Chapter 03
Formalism: Dirac notation and the postulates
Abstracting wavefunctions into vectors in Hilbert space is where the theory takes shape.
- Sections
- 11
- Finalised
- 11/11
- Simulations
- 3
- Estimated time
- 3 hours
- Read first
- Chapter 02
- 3.1From wavefunction to state vectorψ(x) is not the state itself — it is only the state's components in one particular basis. Once you see that, the rest of this chapter falls into place on its own.
- 3.2The complete rules of Dirac notationA notation that compresses "vectors, inner products, projections, changes of basis" into a few angle brackets. Learn it, and the derivations of chapter 2 shrink to a fraction of their length.
- 3.3Hilbert space essentialsThe details mathematicians care about and physicists tend to skip. This section covers only the ones that genuinely affect physical conclusions.
- 3.4Linear operators and the eigenvalue problemAn operator is a machine that maps states to states. In a given basis it is just a matrix, and the eigenvalue problem is the hunt for the directions that get stretched but never turned.
- 3.5Hermitian operators and observablesIn chapter 2, "observables correspond to operators" was a bare decree. This section turns it into a conclusion: only a Hermitian operator can be an observable.
- 3.6Commutators and the generalised uncertainty principleWhether two observables can be sharp at the same time comes down to whether their product cares about the order. One inequality unifies every uncertainty relation.
- 3.7The postulates of quantum mechanicsEverything so far, compressed into five statements. The value of this section is drawing the line: what must be assumed, and what can be derived.
- 3.8Measurement and the projection postulate◈ 1 sim(s)One postulate, decades of argument. First sort out what it says from what it does not say — then see exactly where the argument gets stuck.
- 3.9The time-evolution operator and picturesWhether the time dependence gets booked on the states or on the operators is a free choice. Both bookkeeping schemes give identical physics — but the difficulty can differ enormously.
- 3.10A first look at the density operatorWhen you do not know which state the system is in — or when it has no state of its own at all — the state vector is not enough. The density operator is the tool that is.
- 3.11Rewriting earlier chapters in Dirac notation◈ 2 sim(s)Time to settle the accounts. Redo the derivations that consumed so many pages of chapter 2 in the new language, and see exactly how much was saved.