such as the [[step potential]] and [[particle in a box]] respectively.

As the first order derivatives are arbitrary, the wave function can be a [[smooth function|continuously differentiable function]] of space, since at any boundary the gradient of the wave function can be matched.

On the contrary, wave equations in physics are usually ''second order in time'', notable are the family of classical [[wave equation]]s and the quantum [[Klein–Gordon equation]].

=== Local conservation of probability ===
{{Main|Probability current|Continuity equation}}

The Schrödinger equation is consistent with [[conservation of probability|probability conservation]]. Multiplying the Schrödinger equation on the right by the complex conjugate wave function, and multiplying the wave function to the left of the complex conjugate of the Schrödinger equation, and subtracting, gives the [[Continuity equation#Quantum mechanics|continuity equation]] for probability:<ref name="Quantum Mechanics 2004">''Quantum Mechanics'', E. Abers, Pearson Ed., Addison Wesley, Prentice Hall Inc, 2004, {{isbn|978-0-13-146100-0}}</ref>
:<math>{ \partial \over \partial t} \rho\left(\mathbf{r},t\right) + \nabla \cdot \mathbf{j} = 0, </math>
where 
:<math>\rho=|\Psi|^2=\Psi^*(\mathbf{r},t)\Psi(\mathbf{r},t)\,\!</math>
is the [[probability density function|probability density]] (probability per unit volume, {{math|*}} denotes [[complex conjugate]]), and
:<math> \mathbf{j} = {1 \over 2m} \left( \Psi^*\hat{\mathbf{p}}\Psi  - \Psi\hat{\mathbf{p}}\Psi^* \right)\,\!</math>
is the [[probability current]] (flow per unit area).

Hence predictions from the Schrödinger equation do not violate probability conservation.

=== Positive energy ===
If the potential is bounded from below, meaning there is a minimum value of potential energy, the eigenfunctions of the Schrödinger equation have energy which is also bounded from below. This can be seen most easily by using the [[variational principle]], as follows. (See also below).

For any linear operator {{math|''Â''}} [[Bounded operator|bounded]] from below, the eigenvector with the smallest eigenvalue is the vector {{math|''ψ''}} that minimizes the quantity

:<math> \langle \psi |\hat{A}|\psi \rangle </math>

over all {{math|''ψ''}} which are [[normalizable wave function|normalized]].<ref name="Quantum Mechanics 2004"/> In this way, the smallest eigenvalue is expressed through the [[variational principle]]. For the Schrödinger Hamiltonian {{math|''Ĥ''}} bounded from below, the smallest eigenvalue is called the ground state energy. That energy is the minimum value of

:<math>\langle \psi|\hat{H}|\psi\rangle = \int \psi^*(\mathbf{r}) \left[ - \frac{\hbar^2}{2m} \nabla^2\psi(\mathbf{r}) + V(\mathbf{r})\psi(\mathbf{r})\right] d^3\mathbf{r} = \int \left[ \frac{\hbar^2}{2m}|\nabla\psi|^2 + V(\mathbf{r}) |\psi|^2 \right] d^3\mathbf{r} = \langle \hat{H}\rangle </math>

(using [[integration by parts]]). Due to the [[Absolute value#Complex numbers|complex modulus]] of {{math|''ψ''<sup>2</sup>}} (which is positive definite), the right hand side is always greater than the lowest value of {{math|''V''(''x'')}}. In particular, the ground state energy is positive when {{math|''V''(''x'')}} is everywhere positive.

For potentials which are bounded below and are not infinite over a region, there is a ground state which minimizes the integral above. This lowest energy wave function is real and positive definite – meaning the wave function can increase and decrease, but is positive for all positions. It physically cannot be negative: if it were, smoothing out the bends at the sign change (to minimize the wave function) rapidly reduces the gradient contribution to the integral and hence the kinetic energy, while the potential energy changes linearly and less quickly. The kinetic and potential energy are both changing at different rates, so the total energy is not constant, which can't happen (conservation). The solutions are consistent with Schrödinger equation if this wave function is positive definite.

