Editing Schrödinger equation (section)

=== 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>
|cellpadding
|border
|border colour = rgb(80,200,120)
|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>