Editing Schrödinger equation (section)

=== 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}}