This section is the heart of the state description. Hilbert space is the arena, a basis is a set of possible outcomes for a chosen measurement, and superposition is the statement that a state can be expanded in that basis. The Stern-Gerlach device makes the abstract idea concrete: changing the magnet direction changes the basis used to describe the same spin state.
The book defines Hilbert space as a generalization of Euclidean vector space equipped with an inner product. It may be finite-dimensional or infinite-dimensional. For spin-1/2, the space is finite and two-dimensional, which makes all the basic ideas visible with simple column vectors.
An orthonormal basis \(\{|n\rangle\}\) lets us write any state as
This is the vector version of expanding a wave function in eigenfunctions. Physically, the expansion says that the state is a superposition of eigenstates of the chosen measurement.
The coefficients are not arbitrary labels. Multiply the expansion by \(\langle m|\) on the left:
Using orthonormality, only the term \(n=m\) survives, so
The coefficient is therefore a projection amplitude of the state onto the basis vector \(|m\rangle\).
Substituting the projection formula back into the expansion gives
Because this must hold for any state, the expression in parentheses is the identity operator:
This completeness relation will be used repeatedly to build matrix operators and expectation values.
The Stern-Gerlach experiment gives a two-level system. Along the \(z\)-axis, the basis is chosen as
A different magnet direction corresponds to a different basis. For the \(x\)-axis, a convenient choice is
The normalization and orthogonality conditions are satisfied in both bases. The difference is not the dimension of the space, but the measurement question being asked.
The \(x\)-basis can be written in terms of the \(z\)-basis:
The inverse relations are equally important:
These formulas are what make the later Stern-Gerlach sequences understandable. A state that is definite for SG(Z) becomes a superposition when described in the SG(X) basis.
Superposition is basis-dependent in its appearance. The state \(|+\rangle\) is a single basis vector for a \(z\)-measurement, but it is a balanced superposition for an \(x\)-measurement. This is why a beam prepared by SG(Z) can split again when it enters SG(X).
This page is designed to support short guided exercises on: Orthonormal bases, vector expansion, spin bases along different axes and the meaning of coefficients as projections.
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.