In the position basis the position operator is as simple as an operator can be: it multiplies the wavefunction by $x$. Its average in a state is the centre of mass of the probability density, and its spread is the width of that density. Any function of position, such as a potential energy, acts the same way — by multiplication.

The balance point of a beam with uneven weights is the weighted average of positions. The expectation value of position is the balance point of the probability density.

The density $2\sin^{2}(\pi x)$ of a particle in a box of width 1. Its balance point is $\langle x\rangle=0.5$, the dashed line, and its spread is $\Delta x\approx0.18$ on each side, the green bar.

On wavefunctions the position operator multiplies by the coordinate: $\hat{x}\ket{x}=x\ket{x}$ becomes $(\hat{x}\psi)(x)=x\,\psi(x)$. It is Hermitian because is real. Its expectation value and its moments are integrals of the density.

• $\langle x^{2}\rangle=\int x^{2}|\psi|^{2}dx$ and .

Insert the completeness relation of the position basis next to the operator. Each position state is an eigenvector of $\hat{x}$, so the operator turns into the number $x$, and the remaining brackets are the wavefunction and its conjugate.

Insert $\int\ket{x}\bra{x}dx$ between $\hat{x}$ and the ket.

Each position state is an eigenvector of the position operator.

Pull the eigenvalue out as a number.

$\braket{\psi|x}\braket{x|\psi}=\psi^{*}(x)\psi(x)$.

The expectation value of position is the integral of $x$ times the probability density. A density symmetric about a point balances there.

The spread of position is $\sqrt{\langle x^{2}\rangle-\langle x\rangle^{2}}$.

In the position basis the position operator multiplies the wavefunction by $x$, and a function of position multiplies it by that function.

Multiplying by a real function can be moved to either side of an integral bracket unchanged, so it is a Hermitian operator.

Great work! You now know how to:

• act with the position operator by multiplication
• compute an average position and a spread from a density
• average any function of position against the density