The lack of sign changes also shows that the ground state is nondegenerate, since if there were two ground states with common energy {{math|''E''}}, not proportional to each other, there would be a linear combination of the two that would also be a ground state resulting in a zero solution.

===Analytic continuation to diffusion===
{{see also|path integral formulation#The Schrödinger equation|label 1=Path integral formulation (The Schrödinger equation)}}

The above properties (positive definiteness of energy) allow the [[analytic continuation]] of the Schrödinger equation to be identified as a [[stochastic process]]. This can be interpreted as the [[Huygens–Fresnel principle]] applied to De Broglie waves; the spreading wavefronts are diffusive probability amplitudes.<ref name="Quantum Mechanics 2004"/> 
For a free particle (not subject to a potential) in a [[random walk]], substituting {{math|''τ'' {{=}} ''it''}} into the time-dependent Schrödinger equation gives:<ref>{{cite journal |doi=10.1103/PhysRevE.82.011132 |pmid=20866590|url=http://www.stt.msu.edu/~mcubed/Relativistic.pdf|title=Stochastic models for relativistic diffusion|journal=Physical Review E|volume=82|issue=1 Pt 1|pages=011132|year=2010|last1=Baeumer|first1=Boris|last2=Meerschaert|first2=Mark M.|last3=Naber|first3=Mark|bibcode=2010PhRvE..82a1132B}}</ref>

:<math> {\partial \over \partial \tau} X(\mathbf{r},\tau) = \frac{\hbar}{2m} \nabla ^2 X(\mathbf{r},\tau) \, , \quad X(\mathbf{r},\tau) = \Psi(\mathbf{r},\tau/i)</math>

which has the same form as the [[diffusion equation]], with diffusion coefficient {{math|{{sfrac|''ħ''|2''m''}}}}.

===Regularity===
On the space <math>L^2</math> of square-integrable densities, the Schrödinger semigroup <math>e^{it\hat{H}}</math> is a unitary evolution, and therefore surjective. The flows satisfy the Schrödinger equation <math>i\partial_t u = \hat{H}u</math>, where the derivative is taken in the [[distribution (mathematics)|distribution]] sense. However, since <math>\hat{H}</math> for most physically reasonable Hamiltonians (e.g., the [[Laplace operator]], possibly modified by a potential) is unbounded in <math>L^2</math>, this shows that the semigroup flows lack Sobolev regularity in general. Instead, solutions of the Schrödinger equation satisfy a [[Strichartz estimate]].

==Relativistic quantum mechanics==
[[Relativistic quantum mechanics]] is obtained where quantum mechanics and [[special relativity]] simultaneously apply. In general, one wishes to build [[relativistic wave equations]] from the relativistic [[energy–momentum relation]]

:<math>E^2 = (pc)^2 + (m_0c^2)^2 \, , </math>

instead of classical energy equations. The [[Klein–Gordon equation]] and the [[Dirac equation]] are two such equations. The Klein–Gordon equation,

:<math> \frac {1}{c^2} \frac{\partial^2}{\partial t^2} \psi - \nabla^2 \psi + \frac {m^2 c^2}{\hbar^2} \psi = 0. </math>,

was the first such equation to be obtained, even before the nonrelativistic one, and applies to massive spinless particles. The Dirac equation arose from taking the "square root" of the Klein–Gordon equation by factorizing the entire relativistic wave operator into a product of two operators – one of these is the operator for the entire Dirac equation. Entire Dirac equation:
:<math>\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>

The general form of the Schrödinger equation remains true in relativity, but the Hamiltonian is less obvious. 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>\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|'''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 {{nobreak|[[spin-1/2|spin-{{frac|2}}]]}} particles, and the solutions to the equation are {{nobreak|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 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 use 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 {{nobreak|{{math|2(2''s'' + 1)}}-component}} [[spinor field]]s.

