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Hilbert Space and Dirac's Bra-Ket Language

Lift the theory off of wavefunctions and onto abstract vectors. Kets, bras, inner products and complete bases turn quantum mechanics into clean linear algebra you can actually compute with.

States live in Hilbert space

The wavefunction \psi(x) is only one description of a state — its description in the position basis. The state itself is more abstract: a vector |\psi\rangle living in a complex vector space equipped with an inner product, called a Hilbert space. The same state can be written as \psi(x), or as a momentum-space \tilde\psi(p), or as a column of amplitudes — one arrow, many coordinate systems.

Two ingredients make a Hilbert space the right home for quantum states. An inner product, which lets us define lengths, angles and — as we'll see — probabilities; and completeness, which guarantees that limits of states are again states, a technical necessity once the space is infinite-dimensional. Physical states are always normalized to length one.

Kets, bras, and inner products

Dirac's bra-ket notation is the working language. A state is a ket |\psi\rangle. To each ket there corresponds a bra \langle\psi|, its Hermitian conjugate, which acts as a linear functional. Sliding a bra against a ket produces a single complex number, the inner product \langle\phi|\psi\rangle — the `bracket` that gives the notation its name.

\langle\phi|\psi\rangle=\int_{-\infty}^{\infty}\phi^{*}(x)\,\psi(x)\,dx

The inner product, written out in the position representation — a single complex number built from two states.

\langle\phi|\psi\rangle=\langle\psi|\phi\rangle^{*},\qquad \langle\psi|\psi\rangle=\lVert\psi\rVert^{2}\ge 0,\qquad \langle\psi|\psi\rangle=1

Its defining properties: conjugate-symmetry, a real non-negative norm, and normalization to one for physical states.

The inner product is the machine that converts two states into a probability amplitude. Read \langle\phi|\psi\rangle as `the amplitude to find the system in state \phi given that it is in state \psi`; its squared magnitude |\langle\phi|\psi\rangle|^2 is the probability. This is the Born rule again, now written in a form that no longer mentions position at all.

Bases and completeness

Pick an orthonormal basis \{|n\rangle\}, meaning \langle m|n\rangle=\delta_{mn}. Any state then expands as |\psi\rangle=\sum_n c_n|n\rangle, and each coefficient is a projection, c_n=\langle n|\psi\rangle — the component of the state along |n\rangle, which is exactly the amplitude for the outcome n.

A state |\psi\rangle as a vector, projected onto orthonormal basis directions |n\rangle. The projections \langle n|\psi\rangle are the amplitudes c_n, and normalization means \sum_n|c_n|^2=1: the probabilities of all outcomes sum to one.

|\psi\rangle=\sum_{n}|n\rangle\langle n|\psi\rangle \quad\Longrightarrow\quad \sum_{n}|n\rangle\langle n|=\hat I

The completeness relation (resolution of the identity): summing the projectors |n\rangle\langle n| over a complete basis rebuilds the identity operator.

Representations: position and momentum

Now the wavefunction reveals its true identity: it is a projection. \psi(x)=\langle x|\psi\rangle is the amplitude to find the particle at position x, i.e. the component of |\psi\rangle along the position eigenstate |x\rangle. Because x is continuous, this basis uses an integral in place of a sum. The momentum representation \tilde\psi(p)=\langle p|\psi\rangle is the Fourier transform of \psi(x) — the same state, a different basis.

\psi(x)=\langle x|\psi\rangle,\qquad \hat p\;\longrightarrow\;-i\hbar\,\frac{d}{dx}\ \ (\text{position basis})

The wavefunction is the position-basis component of the abstract state; in that basis, momentum acts as a derivative.

This is the real dividend of the abstraction. A result proved with |\psi\rangle and operators holds in every representation at once — you may compute in whichever basis is easiest, and the physics (expectation values, probabilities, spectra) comes out basis-independent. Those observables and the operators that represent them are the subject of the next guide.