After solving one model, the chapter states the general rules. Keep the postulates separate: states say what describes the system; observables say what can be measured; probabilities say how likely outcomes are; measurement says what state remains after a result.
The infinite well shows how one Hamiltonian produces eigenfunctions and discrete energies. The next question is broader: what do wave functions, operators, probabilities and measurements mean for any quantum system?
The postulates give the translation rules between mathematical objects and experimental predictions.
| Postulate | Core statement | Question answered |
|---|---|---|
| Quantum states | A state is represented by a wave function or state vector. | What describes the system? |
| Observables | A measurable quantity is represented by a Hermitian operator. | What can be measured? |
| Probability | Expansion coefficients determine measurement probabilities. | How likely is each result? |
| Measurement | A definite outcome leaves the system in the corresponding eigenstate. | What happens after a result? |
The well eigenfunctions \(\psi_n(x)\) are not only solutions of a particular problem. They illustrate a general idea: a state can often be expanded in a basis of allowed states.
The coefficients \(C_n\) become the bridge between the state and the probabilities of measurement outcomes.
Each postulate should be read with two questions in mind: what mathematical object is being introduced, and what experimental prediction does it make possible?
This page is designed to support short guided exercises on: Bridge card: after solving one model, the chapter states the rules for states, observables, probabilities and measurement.
Use these anchors to design compact exercises. The exercise should be answerable from this page plus standard algebra, while longer derivations, full worked examples and broader context should point back to the original book.