Editing Schrödinger equation (section)

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

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

Physical quantities of interest – position, momentum, energy, spin – are represented by [[observable]]s, which are [[self-adjoint operator]]s acting on the Hilbert space. A wave function can be an [[eigenvector]] of an observable, in which case it is called an [[eigenstate]], and the associated [[eigenvalue]] corresponds to the value of the observable in that eigenstate. More generally, a quantum state will be a linear combination of the eigenstates, known as a [[quantum superposition]]. When an observable is measured, the result will be one of its eigenvalues with probability given by the [[Born rule]]: in the simplest case the eigenvalue <math>\lambda</math> is non-degenerate and the probability is given by <math>|\langle \lambda | \psi\rangle|^2</math>, where <math> |\lambda\rangle</math> is its associated eigenvector. More generally, the eigenvalue is degenerate and the probability is given by <math>\langle \psi | P_\lambda |\psi\rangle</math>, where <math>P_\lambda</math> is the [[Projection-valued measure#Application in quantum mechanics|projector]] onto its associated eigenspace.{{refn|group=note|This rule for obtaining probabilities from a state vector implies that vectors that only differ by an overall phase are physically equivalent; <math>|\psi\rangle</math> and <math>e^{i\alpha}|\psi\rangle</math> represent the same quantum states. In other words, the possible states are points in the [[projective space]] of a Hilbert space, usually called the [[projective Hilbert space]].}}

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

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

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

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

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

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