==Quantum field theory==
The general equation is also valid and used in [[quantum field theory]], both in relativistic and nonrelativistic situations. However, the solution {{math|''ψ''}} is no longer interpreted as a "wave", but should be interpreted as an operator acting on states existing in a [[Fock space]].{{Citation needed|date=September 2015}}

==First order form==
The Schrödinger equation can also be derived from a first order form<ref  name="one">{{cite journal|last1=Ajaib|first1=Muhammad Adeel|title=A Fundamental Form of the Schrödinger Equation|journal=Found. Phys. |volume=45 |issue=12 |pages=1586–1598|doi=10.1007/s10701-015-9944-z|year=2015|bibcode=2015FoPh...45.1586A|arxiv=1502.04274}}</ref><ref name="two">{{cite journal|last1=Ajaib|first1=Muhammad Adeel|title=Non-Relativistic Limit of the Dirac Equation|journal=International Journal of Quantum Foundations|url=http://www.ijqf.org/archives/3574|year=2016}}</ref><ref name="three">{{cite journal|last1=Lévy-Leblond|first1=J-.M.|title=Nonrelativistic particles and wave equations|journal=Commun. Math. Phys.|volume=6 |number=4 |pages=286–311|year=1967|url=https://projecteuclid.org/euclid.cmp/1103840281|bibcode=1967CMaPh...6..286L|doi=10.1007/BF01646020}}</ref> similar to the manner in which the [[Klein–Gordon equation]] can be derived from the [[Dirac equation]]. In 1D the first order equation is given by

:<math display="block">
\begin{align}
-i \partial_z \psi = (i  \eta \partial_t  + \eta^\dagger m) \psi 
\end{align}
</math>

This equation allows for the inclusion of spin in nonrelativistic quantum mechanics. Squaring the above equation yields the Schrödinger equation in 1D. The matrices <math> \eta </math> obey the following properties

:<math display="block">
\begin{align}
\eta^2=0  \\
(\eta^\dagger)^2=0  \\
\left\lbrace \eta, \eta^\dagger \right\rbrace= 2 I
\end{align}
</math>

The 3 dimensional version of the equation is given by
 
:<math display="block">
\begin{align}
-i \gamma_i \partial_i \psi = (i  \eta \partial_t  + \eta^\dagger m) \psi 
\end{align}
</math>

Here <math> \eta=(\gamma_0+i \gamma_5)/\sqrt{2} </math> is a <math>4 \times 4</math> [[nilpotent matrix]] and <math> \gamma_i </math> are the Dirac [[gamma matrices]] (<math>i=1,2,3</math>). The Schrödinger equation in 3D can be obtained by squaring the above equation. 
In the nonrelativistic limit <math>E-m \simeq E'</math> and <math>E+m \simeq 2m</math>, the above equation can be derived from the Dirac equation.<ref name="two" />

==See also==
{{div col}}
*[[Eckhaus equation]]
*[[Fractional Schrödinger equation]]
*[[List of quantum-mechanical systems with analytical solutions]]
*[[Logarithmic Schrödinger equation]]
*[[Nonlinear Schrödinger equation]]
*[[Quantum carpet]]
*[[Quantum revival]]
*[[Relation between Schrödinger's equation and the path integral formulation of quantum mechanics]]
*[[Schrödinger field]]
*[[Schrödinger picture]]
*[[Schrödinger's cat]]
*[[Theoretical and experimental justification for the Schrödinger equation]]
{{div col end}}

