The equation of motion for the wavefunction
Newton had F=ma to march x(t) forward in time. Quantum mechanics has the time-dependent Schrödinger equation to march \Psi forward. It is first order in time: hand it \Psi now and it delivers \Psi at every future instant. The evolution of the amplitude is perfectly deterministic — all the randomness in quantum mechanics lives at the moment of measurement, not in this equation.
The time-dependent Schrödinger equation. \hbar is Planck's constant divided by 2\pi, and \hat H is the energy operator.
\hat H is the Hamiltonian operator — the classical energy p^2/2m+V with momentum promoted to an operator, p\to -i\hbar\,\partial/\partial x. This promotion of a classical quantity to an operator is canonical quantization, the recipe that turns a classical system into its quantum counterpart.
The Hamiltonian operator = kinetic term (the second derivative) + potential energy V(x).
Stationary states
A special family of solutions has the form \Psi(x,t)=\psi(x)\,e^{-iEt/\hbar}: a fixed spatial shape times a rotating phase. Substitute it into the Schrödinger equation and the time drops out, leaving the time-independent Schrödinger equation. These are the stationary states, so named because |\Psi|^2=|\psi|^2 carries no time — the probability cloud stands perfectly still even as its phase whirls.
The time-independent Schrödinger equation is an eigenvalue equation: find the \psi's the operator \hat H merely rescales, and their scale factors E.
This is the master equation of applied quantum mechanics. Feed it a potential V(x) and it returns the allowed spatial shapes \psi_n and the allowed energies E_n. The energies are exactly the values a measurement of energy can return. And crucially, it is the boundary conditions — not the differential equation itself — that do the quantizing.
Particle in a box: quantization from a boundary
Take the simplest bound system: a particle in a box, with V=0 inside $0<x<L$ and infinitely hard walls outside. The idealization here is those infinitely hard walls, which force \psi to vanish at x=0 and x=L. Inside, the equation is just that of a sine wave.
- Inside the well V=0, so the equation becomes \psi''=-k^2\psi with k=\sqrt{2mE}/\hbar. Its general solution is \psi(x)=A\sin kx+B\cos kx.
- Left wall. \psi(0)=0 forces B=0, leaving \psi(x)=A\sin kx.
- Right wall. \psi(L)=0 forces \sin kL=0, so kL=n\pi with n=1,2,3,\dots — only whole numbers of half-wavelengths fit inside. This is the quantization condition.
- Energies. From E=\hbar^2 k^2/2m and k=n\pi/L: E_n=n^2\pi^2\hbar^2/(2mL^2). Energy is discrete precisely because k is.
- Normalize. \int_0^L|\psi|^2\,dx=1 gives A=\sqrt{2/L}, so \psi_n(x)=\sqrt{2/L}\,\sin(n\pi x/L) — a standing wave with n-1 interior nodes.
The quantized energies of the box. They grow as n^2 and squeeze together as the box L widens.
Dynamics: superposing stationary states
A single stationary state is dull — nothing observable ever changes. Real dynamics comes from superposing them. Because the Schrödinger equation is linear, the general solution is a sum over stationary states, each ticking along with its own phase e^{-iE_n t/\hbar}. The coefficients c_n are fixed once, by the initial state.
The general solution: any state is a superposition of energy eigenstates, each rotating at its own rate E_n/\hbar.
Because different E_n rotate at different rates, the relative phases drift, the cross terms breathe, and |\Psi|^2 sloshes around the box — this is how a wave packet moves and spreads. Energy eigenstates are the `normal modes` of quantum evolution, in exact analogy with the normal modes of a vibrating string: the stationary states oscillate simplest, and everything else is a chord of them.