Ask a classical physicist "what is the energy of this system?" and the question has a plain answer: a number, sitting there waiting to be read off. Quantum mechanics refuses to play along. In the quantum world a physical quantity you can measure — position, momentum, energy, spin — is not a number attached to the system at all. It is a question you put to the system, and the machinery that turns that question into possible answers is a mathematical object called an operator.

This page is about the bridge between the abstract wavefunction and the numbers a laboratory dial actually shows. Three ideas do all the work, and they lock together tightly: every measurable quantity (an observable) is represented by a special kind of operator; the numbers a single measurement can possibly return are that operator's eigenvalues; and when the answer is uncertain, the long-run average of many identical measurements is the expectation value. Get these three straight and the strange bookkeeping of quantum measurement — probabilities, collapse, "why is the average not a possible answer?" — stops being mysterious and starts being arithmetic.

Observables are operators

An operator is just a rule that eats a function and hands back another function. In quantum mechanics each observable comes with its own operator that acts on the wavefunction \psi(x). The three you meet first are:

• Position \hat{x} — the simplest of all: it just multiplies the wavefunction by x, \hat{x}\,\psi(x) = x\,\psi(x).
• Momentum \hat{p} — this one differentiates: \hat{p} = -i\hbar\,\dfrac{d}{dx}. The factor of i is not decoration; it is exactly what is needed to make the operator well behaved (Hermitian), as we will see.
• Energy \hat{H} — the Hamiltonian, built from the other two just as classical energy is kinetic plus potential: \hat{H} = \dfrac{\hat{p}^2}{2m} + V(\hat{x}) = -\dfrac{\hbar^2}{2m}\dfrac{d^2}{dx^2} + V(x).

These operators are not arbitrary. Every operator standing for a real physical observable must be Hermitian (self-adjoint) — meaning, loosely, that it equals its own conjugate transpose, \hat{A}^\dagger = \hat{A}. That single algebraic condition turns out to guarantee exactly the property a measurable quantity must have: the numbers it can produce are always real. A dial never reads 3 + 2i volts.

What a measurement can return: eigenvalues

Occasionally an operator acts on a special function and gives that same function straight back, merely scaled by a number. Such a function is an eigenstate (or eigenfunction), and the scaling number is its eigenvalue:

The a_n are the heart of the whole theory, because of the measurement postulate: the only possible outcomes of measuring the observable A are the eigenvalues of \hat{A} — nothing in between, ever. If the system happens to be in the eigenstate \psi_n, you are guaranteed to get a_n, dead certain, every time. If it is in some other state you will get one of the a_n, but which one is a matter of probability.

Let us check an eigenvalue by hand. Take the momentum operator \hat{p} = -i\hbar\,\dfrac{d}{dx} and feed it a plane wave \psi(x) = e^{ikx}:

Out came the same function times \hbar k, so e^{ikx} is a momentum eigenstate with the real eigenvalue p = \hbar k — the de Broglie momentum. (Notice how the -i and the i from the derivative multiplied to give +1: that is the i earning its keep, making the eigenvalue real rather than imaginary.)

The expectation value: a weighted average of outcomes

Suppose the state is not an eigenstate, so a single measurement is a gamble among the eigenvalues. Repeat the experiment on a huge number of identically-prepared copies and average the readings. That average is the expectation value \langle A \rangle. It is a probability-weighted mean, exactly like the expected value of a die roll:

Here c_n is the amplitude of the eigenstate \psi_n in the expansion \psi = \sum_n c_n \psi_n, and the Born rule says the probability of getting a_n is p_n = |c_n|^2. There is an equivalent, all-in-one formula that never mentions the individual outcomes — the "sandwich":

The two expressions are the same number written two ways. Below, an observable has just two possible outcomes, a_1 = 2 and a_2 = 6. Drag the slider to change the probability p of getting a_1 (so a_2 has probability 1-p) and watch the expectation value \langle A \rangle = p\,a_1 + (1-p)\,a_2 slide smoothly between the two rungs — landing on an eigenvalue only at the very ends.

Worked examples

Example 1 — an energy measurement as a weighted average. A particle is prepared so that a measurement of its energy returns E_1 = 1\ \text{eV} with probability p_1 = \tfrac14 and E_2 = 3\ \text{eV} with probability p_2 = \tfrac34. What is \langle E \rangle?

The average is 2.5\ \text{eV} — and here is the punchline that trips people up: a single measurement can never give 2.5\ \text{eV}. Every actual reading is either 1 or 3; the observable has no eigenvalue at 2.5. The expectation value is a statement about the whole ensemble, not about any one experiment.

Example 2 — from amplitudes to the average. A state is written as a superposition of three energy eigenstates,

with eigenvalues a_1 = 2,\ a_2 = 5,\ a_3 = 11. First turn amplitudes into probabilities with the Born rule, p_n = |c_n|^2:

Then average:

The amplitudes squared must sum to 1 — that is just the statement that some outcome is certain to occur. Always sanity-check normalisation before you trust an expectation value.

Example 3 — checking a Hamiltonian eigenvalue. Take the free-particle Hamiltonian \hat{H} = -\dfrac{\hbar^2}{2m}\dfrac{d^2}{dx^2} (so V = 0) and the plane wave \psi = e^{ikx}. Two derivatives bring down (ik)^2 = -k^2:

So e^{ikx} is an energy eigenstate with eigenvalue E = \hbar^2 k^2 / 2m — precisely the kinetic energy p^2/2m of a particle with momentum p = \hbar k, real and positive as an energy must be.

The spectral theorem: why this all hangs together

There is a deep reason the bookkeeping works. Because an observable's operator is Hermitian, the spectral theorem guarantees two priceless facts at once: its eigenvalues are real, and its eigenstates form a complete orthonormal basis for the space of states. Orthonormal means the eigenstates are mutually perpendicular and unit length, \langle \psi_m | \psi_n \rangle = \delta_{mn}; complete means any state can be written as a combination of them,

This is the spectral decomposition, and it is what makes measurement well-defined: because the eigenstates are a basis, every possible state can be resolved into eigen-components, each carrying a probability |c_n|^2 of being the outcome. Substituting the expansion into the sandwich formula reproduces the weighted-average form, \langle A \rangle = \sum_n |c_n|^2 a_n — the two definitions are one and the same, guaranteed by the spectral theorem.