Quantum Mechanics · Lesson 05

Course review

The whole course in one sitting: states and amplitudes, operators and measurement, wavefunctions and their evolution, the problems that can be solved exactly — the well, the oscillator, hydrogen — and the methods for those that cannot, symmetry and angular momentum, spin, perturbations fixed and changing, density matrices, entanglement and open systems, driven atoms, fields and identical particles, scattering, paths, relativity and second quantisation. Each question names its own ground. No worked example sits above the answers.

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i ℏ \frac{d}{d t} ∣ ψ ⟩ = H^∣ ψ ⟩

P (a) = ∣ ⟨ a ∣ ψ ⟩ ∣^{2}

[G^, H^] = 0

∣ ψ ⟩ = n \sum c_{n} ∣ n ⟩

Chapter navigation

Chapter practice

Intuition

The whole course in one sitting: states and amplitudes, operators and measurement, wavefunctions and their evolution, the problems that can be solved exactly — the well, the oscillator, hydrogen — and the methods for those that cannot, symmetry and angular momentum, spin, perturbations fixed and changing, density matrices, entanglement and open systems, driven atoms, fields and identical particles, scattering, paths, relativity and second quantisation. Each question names its own ground. No worked example sits above the answers.

At the end of a long walk, a look back from the hilltop: the whole path in view at once, and each stretch recognisable.

Before you review

Ask three questions of every problem. What is the state — a vector, a density matrix, a field? What moves it — which Hamiltonian, and is it solvable exactly or only approximately? And what is measured — which observable, whose eigenvalues and Born probabilities are the answer?

\text{state}\ \longrightarrow\ \text{evolution}\ \longrightarrow\ \text{measurement}

What to reach for

Exactly solvable: the well, $E_{n}\propto n^{2}$ ; the oscillator, $\hbar\omega(n+\frac12)$ ; hydrogen, $-13.6\ \text{eV}/n^{2}$ ; spin, with Pauli matrices.
Approximate: the variational bound, WKB, perturbation theory fixed and in time, the golden rule, the Born approximation.
Structure: commutators, symmetries and their conserved quantities, the addition of angular momenta, the exchange symmetry of identical particles.
Beyond one pure state: density matrices, partial traces, entanglement and Bell’s inequality, channels and the Lindblad equation.
Beyond the Schrödinger equation: the path integral, the Klein–Gordon and Dirac equations, and fields whose quanta are particles.

Applications

Practice

Try it

A qubit is in $\frac{1}{\sqrt3}\ket{0}+\sqrt{\frac23}\,\ket{1}$ . What is the probability of the outcome 1? Give three decimal places.