Expectation Values

Introduction

Quantum mechanics rarely predicts the result of a single measurement with certainty. If you measure the position of an electron in a molecule, you might find it here on one occasion and there on the next. What the theory does predict precisely is the average you would obtain from a very large number of measurements on identically prepared systems. This average is called the expectation value . It is the bridge between the abstract wavefunction and the numbers that chemists record — mean bond lengths, dipole moments and average energies.

Core explanation

Averages from a probability distribution. In ordinary statistics, if a quantity x takes values with probability density P(x), its mean is ⟨x⟩ = ∫x P(x) dx. The Born interpretation says that the probability density for position is ψ(x) ² = ψ ψ, so for a normalised wavefunction

⟨x⟩ = ∫ψ (x) x ψ(x) dx

The general rule. For any observable Ω with operator Ω̂, the expectation value is the "sandwich" integral

⟨Ω⟩ = ∫ψ Ω̂ ψ dτ

where the integral runs over all space and ψ is normalised. If ψ is not normalised, divide by ∫ψ ψ dτ. The order matters: the operator acts on ψ first, the result is multiplied by ψ , and then everything is integrated. For multiplicative operators such as x or V(x) the order is unimportant, but for derivative operators such as momentum or kinetic energy it is essential.

Eigenstates give sharp values. If ψ is an eigenfunction of Ω̂ with eigenvalue ω, then Ω̂ψ = ωψ and

⟨Ω⟩ = ∫ψ ω ψ dτ = ω ∫ψ ψ dτ = ω

Every single measurement then gives ω, so the average equals that value and the spread is zero. For example, a system in an energy eigenstate has a definite energy, and ⟨E⟩ is simply that energy.

Non-eigenstates give a spread. If ψ is not an eigenfunction of Ω̂, individual measurements give different eigenvalues with various probabilities; ⟨Ω⟩ is their weighted mean. The expectation value need not be one of the possible outcomes — just as the average score on a die, 3.5, is never rolled.

Measuring the spread. The variance is σ² = ⟨Ω²⟩ − ⟨Ω⟩², and σ, its square root, is the uncertainty ΔΩ. This is how uncertainties such as Δx and Δp are defined quantitatively, and it leads directly to the uncertainty principle.

Real results. Because operators for observables are Hermitian, their expectation values are always real, as measured averages must be.

Formulae

⟨Ω⟩ = ∫ψ Ω̂ ψ dτ (ψ normalised). For an unnormalised ψ: ⟨Ω⟩ = ∫ψ Ω̂ ψ dτ / ∫ψ ψ dτ. Uncertainty: ΔΩ = (⟨Ω²⟩ − ⟨Ω⟩²)^½.

Step-by-step reasoning

To calculate an expectation value:

1. Check that ψ is normalised; if not, find the normalisation integral. 2. Write down the operator Ω̂ for the observable. 3. Apply Ω̂ to ψ. 4. Multiply the result by ψ (the complex conjugate). 5. Integrate over all space, dividing by the normalisation integral if needed.

Visual explanation

Picture ψ ² as a hill-shaped distribution along the x-axis. The expectation value ⟨x⟩ is the balance point of the hill — the position where a cardboard cut-out of the curve would balance on a pencil. A symmetrical hill balances at its centre; a lopsided one balances nearer its heavier side.

Real-world analogy

A teacher cannot predict any single student's test mark, but after marking a class of a thousand she can state the average mark with confidence. The expectation value is that class average: exact and reproducible, even though each individual result is uncertain.

Real-world example

The dipole moment of a molecule such as HCl, measured from its behaviour in an electric field, is an expectation value. It is the average of the dipole operator over the electronic wavefunction, reflecting where the electron density sits on average — displaced towards chlorine — rather than any single snapshot of the electrons.

Why?

Why must the operator be sandwiched between ψ and ψ? Because ψ ² = ψ ψ is the probability density, and weighting each value of the observable by its probability requires both factors. Placing the operator between them ensures it acts on ψ, which is necessary when the operator involves derivatives.

Common misconception

"The expectation value is the most likely result of a measurement." It is not. It is the mean of many results. For an electron in the 1s orbital, the mean distance from the nucleus is 1.5a₀, but the most probable distance is a₀ — the two differ.

Worked example

Question: A particle in the region 0 ≤ x ≤ L has the normalised wavefunction ψ = (2/L)^½ sin(πx/L). Find ⟨x⟩.

Reasoning: ⟨x⟩ = (2/L) ∫₀ᴸ x sin²(πx/L) dx. Using sin²θ = ½(1 − cos 2θ), the integral of x/2 from 0 to L is L²/4, and the integral of (x/2) cos(2πx/L) over a whole number of periods is zero. So ⟨x⟩ = (2/L)(L²/4) = L/2.

Answer: ⟨x⟩ = L/2, the centre of the region, as the symmetry of ψ ² about the midpoint suggests.

Quick check

1. If ψ is an eigenfunction of Ĥ with eigenvalue E, what is ⟨E⟩ and what is the uncertainty in the energy? Answer: ⟨E⟩ equals E exactly, and the uncertainty ΔE is zero because every measurement gives the same value.

Exam focus

Know the sandwich formula, remember to normalise first, and keep the operator between ψ and ψ. Exploit symmetry: if ψ ² is symmetric about a point, ⟨x⟩ is that point, and ⟨p⟩ is zero for any real, bound wavefunction.

Advanced insight

For any real bound-state wavefunction, ⟨pₓ⟩ = 0, because −iħ∫ψ dψ/dx dx = −(iħ/2)[ψ²] evaluated between limits where ψ vanishes. The particle moves, but it is equally likely to move left or right. Ehrenfest's theorem shows that expectation values obey classical equations of motion, such as d⟨x⟩/dt = ⟨p⟩/m.

Summary

The expectation value ⟨Ω⟩ = ∫ψ Ω̂ψ dτ is the average of many measurements on identically prepared systems. For eigenstates it equals the eigenvalue and the spread is zero; otherwise individual results scatter around it. It need not be a possible single result, and it is not the most probable value. The variance ⟨Ω²⟩ − ⟨Ω⟩² quantifies the spread.

Practice questions

1. Write the expression for ⟨p⟩ in one dimension for a normalised wavefunction. Answer: ⟨p⟩ = ∫ψ (−iħ dψ/dx) dx over all x. 2. Why can an expectation value be a number that is never obtained in a single measurement? Answer: Because it is a weighted average of the possible eigenvalues, and an average can lie between them, like 3.5 for a fair die. 3. A normalised state has ⟨x⟩ = 0 and ⟨x²⟩ = 4.0 × 10⁻²⁰ m². What is Δx? Answer: Δx = (⟨x²⟩ − ⟨x⟩²)^½ = (4.0 × 10⁻²⁰)^½ = 2.0 × 10⁻¹⁰ m. 4. Explain why the uncertainty in an observable is zero for an eigenstate of its operator. Answer: Every measurement gives the same eigenvalue ω, so ⟨Ω²⟩ = ω² and ⟨Ω⟩² = ω², making the variance zero.