Expectation Value
Definition and meaning of Expectation Value in chemistry.
Expectation value is the average result quantum mechanics predicts for a measurable property (observable) when the same measurement is performed once on each of many identically prepared systems, calculated as ⟨A⟩ = ∫ψ*Âψ dτ.
In more detail
Because a quantum system generally has no single definite value for an observable until it is measured, the wavefunction ψ only gives a probability distribution (|ψ|²) over possible outcomes. The expectation value is the mean of that distribution, found by sandwiching the observable's operator between ψ* and ψ and integrating over all space, using a normalized wavefunction. If ψ happens to be an eigenfunction of the operator Â, every individual measurement returns the same eigenvalue exactly, so the expectation value equals that eigenvalue with zero spread.
Key facts
| Field | Physical Chemistry |
|---|---|
| Formula | ⟨A⟩ = ∫ψ*Âψ dτ |
| Requires | Normalized wavefunction: ∫ψ*ψ dτ = 1 |
| Special case | Equals the eigenvalue exactly if ψ is an eigenfunction of  |
For a particle in a one-dimensional box of length L in its ground state, the expectation value of position is ⟨x⟩ = L/2 (the box's midpoint), even though a single position measurement could locate the particle anywhere in the box, with lowest probability near the walls.
Frequently asked questions
Is the expectation value the same as the most likely measured outcome?
No. It is the probability-weighted average over all possible outcomes, not necessarily the most likely single result. It can even land on a value that a single measurement could never return: for the particle in a box's first excited state (n = 2), the expectation value of position is still L/2, yet the wavefunction has a node exactly at that point, so the probability of actually measuring the particle there is zero.
Why do chemists care about expectation values?
They connect the abstract wavefunction to measurable, comparable quantities such as average bond length, dipole moment, or energy, which is how quantum calculations are validated against experiment.