= Time-independent Schrödinger equation
{title2=$H\psi=E\psi$}
For a time-independent <Hamiltonian operator> $H$, an <energy eigenstate> satisfies
$$
H\psi=E\psi.
$$
For a one-dimensional particle of <mass> $m$ in a real <potential energy> $U(x)$,
$$
-\frac{\hbar^2}{2m}\psi''+U(x)\psi=E\psi.
$$
Its full time-dependent <wavefunction> is $\Psi(x,t)=\psi(x)e^{-iEt/\hbar}$: substitution in the <Schrödinger equation> checks both the minus sign and the factor $\hbar$. Thus an <energy eigenstate> has stationary position <probability density>, even though its <quantum phase> evolves. The <boundary conditions> and <operator domain> are part of this <eigenvalue problem>, selecting physically allowed solutions; the <infinite square well> and <Radial Schrodinger equation for the hydrogen atom> give complementary confined examples.
For a complete discrete <orthonormal basis> of <energy eigenstates>, <linearity> gives $\Psi(x,t)=\sum_n c_n\psi_n(x)e^{-iE_nt/\hbar}$, with $\sum_n|c_n|^2=1$. Cross terms in $|\Psi|^2$ oscillate at <angular frequencies> $(E_n-E_m)/\hbar$. A <continuous spectrum> requires the corresponding spectral integrals; for a <free particle> on the whole line, individual <plane waves> are generalized <eigenfunctions>, and a normalized <Gaussian wave packet> is their integral <quantum superposition>. Adding a spatially constant real $C$ to $U$ shifts every <energy eigenvalue> by $C$ and multiplies the full state by $e^{-iCt/\hbar}$, a <global phase>; <energy> differences, position <probabilities> and <probability current> are unchanged.
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