Continuous Bases (Position)

So far our bases have been discrete — countable sets like $\{|0\rangle, |1\rangle\}$ . A particle moving on a line, however, has a position that varies continuously, and its natural basis is uncountable: one ket $|x\rangle$ for every real point $x$ . Extending the formalism to this case is the bridge from qubits to wave mechanics.

The position basis

For a particle on a line we postulate a continuum of states $\{|x\rangle : x \in \mathbb{R}\}$ , where $|x\rangle$ describes the particle located exactly at position $x$ . These are the eigenstates of the position operator $\hat{x}$ :

\hat{x}\,|x\rangle = x\,|x\rangle.

Because the label $x$ ranges over a continuum, sums become integrals and the Kronecker delta becomes the Dirac delta.

Delta normalization

Discrete orthonormality $\langle e_i | e_j \rangle = \delta_{ij}$ generalizes to

\langle x | x' \rangle = \delta(x - x'),

the Dirac delta function, defined by the property that for any smooth $f$ ,

\int_{-\infty}^{\infty} \delta(x - x')\, f(x')\, dx' = f(x).

The delta is not an ordinary function — it is a distribution, zero everywhere except $x = x'$ and with unit integral. Position eigenstates are therefore not normalizable in the usual sense ( $\langle x | x \rangle = \delta(0)$ is infinite), so they are idealizations rather than genuine physical states. Real states are smooth superpositions of them.

The continuous completeness relation

The discrete resolution of the identity $\sum_k |e_k\rangle\langle e_k| = I$ becomes an integral:

\int_{-\infty}^{\infty} |x\rangle\langle x|\, dx = I.

Inserting this into any expression is the continuous version of the "insert a 1" trick. Applied to a state $|\psi\rangle$ ,

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

The wavefunction is a component in this basis

The crucial identification is

\psi(x) = \langle x | \psi \rangle.

The wavefunction $\psi(x)$ is nothing more than the component of the abstract state $|\psi\rangle$ along the position eigenket $|x\rangle$ — the continuous analogue of the coefficient $c_k = \langle e_k | \psi \rangle$ from the discrete case. Everything you learned about basis expansions carries over with sums replaced by integrals.

Inner products and the Born rule

The inner product of two states becomes an integral over the shared position label,

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

recovering the $L^2(\mathbb{R})$ inner product from the Hilbert-spaces lesson. Normalization is the continuous Parseval identity,

\langle\psi|\psi\rangle = \int_{-\infty}^{\infty} |\psi(x)|^2\, dx = 1,

and $|\psi(x)|^2\, dx$ is the Born-rule probability of finding the particle in the infinitesimal interval $[x, x + dx]$ — a probability density rather than a discrete probability.

The takeaway

A continuous basis replaces sums with integrals, the Kronecker delta with the Dirac delta, and discrete components with the wavefunction $\psi(x) = \langle x|\psi\rangle$ . Position eigenstates are non-normalizable idealizations, but the completeness relation $\int |x\rangle\langle x|\,dx = I$ lets the entire Dirac formalism operate unchanged in the continuum.