==Notes==
<references group="note"/>

== References ==
{{Reflist|30em}}

== Further reading ==
* {{cite book | author= P. A. M. Dirac |author-link=Paul Dirac |title= The Principles of Quantum Mechanics | edition = 4th | publisher= Oxford University Press |year=1958 |title-link=Principles of Quantum Mechanics }} {{ISBN|0-198-51208-2}}.
* {{cite book |author1=B.H. Bransden  |author2=C.J. Joachain  |lastauthoramp=yes |title= Quantum Mechanics | edition = 2nd | publisher= Prentice Hall PTR |year=2000 | isbn = 978-0-582-35691-7 }}
* {{cite book | author = David J. Griffiths |  title = Introduction to Quantum Mechanics  | edition = 2nd | year = 2004 | publisher = Benjamin Cummings |  isbn = 978-0-13-124405-4 | author-link = David J. Griffiths }}
*{{citation|first=Brian C.|last=Hall|title=Quantum Theory for Mathematicians|series=Graduate Texts in Mathematics|volume=267 |publisher=Springer|year=2013| isbn=978-1461471158}}
* {{cite book | author = David Halliday | title = Fundamentals of Physics | edition = 8th | year = 2007 | publisher = Wiley | isbn = 978-0-471-15950-6 | url = https://archive.org/details/studentscompanio00davi }}
* {{cite book | author = Richard Liboff | title = Introductory Quantum Mechanics | edition = 4th | year = 2002 | publisher = Addison Wesley | isbn = 978-0-8053-8714-8 | author-link = Richard Liboff }}
* {{cite book | author = Serway, Moses, and Moyer | title = Modern Physics | edition = 3rd | year = 2004 | publisher = Brooks Cole | isbn = 978-0-534-49340-0}}
* {{cite journal | author = Schrödinger, Erwin |date=December 1926 | title = An Undulatory Theory of the Mechanics of Atoms and Molecules | journal = Phys. Rev. | pages = 1049–1070 | doi = 10.1103/PhysRev.28.1049 | volume = 28 |bibcode = 1926PhRv...28.1049S | issue = 6 }}
* {{cite book| last = Teschl| given = Gerald|authorlink=Gerald Teschl| title=Mathematical Methods in Quantum Mechanics; With Applications to Schrödinger Operators| publisher=[[American Mathematical Society]]| place = Providence, Rhode Island| year=2009 |url=http://www.mat.univie.ac.at/~gerald/ftp/book-schroe/ |isbn=978-0-8218-4660-5 }}

==External links==
* {{springer|title=Schrödinger equation|id=p/s083410}}
* [http://www.lightandmatter.com/html_books/0sn/ch13/ch13.html Quantum Physics] – textbook by Benjamin Crowell with a treatment of the time-independent Schrödinger equation
* [http://eqworld.ipmnet.ru/en/solutions/lpde/lpde108.pdf Linear Schrödinger Equation] at EqWorld: The World of Mathematical Equations.
* [http://eqworld.ipmnet.ru/en/solutions/npde/npde1403.pdf Nonlinear Schrödinger Equation] at EqWorld: The World of Mathematical Equations.
* [https://web.archive.org/web/20060407153923/http://www.colorado.edu/UCB/AcademicAffairs/ArtsSciences/physics/TZD/PageProofs1/TAYL07-203-247.I.pdf The Schrödinger Equation in One Dimension] as well as the [https://web.archive.org/web/20060524165051/http://www.colorado.edu/UCB/AcademicAffairs/ArtsSciences/physics/TZD/PageProofs1/ directory of the book].
* [http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html All about 3D Schrödinger Equation ]
*Mathematical aspects of Schrödinger equations are discussed on the [http://wiki.math.toronto.edu/DispersiveWiki/index.php/Main_Page Dispersive PDE Wiki].
* [http://www.nanotechnology.hu/online/web-schroedinger/index.html Web-Schrödinger: Interactive solution of the 2D time-dependent and stationary Schrödinger equation ]
* [http://behindtheguesses.blogspot.com/2009/06/schrodinger-equation-corrections.html An alternate reasoning behind the Schrödinger Equation ]
* Online software-[http://nanohub.org/resources/3847 Periodic Potential Lab] Solves the time-independent Schrödinger equation for arbitrary periodic potentials.
*[https://web.archive.org/web/20180926125403/http://www.felderbooks.com/papers/psi.html What Do You Do With a Wavefunction?]
*[http://www.felderbooks.com/papers/quantum.html The Young Double-Slit Experiment]
*[https://github.com/LaurentNevou Schrodinger solver in 1, 2 and 3d]

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