The Schrödinger Equation

Every physics revolution gets its equation. Newton’s tells forces how to move things; Maxwell’s tells light how to travel. Quantum mechanics got its own in early 1926, when Erwin Schrödinger – challenged, the story goes, by a colleague who scoffed that a proper wave deserves a proper wave equation – went on a winter retreat in the Alps and came back with the law that drives $\psi$.

You have already watched it at work without knowing: every packet, fringe and corkscrew in the last three chapters obeyed it. This chapter puts it in charge openly, in three classic arenas: a particle trapped in a box, a particle in a valley, and – the showstopper – a particle walking through a wall.

Before solving anything, one idea unlocks all three arenas. Ask: are there wave shapes that don’t change as time passes – shapes where the arrows underneath all turn in step, so the strength $|\psi|^2$ freezes? There are. They are called stationary states, and each one comes with a definite energy: the faster its arrows turn, the higher the energy. They are the quantum version of a guitar string’s pure notes – §1.4’s hunch, now made exact.

Confine a wave – in a box, in a valley – and only certain notes fit: energy comes in rungs. Every other motion, it turns out, is just a chord: several notes sounding at once, beating against each other. Find the notes and you have solved the system completely.

The simplest cage: a particle stuck between two impenetrable walls. The wave must vanish at both walls, so – exactly like a guitar string pinned at both ends – a whole number of half-waves must fit. One bump, two bumps, three: that integer $n$ is the only choice left, and each choice has its energy rung. Notice the rungs in the figure spread apart as you climb – and notice there is no rung at zero. A confined quantum particle cannot sit still: perfect stillness would mean perfect position and perfect (zero) speed, which Chapter 3’s uncertainty floor forbids.

Then tick the superposition box. One pure note holds perfectly still; two notes at once slosh side to side. That slosh is what “motion” is in quantum mechanics.

Replace the box with a smooth valley – a marble in a bowl, a mass on a spring. This one potential is physics’ favourite, because near the bottom every smooth valley looks like it: vibrating molecules, atoms in crystals, even light waves are “springs” in disguise. Its rungs hold a surprise: they are perfectly evenly spaced. Climbing any rung costs exactly the same energy packet, $\hbar\omega$ – and that is where Chapter 0’s light-comes-in-packets rule was hiding all along.

Then switch to the coherent state: a lump displaced up the valley wall and released. It swings back and forth exactly in step with a classical marble – the red dot – and never spreads out. After four chapters of quantum strangeness, this is the familiar world being handed back: it was a special superposition all along.

Now the flagship. Roll a ball at a hill too tall for its speed and it comes back – always, guaranteed. Send a quantum particle at a wall too tall for its energy and something impossible happens: a sliver of the wave seeps through, and the particle is sometimes found on the far side, having crossed a region it never had the energy to be in. Inside the wall the wave cannot ripple, so it fades – but a wall of finite thickness can’t quite fade it to nothing.

This is not a curiosity; it runs the universe. The Sun shines because protons tunnel through their electric repulsion. Radioactive atoms decay when a fragment tunnels out of the nucleus. Your phone’s flash memory writes bits by pushing electrons through an insulating wall. Watch the split in the figure – and mind how violently the leak responds to the width slider.

The machine is complete. A wave of arrows (Chapter 2), odds from its strength (Chapter 3), and now the engine that drives it – notes, chords, and walls that leak. With these four chapters you can honestly say you know how quantum mechanics works. What’s left is what the world does with it: the strange two-valued compass inside every electron called spin, measurements that scramble each other, and – at the end of the road – the full quantum atom.