the time-independent Schrödinger equation
/ SHROH-ding-er /
Many quantum problems ask a standing-wave question: for a given potential, what wave patterns are self-consistent and sit still, and what energies do they carry? The time-independent Schrodinger equation is the eigenvalue problem that answers this. It is where the famous quantization of energy — discrete allowed levels rather than a continuum — actually comes from.
It is obtained by separating variables. Writing the full state as Psi(x, t) = psi(x) e^{-i E t / hbar} and inserting it into the time-dependent Schrodinger equation collapses the time part into a phase and leaves H psi = E psi. For one particle in a potential this is -(hbar^2 / 2m) d^2 psi/dx^2 + V(x) psi = E psi. It says psi is an eigenfunction of the Hamiltonian operator H with eigenvalue E, the energy. Only certain E admit solutions that satisfy the boundary conditions (finite, normalizable, continuous), and those admissible E are the energy spectrum.
Why it matters: solving it for a given V(x) gives you the allowed energies and the stationary states, and any general time-dependent state is a superposition of these with phases e^{-i E_n t / hbar} attached. Discreteness is not built in by hand — it emerges from demanding that psi be well-behaved in a confining potential, exactly as the allowed frequencies of a guitar string emerge from its fixed ends.
For an infinite square well of width L the solutions are psi_n = sqrt(2/L) sin(n pi x / L) with energies E_n = n^2 pi^2 hbar^2 / (2 m L^2); the integer n labels how many half-wavelengths fit inside.
Quantized energies E_n ~ n^2 fall out of the boundary conditions, not from any extra postulate.
It only holds when the potential V does not depend on time; a stationary state is not a static wavefunction but one whose time dependence is a pure phase e^{-iEt/hbar}.