Ask a chemist why gold is yellow, why a laser is the colour it is, or why one atom bonds to another and not a third, and every honest answer bottoms out in the same place: solve the Schrödinger equation for the electrons and read off what it says. It is the single equation from which the whole periodic table, the colours of the world, and the behaviour of every atom and molecule ultimately follow. If Newton's F = ma is the master law of the everyday world of pushes and orbits, the Schrödinger equation is its replacement for the world of the very small — the master law of quantum mechanics.

Here is the difference in one breath. Newton's law tells you how a particle's position changes in time. Schrödinger's law tells you how a particle's wavefunction \psi — the spread-out amplitude whose squared size |\psi|^2 gives the probability of finding the particle — changes in time. It is a wave equation for \psi. Feed it the forces (packaged as a potential energy V) and it grinds out the future of the wave, exactly and deterministically. The randomness of quantum mechanics is not in this equation; \psi itself evolves as smoothly and predictably as a ripple on a pond.

The time-dependent equation

For a single particle of mass m moving in one dimension in a potential V(x), the law of motion for its wavefunction \psi(x,t) is the time-dependent Schrödinger equation:

The object \hat{H} is the Hamiltonian — the operator that stands for the total energy. It has two pieces that mirror the familiar E = \text{kinetic} + \text{potential} of classical mechanics:

• the kinetic term -\dfrac{\hbar^2}{2m}\,\dfrac{\partial^2}{\partial x^2}, which measures the curvature of the wave — how sharply \psi bends. A wildly wiggling wavefunction is a high-energy, fast-moving particle; a gently curved one is slow;
• the potential term V(x), which simply multiplies \psi by the potential energy at each point.

The i = \sqrt{-1} on the left is not decoration: it is the mathematical seed of all the wave-like behaviour. Because of it, solutions do not just grow or decay like an ordinary diffusion — they oscillate, carrying phase around and around, and it is that turning phase that produces interference, energy levels, and everything else quantum.

Where does the kinetic term's peculiar shape come from? From de Broglie. A free particle of momentum p is a wave of wavelength

the celebrated de Broglie relation. A shorter wavelength means a faster wiggle means more curvature — and, through the kinetic operator, more energy. Curvature is kinetic energy. Hold on to that picture; the interactive graphs below make it visible.

A wave carrying momentum

The simplest wavefunction of all is a plane wave, \psi = e^{i(kx - \omega t)}, whose real part is the ordinary cosine drawn below. The number k = 2\pi/\lambda is the wavenumber, and de Broglie's relation becomes the tidy p = \hbar k. Slide k up and watch the wave crowd together: the wiggles get tighter, the curvature grows, and the particle it describes carries more momentum and more kinetic energy.

You can check this wave solves the free-particle equation (with V = 0) directly. Differentiating e^{i(kx-\omega t)} twice in x pulls down (ik)^2 = -k^2, so the kinetic operator returns \dfrac{\hbar^2 k^2}{2m}\,\psi; differentiating once in t pulls down -i\omega, so the left-hand side is \hbar\omega\,\psi. The equation holds precisely when \hbar\omega = \dfrac{\hbar^2 k^2}{2m} — which, with E = \hbar\omega and p = \hbar k, is nothing but E = \dfrac{p^2}{2m}, the everyday kinetic energy. The Schrödinger equation knows classical physics is hiding inside it.

Splitting time from space: stationary states

Most problems have a potential that does not change with time — V(x), not V(x,t). For those, a beautiful simplification is available. Look for solutions in which the space part and the time part come apart into a product:

This is separation of variables. Substitute it into the time-dependent equation. The time derivative on the left brings down -iE/\hbar, and the i\hbar out front turns it into a plain E; the wobbling exponential appears on both sides and cancels clean away. What is left is an equation for the space part alone:

This is the equation you actually solve in practice. Give it a potential — a box, a spring, a hydrogen nucleus — and it hands you the allowed energies and the shapes of the states that live at them. It is an eigenvalue problem in exactly the sense of linear algebra: apply the operator \hat{H} to the special functions \psi and you get the same function back, merely scaled by its energy.

Energy eigenstates in a box

The cleanest example is a particle trapped between two hard walls a distance L apart — the infinite square well, where V = 0 inside and the wave is forced to zero at each wall. Solving \hat{H}\psi = E\psi there gives a ladder of standing-wave eigenstates, one for each whole number n = 1, 2, 3, \dots:

Each state fits a whole number of half-wavelengths across the box, just like a plucked guitar string. A higher n packs in more wiggles — more curvature — and so, through the kinetic term, a higher energy, climbing as n^2. Drag the slider to walk up the ladder and watch each eigenstate ride at its own energy level.

This one picture holds the essence of why atoms have sharp energy levels at all: confine a wave, and only certain wavelengths fit; only certain wavelengths means only certain energies. The colours of a neon sign are its trapped electrons stepping between rungs of a ladder just like this one.

Why "stationary" states never sit still

Reassemble a single eigenstate into a full time-dependent solution and it carries the phase factor e^{-iEt/\hbar}. That factor is spinning: as time runs, the complex number \psi(x,t) whirls round and round the origin of the complex plane at angular frequency \omega = E/\hbar. So the state is emphatically not frozen — its phase is in constant motion. Why, then, call it stationary?

Because the one thing you can actually observe — the probability density |\psi(x,t)|^2 — does not move at all. Watch the phase cancel:

The whirling factor has magnitude exactly 1 — it is a pure phase — so it vanishes from |\psi|^2 entirely. The probability of finding the particle anywhere is fixed for all time, along with the energy and every other measurable average. That constancy is what "stationary" means: the observable state stands still even though the wavefunction underneath it is forever turning. An electron in a stationary atomic state can sit there indefinitely without radiating — which is exactly the puzzle the old Bohr model had to postulate by fiat, and which the Schrödinger equation now explains.

Worked examples

Example 1 — verify an eigenstate and read off its energy. Take the ground state of the box, \psi_1(x) = \sqrt{2/L}\,\sin(\pi x/L), with V = 0 inside. Apply the Hamiltonian. Differentiating the sine twice gives

so

The operator returned the same function times a constant — so \psi_1 is indeed an eigenstate, and its energy eigenvalue is E_1 = \pi^2\hbar^2/2mL^2. Every step of that is just "the curvature of the sine sets its energy."

Example 2 — a free-particle plane wave. Check that \psi(x) = e^{ikx} solves the time-independent equation with V = 0. Two x-derivatives give (ik)^2 e^{ikx} = -k^2 e^{ikx}, so

It is an eigenstate with energy E = \hbar^2 k^2/2m = p^2/2m, using p = \hbar k — pure kinetic energy, as it must be for a free particle.

Example 3 — de Broglie wavelength. An electron (m = 9.11\times10^{-31}\,\text{kg}) moves at v = 2.0\times10^{6}\,\text{m/s}. Its momentum is p = mv = 1.82\times10^{-24}\,\text{kg·m/s}, so its wavelength is

That is roughly the spacing between atoms in a crystal — which is exactly why electrons diffract off crystals, the experiment that proved matter really is wavy.