The chapter now changes language. Instead of writing a state as a wave function from the start, it first defines the abstract vector notation that will be used for spin. The notation is compact, but each symbol has a clear role: kets are state vectors, bras are their dual vectors, and inner products measure overlaps.
A ket \(|n\rangle\) is a vector in a \(d\)-dimensional Hilbert space. In a chosen basis, it can be represented as a column vector:
For Stern-Gerlach spin, the relevant Hilbert space is two-dimensional because the chosen spin component has two possible outcomes. That is why spin-1/2 is the natural first example for matrix mechanics.
The corresponding bra is the adjoint of the ket. It is a row vector obtained by transposing and complex conjugating the ket:
This is why Dirac notation is efficient: the symbol \(\langle n|\) automatically reminds us that complex conjugation is part of the dual vector. That matters because quantum amplitudes can be complex.
The product \(\langle m|n\rangle\) is an inner product. It is the matrix-mechanics version of the overlap between two states. For an orthonormal basis, the result is the Kronecker delta:
If \(m=n\), the overlap is one; if the basis states are different, the overlap is zero. In Stern-Gerlach language, this says that the two output beams for a fixed magnet orientation represent mutually exclusive basis states.
A physical state must be normalized, because the total probability of all possible outcomes must be one. In vector notation this reads
This compact condition replaces a familiar wave-mechanics integral. Instead of \(\int |\Psi(x)|^2dx=1\), the finite-dimensional spin example uses a finite sum of squared amplitudes.
For a Stern-Gerlach device aligned along \(z\), it is convenient to choose the two outcome states as the basis:
These vectors are normalized and orthogonal. They are not little arrows in ordinary space; they are abstract labels for the two possible outcomes of the spin measurement along \(z\).
The state postulate can now be stated in matrix language: a quantum state is completely represented by a vector \(|\Psi(t)\rangle\) in Hilbert space. The word completely is important. Once the vector is known, the theory tells us how to compute all probabilities for measurements represented in that space.
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