Expectation Values, Formally

An expectation value is the average result you would get by measuring an observable on many identically prepared copies of a state. It is not (in general) one of the possible outcomes — it is their probability-weighted mean. This lesson derives the formula from the Born rule and computes one by hand.

From outcomes to averages

Measuring $\hat{A} = \sum_k a_k\hat{P}_k$ on $|\psi\rangle$ gives outcome $a_k$ with probability $p(a_k) = \langle\psi|\hat{P}_k|\psi\rangle$ . The average outcome is therefore the ordinary statistical mean,

\langle \hat{A}\rangle \;=\; \sum_k a_k\, p(a_k) \;=\; \sum_k a_k\,\langle\psi|\hat{P}_k|\psi\rangle.

Now use the spectral decomposition $\hat{A} = \sum_k a_k\hat{P}_k$ to pull the sum inside the expectation:

\langle\hat{A}\rangle = \langle\psi|\Big(\sum_k a_k\hat{P}_k\Big)|\psi\rangle = \langle\psi|\hat{A}|\psi\rangle.

This is the compact and famous form of the expectation value:

\boxed{\;\langle\hat{A}\rangle = \langle\psi|\hat{A}|\psi\rangle\;}

The two expressions are identical — the projector form makes the statistical meaning explicit, while $\langle\psi|\hat{A}|\psi\rangle$ is the convenient form for calculation.

A worked qubit example

Take $|\psi\rangle = \cos\theta\,|0\rangle + \sin\theta\,|1\rangle$ and the observable $\hat{Z}$ , which has eigenvalue $+1$ on $|0\rangle$ and $-1$ on $|1\rangle$ . The Born-rule probabilities are

p(0) = |\langle 0|\psi\rangle|^2 = \cos^2\theta, \qquad p(1) = |\langle 1|\psi\rangle|^2 = \sin^2\theta.

So the expectation value is

\langle\hat{Z}\rangle = (+1)\cos^2\theta + (-1)\sin^2\theta = \cos^2\theta - \sin^2\theta = \cos 2\theta.

For $\theta = \pi/6$ this gives $\cos(\pi/3) = \tfrac12$ . You can confirm with the matrix form:

\langle\psi|\hat{Z}|\psi\rangle = \begin{pmatrix} \cos\theta & \sin\theta \end{pmatrix} \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \begin{pmatrix} \cos\theta \\ \sin\theta \end{pmatrix} = \cos^2\theta - \sin^2\theta,

the same answer. The expectation value $\tfrac12$ is not a possible single measurement result — those are only $\pm 1$ — but it is the long-run average.

Properties worth remembering

Real-valued. Because $\hat{A}$ is Hermitian, $\langle\hat A\rangle$ is always real.
Bounded by the spectrum. $\langle\hat A\rangle$ lies between the smallest and largest eigenvalues, since it is a weighted average of them.
Linear in the observable. $\langle\alpha\hat A + \beta\hat B\rangle = \alpha\langle\hat A\rangle + \beta\langle\hat B\rangle$ .

Try it

For $|\psi\rangle = \cos\theta\,|0\rangle + \sin\theta\,|1\rangle$ with $\theta = \pi/6$ , compute the expectation value $\langle\hat Z\rangle = (+1)p(0) + (-1)p(1)$ using the Born-rule probabilities, and return the number.

Run your code to see the quantum state.