= Energy eigenstate
{title2=$H\phi=E\phi$}
An <energy eigenstate> is a <quantum state> represented by an <eigenvector> of the <Quantum Hamiltonian>. For a time-independent <Quantum Hamiltonian>, a normalized <eigenfunction> $\phi$ with <energy eigenvalue> $E$ evolves according to the <Schrödinger equation> as
$$
\psi(x,t)=\phi(x)e^{-iEt/\hbar},\qquad |\psi(x,t)|^2=|\phi(x)|^2.
$$
Here $\hbar=h/(2\pi)$ is the reduced <Planck constant>. The changing factor is a <global phase>, so this is a <stationary state>: its position <probability density> is constant. For example, the second <infinite square well> mode has two density lobes separated by a <node of a standing wave> at $x=L/2$; both lobes stay fixed while its <quantum phase> rotates. An ideal <energy measurement> returns $E$ with probability one, but a <position measurement> still has a distribution. Stationary density does not describe a particle sitting at the peak of a lobe. Distinct <energy eigenvalues> accumulate different <quantum phases>, making their <quantum superposition> behave differently.
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