Editing Schrödinger equation (section)

=== 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>