Introduction: Measuring the Spread
The expectation value tells us the average, but not how spread out the measurements are. The variance and standard deviation quantify the uncertainty in measurement outcomes—they're the \(\Delta A\) in the uncertainty principle.
Definitions
The variance of observable \(\hat{A}\) is:
\[(\Delta A)^2 = \langle(\hat{A} - \langle\hat{A}\rangle)^2\rangle = \langle\hat{A}^2\rangle - \langle\hat{A}\rangle^2\]The standard deviation is:
\[\Delta A = \sqrt{\langle\hat{A}^2\rangle - \langle\hat{A}\rangle^2}\]Properties
- \(\Delta A \geq 0\) always
- \(\Delta A = 0\) if and only if \(|\psi\rangle\) is an eigenstate of \(\hat{A}\)
- Larger \(\Delta A\) means more spread in measurement outcomes
Worked Examples
Example 1: Spin-1/2 in Eigenstate
For \(|+\rangle\) (eigenstate of \(\sigma_z\) with eigenvalue +1):
\(\langle\sigma_z\rangle = +1\) and \(\langle\sigma_z^2\rangle = (+1)^2 = 1\)
\(\Delta\sigma_z = \sqrt{1 - 1^2} = 0\)
No uncertainty—we always get +1.
Example 2: Spin-1/2 in Superposition
For \(|\psi\rangle = \frac{1}{\sqrt{2}}(|+\rangle + |-\rangle)\):
\(\langle\sigma_z\rangle = 0\) (calculated before)
\(\langle\sigma_z^2\rangle = \frac{1}{2}(1) + \frac{1}{2}(1) = 1\)
\(\Delta\sigma_z = \sqrt{1 - 0} = 1\)
Maximum uncertainty for spin-1/2.
Example 3: Position Variance of Gaussian
For \(\psi(x) = \left(\frac{1}{\pi a^2}\right)^{1/4}e^{-x^2/2a^2}\):
\(\langle x\rangle = 0\) (symmetric)
\(\langle x^2\rangle = \frac{a^2}{2}\)
\(\Delta x = \frac{a}{\sqrt{2}}\)
The Quantum Connection
The variance characterizes how "quantum" a measurement is. In an eigenstate, there's no uncertainty (\(\Delta A = 0\))—the outcome is predetermined. In a superposition, there's genuine randomness. The product \(\Delta x \cdot \Delta p\) bounds how localized a state can be simultaneously in both position and momentum.