A particle on a line can be found at any point, so its position has a continuum of possible values. The idealised state of being exactly at a point $x$ is written as a ket of its own, and the amplitude for a state to be found there, as $x$ runs over the whole line, is a function: the wavefunction. Every formula of the first chapter survives, with sums over a basis replaced by integrals over the line.

A list of numbers indexed by 1, 2, 3 is a vector; a function indexed by every point of a line is the same thing with a continuous index. The wavefunction is a state written out in the basis of positions.

Three peaks of area 1, each half as wide and twice as tall as the one before. Their limit is the delta function $\delta(x)$: zero away from the origin and with total area 1, so that $\int\delta(x-a)f(x)\,dx=f(a)$. It is the "wavefunction" of a particle exactly at a point.

The position observable $\hat{x}$ of a particle on a line has an eigenvalue for every real $x$, with eigenstates $\ket{x}$. They are orthonormal in the continuous sense, with the Dirac delta in place of the Kronecker delta, and complete with an integral in place of a sum. The components of a state in this basis form its wavefunction.

• The delta function is defined by what it does inside an integral: $\int\delta(x-a)f(x)\,dx=f(a)$ for every continuous $f$. It is the limit of ever narrower peaks of area one, not an ordinary function.

Insert the completeness relation of the position basis between the bra and the ket. Each bracket with a position state is a value of a wavefunction, and the conjugate comes from reversing a bracket, exactly as in Parseval's identity with the sum replaced by an integral.

Put the completeness relation of the position basis in the middle.

The bra and the ket go inside the integral.

Brackets with position states are values of the wavefunctions, the first one reversed and so conjugated.

So the inner product of two states is the integral of one wavefunction's conjugate times the other.

Integrated against any continuous function, the delta function centred at $a$ returns the value of the function at $a$.

The wavefunction at a point $x$ is the bracket of the position state at $x$ with the state. It is the amplitude for finding the particle at $x$.

Two position states at different points are orthogonal: a particle exactly at one point is certainly not at another.

The bracket of two states is the integral over the line of the first wavefunction’s conjugate times the second.

Great work! You now know how to:

• write a state in the position basis as its wavefunction
• use the delta function inside integrals
• turn brackets into integrals of wavefunctions