The closing ideas of the chapter and appendices show how matrix mechanics becomes a flexible computational language. Spectral functions make time-evolution operators precise, while the correspondence with wave mechanics shows that kets and wave functions are two representations of the same state.
If an operator has spectral decomposition
then a function of the operator is defined by applying the function to its eigenvalues:
This rule makes expressions such as \(e^{-i\hat Ht/\hbar}\) precise. In the energy basis, the exponential simply multiplies each energy eigenstate by a phase.
Appendix 3.B connects matrix mechanics back to wave mechanics. A wave function is the projection of a state vector onto the position basis:
The position basis is continuous, so orthogonality and completeness use the Dirac delta and an integral:
Insert the continuous completeness relation into \(\langle n|n\rangle\):
Using \(\psi_n(x)=\langle x|n\rangle\), this becomes
This equation shows explicitly that the vector notation and the wave-function notation describe the same normalization condition.
| Wave mechanics | Matrix mechanics | Physical meaning |
|---|---|---|
| Wave function \(\psi(x)\) | Ket \(|\Psi\rangle\) | State of the system |
| Integral overlap | Inner product | Projection amplitude |
| Operator acting on functions | Matrix/operator acting on kets | Observable or transformation |
| Expansion in eigenfunctions | Expansion in basis kets | Superposition |
| TDSE | Unitary evolution | Time dependence |
Chapter 3 turns the postulates into a compact matrix language. A quantum state is a ket in Hilbert space. An observable is a Hermitian operator. Possible results are eigenvalues. Probabilities are squared projection amplitudes. Expectation values are bracket expressions. Measurement prepares new states. Time evolution is unitary. Basis changes are unitary transformations.
The Stern-Gerlach experiment is the running laboratory in which these ideas become visible: changing magnet orientation changes the basis, blocking a beam prepares a state, and repeated devices reveal the consequences of non-commuting observables.
This page is designed to support short guided exercises on: Spectral functions of operators and the correspondence between wave mechanics and matrix mechanics.
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.