Ψ|Ψ|² THE SCHRÖDINGER EQUATION
01 — The Equation

What the equation actually says

The version on the homepage is the time-dependent Schrödinger equation. It's a single statement — but every symbol in it is doing real work. Here's what each piece means, and where the equation goes from here.

The Time-Dependent Form

This is the equation as it appears on the homepage — the general rule for how a quantum system's wave function changes moment to moment:

iℏ ∂Ψ(r,t)/∂t = Ĥ Ψ(r,t)

Read left to right: the rate of change of the wave function over time (∂Ψ/∂t), scaled by iℏ, equals the Hamiltonian operator (Ĥ) acting on that same wave function. It's not solving for a number — it's a rule that, given the wave function right now, tells you the wave function an instant later. Apply it continuously, and you get the system's entire evolution through time.

iThe imaginary unit. Without it, this equation would describe simple decay, not oscillation — i is what lets Ψ rotate through phase rather than just shrink.
ℏReduced Planck's constant — the fundamental scale of quantum action. Tiny in everyday units, which is why quantum effects are invisible at ordinary scales.
Ψ(r,t)The wave function itself — a complex-valued function whose squared magnitude, |Ψ|², gives the probability of finding the particle at position r and time t.
ĤThe Hamiltonian — the total energy operator, combining kinetic and potential energy. It's what actually drives the system's evolution.

The Time-Independent Form

Many of the systems worth solving — an electron in an atom, a particle in a box — settle into states whose probability distribution, |Ψ|², doesn't change over time, even though the wave function's phase still does. These are called stationary states, and they satisfy a simpler version of the equation:

Ĥ ψ = E ψ

This is an eigenvalue equation: apply the Hamiltonian to ψ, and you get the same ψ back, scaled by a number E — the system's energy. Solving this equation for a given potential tells you exactly which energies the system is allowed to have, and what its wave function looks like at each one. The full time-dependent solution can always be built by combining these stationary states.

A Worked Example: The Particle in a Box

The simplest system you can solve exactly is a particle confined to a region of space it can't escape — infinite walls on either side, zero potential in between. Solving Ĥψ = Eψ for this setup gives a striking result: the particle can't have just any energy. Only a discrete set of energies are allowed, indexed by a whole number n = 1, 2, 3…

Each allowed energy corresponds to a wave function that looks like a standing wave — the same shape you'd get from a vibrating guitar string fixed at both ends, with n − 1 points where the wave function crosses zero. This is quantization falling directly out of the mathematics, not imposed on it: confine a wave, and only certain shapes fit.

It's a toy model, but the same logic — solve for the allowed standing waves — is what eventually gives you the discrete energy levels of a real hydrogen atom.

What the Equation Can't Describe

The Schrödinger equation is non-relativistic — it assumes particles move much slower than light, and it has no mechanism for particles being created or destroyed. That's fine for chemistry, atomic physics, and most of what shows up in Quantum 101, but it breaks down at high energies, where relativistic effects and particle creation matter. What replaces it there is covered on The Frontier.

The equation also doesn't say what "measurement" physically is, or resolve why a spread of possibilities becomes one definite outcome. That gap — the measurement problem — is covered honestly on The Weird Stuff, labeled for exactly the kind of open question it is.