Introduction: Where is the Particle?
The position operator \(\hat{x}\) represents the observable "where is the particle?" Its eigenstates are states of definite position, and its action on wavefunctions is remarkably simple: multiplication.
Definition and Action
The position operator in position representation is:
\[\hat{x}\psi(x) = x \cdot \psi(x)\]It simply multiplies the wavefunction by \(x\).
The eigenstates satisfy:
\[\hat{x}|x_0\rangle = x_0|x_0\rangle\]The eigenvalue is the position itself.
Properties of Position Eigenstates
- Eigenvalue spectrum: Continuous, all real numbers \(x_0 \in \mathbb{R}\)
- Normalization: \(\langle x'|x\rangle = \delta(x - x')\)
- Completeness: \(\int_{-\infty}^{\infty} |x\rangle\langle x| \, dx = \hat{I}\)
- Wavefunction: \(\langle x|x_0\rangle = \delta(x - x_0)\)
Worked Examples
Example 1: Applying the Position Operator
Let \(\psi(x) = e^{-x^2}\). Then:
\[\hat{x}\psi(x) = x \cdot e^{-x^2}\]The result is a new function \(xe^{-x^2}\).
Example 2: Expectation Value of Position
For a normalized state \(\psi(x)\):
\[\langle\hat{x}\rangle = \langle\psi|\hat{x}|\psi\rangle = \int_{-\infty}^{\infty} \psi^*(x) \cdot x \cdot \psi(x) \, dx = \int_{-\infty}^{\infty} x|\psi(x)|^2 \, dx\]This is the probability-weighted average position.
Example 3: Position of a Gaussian
For \(\psi(x) = \left(\frac{1}{\pi a^2}\right)^{1/4} e^{-(x-x_0)^2/2a^2}\):
The expectation value is \(\langle\hat{x}\rangle = x_0\) (centered at \(x_0\)).
The Quantum Connection
Position eigenstates \(|x\rangle\) are idealizations—they're not normalizable (delta function normalization). No physical particle can be in a perfect position eigenstate. Real particles are always in superpositions \(|\psi\rangle = \int \psi(x)|x\rangle \, dx\), spread over some region. The position operator extracts the location information from these superpositions.