Once states are vectors, observables must be represented by objects that act on vectors. In a chosen basis, those objects are matrices. This section follows one continuous chain: construct the matrix elements, identify the eigenstates with definite outcomes, and then rebuild the operator from those outcomes through its spectral decomposition.
The expression \(\langle n|\Psi\rangle\) is an inner product: it returns a number, the projection amplitude. The expression \(|n\rangle\langle n|\) is an outer product: it returns an operator, or matrix, that projects onto the state \(|n\rangle\).
This distinction is essential. Projection amplitudes become probabilities, while projectors become the building blocks of observables and measurements.
Insert the completeness relation on both sides of an operator:
Expanding this expression gives
The numbers \(A_{mn}=\langle m|\hat A|n\rangle\) are the matrix elements of the operator in the basis \(\{|n\rangle\}\).
In the standard basis, each outer product \(|m\rangle\langle n|\) contributes to one row and one column of the matrix. Thus a \(d\)-dimensional operator can be written as
This is the formal reason matrix mechanics is called matrix mechanics: the observable is represented by the table of all its actions between basis states.
The postulate for observables says that every measurable quantity is represented by a Hermitian operator. In the Stern-Gerlach case, the measurable quantity is a spin projection along the axis selected by the magnet. The possible results are obtained from the eigenvalue equation.
If the incoming state is \(|a_n\rangle\), the measurement gives \(a_n\) with certainty. Thus the eigenvectors are the definite-value states and the eigenvalues are the numbers that the apparatus can display.
If the basis vectors are eigenvectors of \(\hat A\), then \(\hat A|a_n\rangle=a_n|a_n\rangle\). Substituting this into the matrix expansion collapses the double sum into a simpler expression:
This spectral form is the bridge to measurement. The projectors select the outcome channels; the numbers \(a_n\) label the values displayed by the apparatus.
For SG(Z), the basis \(\{|+\rangle,|-\rangle\}\) is already the eigenbasis of the measured observable. For SG(X), the eigenbasis is \(\{|+\rangle_x,|-\rangle_x\}\). The same physical spin space is being used; the operator changes because the measurement axis changes.
A matrix element \(A_{mn}=\langle m|\hat A|n\rangle\) tells how much of the output along \(|m\rangle\) is produced when \(\hat A\) acts on the input basis state \(|n\rangle\).
Columns therefore describe the transformed basis vectors. This interpretation helps students read matrices physically rather than treating them as arrays of disconnected numbers.
For observables, the special basis is the eigenbasis, where the matrix becomes diagonal and the diagonal entries are possible measurement outcomes.
This page is designed to support short guided exercises on: matrix representations, definite-value states and spectral decomposition.
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.