= Solution
The <time-independent Schrödinger equation> is the <energy eigenvalue> equation for the <Hamiltonian operator>:
$$
\boxed{-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2}+V(x)\psi(x)=E\psi(x).}
$$
Here $\hbar$ is the reduced <Planck constant>. A corresponding stationary <wavefunction> evolves as $\Psi(x,t)=\psi(x)e^{-iEt/\hbar}$, which follows by substitution in the <Time-dependent Schrödinger equation>. Its position <probability density> is time independent because the time factor has unit modulus. Physical <energy eigenstates> must also satisfy the system's <boundary conditions>; for a <bound state>, <wavefunction normalization> requires $\int_{-\infty}^{\infty}|\psi(x)|^2\,dx=1$.
Back to article page
