Quantum Mechanics · Chapter 05

Time evolution

How a quantum state changes when nobody measures it. The Schrödinger equation, the Hamiltonian, stationary states and the beating of their superpositions, the evolution operator, the Heisenberg picture in which operators move instead of states, and what stays constant. The chapter runs beside the one on position and momentum, and meets it for the particle in a potential, the probability current and Ehrenfest’s theorem.

i ℏ \frac{d}{d t} ∣ ψ ⟩ = H^∣ ψ ⟩

H^∣ E ⟩ = E ∣ E ⟩

U^(t) = e^{- \frac{i H ^ t}{ℏ}}

ψ (t) = e^{- \frac{i E t}{ℏ}} ψ (0)

Lessons

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05

The evolution operatorEvery solution of the Schrödinger equation is the initial state acted on by one operator, the evolution operator. For a Hamiltonian that does not change in time it is $e^{-i\hat{H}t/\hbar}$. It is unitary, which is the Schrödinger equation’s promise that no information is lost, and evolving for two stretches of time is evolving for their sum.

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06The Heisenberg pictureSo far the state has moved and the observables have stood still. The opposite bookkeeping works just as well: freeze the state at its initial value and let every observable move instead. This is the Heisenberg picture, and its equations of motion look like the classical ones — the rate of change of an observable is its commutator with the Hamiltonian.Read lesson →

07Conserved quantitiesA quantity is conserved when nothing about it changes as the system evolves: not its average, not its spread, not the probability of any of its values. The test is a single commutator — an observable that commutes with the Hamiltonian is conserved, in every state. Energy always is. And the energy–time relation says how fast anything can change when the energy is spread by a given amount.Read lesson →

08A particle in a potentialFor a particle on a line the Hamiltonian is kinetic energy plus potential energy, and the Schrödinger equation becomes a partial differential equation for the wavefunction. For a free particle its solutions are plane waves whose frequency grows as the square of the wavenumber, and a packet built from them drifts at the classical velocity while it spreads out — slowly for a grain of sand, astonishingly fast for an electron.Read lesson →

09Probability currentProbability can move but cannot vanish: whatever leaves one region must flow into a neighbouring one. That flow is the probability current, built from the wavefunction and its slope. With it, the conservation of total probability becomes a local statement, the continuity equation — the same equation that says charge or water is conserved in a pipe.Read lesson →

10The Ehrenfest theoremWhere is classical mechanics hiding in all this? In the averages. The average position moves with the average momentum over the mass, and the average momentum changes at the rate of the average force. These are Newton’s laws for expectation values — exactly Newton’s laws when the force is the same across the whole packet, and nearly so whenever the packet is small compared with how fast the force changes.Read lesson →

11Chapter practiceBring the chapter together: the Schrödinger equation and its conservation of probability, the Hamiltonian and its levels, stationary states and the beats between them, the evolution operator, the Heisenberg picture, conserved quantities, the free particle, the probability current and Ehrenfest’s theorem. No worked example sits above the answers.Read lesson →

Mapping the lesson.

Time evolution

Build the foundation