Introduction: What Will We Measure on Average?
Since quantum mechanics is probabilistic, single measurements give random outcomes. The expectation value is the average result over many identical measurements. It connects quantum formalism to experimental predictions.
Definition
The expectation value of observable \(\hat{A}\) in state \(|\psi\rangle\) is:
\[\langle\hat{A}\rangle = \langle\psi|\hat{A}|\psi\rangle\]In position representation:
\[\langle\hat{A}\rangle = \int_{-\infty}^{\infty} \psi^*(x) \hat{A}\psi(x) \, dx\]Physical Interpretation
If we expand \(|\psi\rangle\) in eigenstates: \(|\psi\rangle = \sum_n c_n |a_n\rangle\) where \(\hat{A}|a_n\rangle = a_n|a_n\rangle\):
\[\langle\hat{A}\rangle = \sum_n |c_n|^2 a_n\]The expectation value is the probability-weighted average of eigenvalues.
Worked Examples
Example 1: Position Expectation Value
For a symmetric wavefunction \(\psi(x) = \psi(-x)\) centered at origin:
\[\langle\hat{x}\rangle = \int_{-\infty}^{\infty} x|\psi(x)|^2 \, dx = 0\]The average position is the center of symmetry.
Example 2: Momentum Expectation Value
\[\langle\hat{p}\rangle = \int_{-\infty}^{\infty} \psi^*(x)\left(-i\hbar\frac{d\psi}{dx}\right) dx\]For a stationary real wavefunction: \(\langle\hat{p}\rangle = 0\).
Example 3: Spin Measurement
For spin state \(|\psi\rangle = \frac{1}{\sqrt{2}}(|+\rangle + |-\rangle)\):
\[\langle\sigma_z\rangle = |\frac{1}{\sqrt{2}}|^2(+1) + |\frac{1}{\sqrt{2}}|^2(-1) = \frac{1}{2} - \frac{1}{2} = 0\]On average, equal numbers of +1 and -1 measurements.
The Quantum Connection
Expectation values are what experiments actually measure (in the limit of many trials). The connection \(\langle\hat{A}\rangle = \text{Tr}(\rho\hat{A})\) using the density matrix \(\rho\) extends this to mixed states. Classical physics emerges when quantum fluctuations become negligible and only expectation values matter.