The state postulate is the conceptual center of wave mechanics. A state can be a superposition of several basis states, and the coefficients are not optional bookkeeping: they are the amplitudes that later become probabilities.
A physical state is represented by a wave function \(\Psi(x,t)\), or more abstractly by a state vector. The wave function contains all the information needed to predict the probabilities of possible measurement outcomes.
Two valid state solutions can be combined to form another valid state whenever the Schrodinger equation is linear.
The functions \(\psi_n\) are basis states. The complex numbers \(C_n\) are amplitudes. The state is not secretly one term before measurement; it is the full superposition until a measurement selects an outcome.
For a convenient basis, the states are orthonormal:
Multiplying \(\Psi=\sum_n C_n\psi_n\) by \(\psi_m^*\) and integrating isolates one coefficient:
In the infinite well, a state such as
is not an eigenstate of energy. It is a normalized superposition of two energy eigenstates, so an energy measurement can return either \(E_1\) or \(E_2\).
The state postulate becomes useful when a state is expanded in a basis adapted to the question being asked:
The basis functions define the possible alternatives; the coefficients define how much of each alternative is present in the state. Orthonormality lets one isolate a coefficient by projection:
This is the minimum structure needed for probability exercises in the next pages.
This page is designed to support short guided exercises on: Postulate for quantum states: a physical state is represented by a wave function and can be expanded in an orthonormal basis.
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.