📜 Module 3 · Lesson 1/3

〰️The Wave Function

Meet ψ (psi): the “map of possibilities” at the core of quantum physics.

⏱️ 18 minStart

🎯 By the end you will…

✓Say what the wave function ψ describes
✓Turn ψ into probabilities with the Born rule
✓Read the Schrödinger equation like a sentence

If an electron is a wave, a wave of what? In 1926 Erwin Schrödinger wrote the equation that tells this wave how to move, and he named the wave ψ (the Greek letter psi). The same year Max Born figured out what ψ means: it is a map of probabilities. Where the wave is big, you are likely to find the particle; where it is zero, you never will.

🎛️Interactive experiment

A “particle in a box”: like a guitar string, only whole-number wave patterns fit. Switch between ψ (the wave) and |ψ|² (the probability of finding the particle).

🧮Equation, decoded

The Born rule

P (x) = ∣ ψ (x) ∣^{2}

What each symbol means

$P (x)$ The chance of finding the particle at place x
$ψ (x)$ The height of the wave at x (it can be negative!)
$∣ ∣^{2}$ “Size, squared” — this makes every value positive, like a real probability must be

📖 Say it like a story

ψ is the treasure map, and squaring it tells you how much treasure is buried at each spot. Squaring is also why waves can cancel: +1 and −1 add to 0, and 0² is still 0 — a dark stripe in the double-slit!

🧮Equation, decoded

The Schrödinger equation

i ℏ \frac{\partial ψ}{\partial t} = H^ψ

What each symbol means

$\frac{\partial ψ}{\partial t}$ How fast the wave changes as time ticks
$H^$ “H-hat”, the energy recipe: movement energy + stored energy
$i$ The imaginary unit (√−1). It makes ψ spin like a clock hand instead of just growing or shrinking
$ℏ$ Planck's constant, setting the quantum scale

📖 Say it like a story

Read it as: “The way ψ changes in time is decided by its energy.” It's the quantum version of Newton's F = ma. You don't need to solve it — computers do that — but now you know what it says.

✅ Check your understanding

Answer all questions to complete the lesson and earn XP.

1. What does |ψ|² tell you?
2. Why do only certain patterns fit in a box?
3. The Schrödinger equation plays the role of which classical law?

The Born rule

The Schrödinger equation

✅ Check your understanding

📚 Sources & further reading