Two very different questions
Quantum mechanics really asks two separate questions, and people often blur them together. The first is: what does a system do when no one is looking? The second is: what happens at the instant you measure it? This rung is entirely about the first question — the quiet, between-measurement life of a quantum system. And here is the reassuring news: that quiet life is smooth, predictable, and follows a single clean rule. The wild, jumpy, dice-rolling reputation of quantum physics belongs almost entirely to the second question, which we set aside for now.
When physicists say time evolution, they mean exactly this between-measurement flow: take the system's full description now, and figure out its full description a little later. Nothing random happens in between. If you know the quantum state today and leave the system undisturbed, the state tomorrow is completely fixed. It is as deterministic as a planet circling the sun.
The state is the thing that changes
In everyday physics, the thing that changes over time is a position and a velocity: where the ball is, how fast it moves. In quantum mechanics, the thing that changes is the whole wavefunction — a smooth, spread-out description that, loosely speaking, tells you the chance of finding the particle at each place. As time passes, this whole shape flows and reshapes itself, like a ripple sliding across a pond. The particle does not have a single secret location that moves; instead, the entire cloud of possibilities evolves.
Because the wavefunction can be a superposition — several possibilities present at once — time evolution can do something with no classical cousin: it can let those possibilities interfere as they flow, sometimes reinforcing, sometimes cancelling. A great deal of quantum behaviour, from how molecules vibrate to how light is absorbed, is really just this: a superposition evolving and the pieces dancing in and out of step.
One equation runs the whole show
Just as Newton's law tells a ball how to move next, one equation tells a quantum state how to change next: the time-dependent Schrödinger equation. You do not need the mathematics to grasp its spirit. It says the rate at which the state changes, right now, is set by the system's energy — specifically by an object called the Hamiltonian, which is just the rulebook for that system's total energy. In plain words: energy is what makes a quantum state tick.
This has a lovely consequence. Some special states have a single, definite energy and nothing else mixed in. These are called stationary states, and they earn the name: their measurable properties — where the particle is likely to be, how much momentum it has — simply do not change with time. The state's internal clock keeps ticking, but everything you could actually observe sits still. Most of the interesting motion in quantum mechanics comes from blending several such steady states together; the blend, unlike each piece, does move.
What the rest of this rung covers
With the big picture in place, the next four guides zoom in. We will see why time evolution always preserves total probability — the system never "leaks" certainty — and meet the single operator that carries a state forward in time. We will learn that you can watch the same physics from three different vantage points, all of which agree. We will meet the most-used formula in all of applied quantum mechanics, for how fast a system hops from one state to another. And finally we will use all of it to explain how atoms give off and soak up light — the everyday miracle behind every glowing thing you have ever seen.