[[Quantum mechanics MOC]]
# Schrödinger equation

The **Schrödinger equation** governs the time evolution quantum mechanical [[wavefunction]] in the [[Schrödinger picture]].
If $\ket{\Psi(t)}$ is the state of a system at time $t$, then ^14a666
$$
\begin{align*}
i \hbar \frac{d}{dt} \ket{\Psi(t)} = \hat{H}(t)\ket{\Psi(t)} 
\end{align*}
$$
where $\hat{H}(t)$ is the [[Hamiltonian operator]] for the system.
This is a linear [[differential system]] and thus solutions are given by a [[Quantum time evolution operator]].

## Time-independent Schrödinger equation

A **stationary state** is one for which every [[Observable]] is independent of time.
They are precisely the solutions to the **time-independent Schrödinger equation** as obtained from [[Separation of variables]]
$$
\begin{align*}
\hat{H} \ket{\Psi(t)} &= E \ket{\Psi(t)} = i\hbar \frac{d}{dt} \ket{\Psi(t)} 
\end{align*}
$$
and are thus are energy eigenstates.[^2018]

[^2018]: 2018\. [[Sources/@griffithsIntroductionQuantumMechanics2018|Introduction to Quantum Mechanics]], §2.1, pp. 27–28

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