Quantum mechanics: revision notes for the first course
What examiners actually test in an introductory QM course, condensed from several years of tutoring. Not a textbook — a map of what to know cold, what to be able to derive, and where marks are lost.
How to use these notes. Each section separates state (results you must quote instantly), derive (arguments you must reproduce in under a page) and recognise (patterns in problems). Revise with past papers open; the notes only tell you what to look for.
Postulates and formalism
- State. A system is described by a normalised vector |ψ⟩ in a Hilbert space; observables are Hermitian operators; possible outcomes are eigenvalues; the probability of outcome
ais |⟨a|ψ⟩|²; after measurement the state collapses to |a⟩. - Dynamics. The time-dependent Schrödinger equation
iℏ ∂|ψ⟩/∂t = Ĥ|ψ⟩; stationary states|ψ(t)⟩ = e^{-iEt/ℏ}|E⟩; expansion of an arbitrary state in energy eigenstates and how each phase evolves. - Commutators.
[x, p] = iℏ,[L_i, L_j] = iℏ ε_{ijk} L_k, and the generalised uncertainty relationΔA ΔB ≥ ½|⟨[A,B]⟩|. Be able to derive the position–momentum case from it. - Representations. Wavefunction in position and momentum space; the Fourier transform between them;
p = -iℏ∇in position representation and why. - Ehrenfest.
d⟨A⟩/dt = (i/ℏ)⟨[Ĥ,A]⟩ + ⟨∂A/∂t⟩. Derive it; apply it to recover Newton's second law for expectation values.
Standard problems you must be able to solve completely
| System | Know | Typical twist |
|---|---|---|
| Infinite square well | Energies E_n = n²π²ℏ²/2mL², eigenfunctions, orthonormality | Sudden change of well width; probability of remaining in the ground state |
| Finite well and step | Matching conditions for ψ and ψ′, transmission and reflection, tunnelling coefficient in the WKB-like limit | Reflection above the barrier; delta-function potential as a limit |
| Harmonic oscillator | Ladder operators a, a†, E_n = ℏω(n+½), matrix elements of x and p | Coherent states; expectation values in a superposition of |0⟩ and |1⟩ |
| Angular momentum | Spectrum of L² and L_z, spherical harmonics for l ≤ 2, raising and lowering | Measuring L_x in an L_z eigenstate |
| Hydrogen atom | Energy levels -13.6 eV / n², degeneracy n², quantum numbers and their ranges, radial probability density | Most probable radius vs expectation value of r; selection rules |
| Spin-½ | Pauli matrices, eigenstates of S_x, S_y, S_z, Stern–Gerlach sequences, Larmor precession | Successive measurements along different axes; time evolution in a magnetic field |
| Perturbation theory | First- and second-order energy shifts, first-order state correction, degenerate case | Choosing the right unperturbed basis in the degenerate case |
Derivations that appear every year in some form: the harmonic oscillator via ladder operators;
[L_x, L_y] = iℏL_z from the definition; the time-independent perturbation formulas to second order; the transmission coefficient for a rectangular barrier. Each should take less than a page.
Where marks are lost
- Forgetting to normalise a superposition before computing probabilities.
- Treating
⟨x²⟩as⟨x⟩²; variance and expectation value are different things. - Adding probabilities where amplitudes should be added (and the reverse) — ask whether the paths are distinguishable.
- Applying non-degenerate perturbation theory to a degenerate level.
- Writing the time evolution of a superposition with a common phase factor: each energy eigenstate gets its own
e^{-iE_n t/ℏ}. - Confusing
lwithmwhen counting degeneracies, or lettingmrun beyond±l. - Dropping units in numerical questions. A hydrogen energy in joules with the wrong power of ten scores nothing.
A four-week revision plan
| Week | Focus | Output |
|---|---|---|
| 1 | Formalism: postulates, operators, commutators, Dirac notation | Two-page summary of definitions and key relations |
| 2 | Exactly solvable systems, one per day, derived from scratch | Own worked solutions for well, oscillator, hydrogen, spin |
| 3 | Past papers, untimed, full write-ups | Error log grouped by type of mistake |
| 4 | Past papers under timed conditions, two per week | Marked scripts and an updated error log |
Most students find that three or four recurring errors account for the bulk of lost marks. The error log is there to find yours; the papers are only the means.