Editing Schrödinger equation (section)

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

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

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

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

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

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

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

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

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

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

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

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

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

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