---
title: "Operators, Eigenvalues and Measurement"
description: "Every observable is represented by an operator, and the values a measurement can return are that operator's eigenvalues. A state that is not an eigenstate gives one of those values with a probability "
canonical: https://lightmysky.com/learn/science/operators-eigenvalues-and-measurement-mt_E9C_9DlwD2
source: https://lightmysky.com/learn/science/operators-eigenvalues-and-measurement-mt_E9C_9DlwD2.md
retrieved: 2026-09-12
---

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# Operators, Eigenvalues and Measurement

Every observable is represented by an operator, and the values a measurement can return are that operator's eigenvalues. A state that is not an eigenstate gives one of those values with a probability set by the wavefunction.

Subject: Science · Area: Quantum & Modern Physics · Ages 20 to 22
Page: https://lightmysky.com/learn/science/operators-eigenvalues-and-measurement-mt_E9C_9DlwD2

## Ready when they can

- Applies the position, momentum and energy operators to a stated wavefunction
- Identifies eigenstates and reads off the eigenvalue
- Explains why a superposition returns one of several results rather than an average

## Lesson: Operators ask, eigenvalues answer: quantum measurement

Operators turn physical quantities into actions on the wavefunction. Position multiplies by x, momentum differentiates in space scaled by a constant, and energy drives time change through the Hamiltonian. The Schrodinger equation itself is an energy eigenvalue problem: H acting on psi returns E times psi. Each solution pairs one pattern with one energy, like the even oscillator ladder you met.

**Example.** Test any wavefunction with the apply-and-compare move. Apply the operator and check what comes back. Getting the same function times a number means you hold an eigenstate, and the number is the eigenvalue. If H applied to psi returns 5 eV times psi, then psi is an energy eigenstate with eigenvalue 5 eV. Anything else means it is not an eigenstate.

Measurement acts on the system. A system in an eigenstate of the measured operator returns its eigenvalue with certainty. A superposition returns one eigenvalue at random, with odds given by the squared amplitudes, and the state settles onto the winner. Averages over many trials give the expectation value. No pair like position and momentum shares an eigenstate, so sharpening one blurs the other.

**Tip.** For measurement questions, ask eigenstate of what, then apply and compare. Never average eigenvalues into a single result: one trial gives one eigenvalue. A single result says nothing about the past, since repeats follow the odds and average to the expectation value.

**Recap.** Operators act on wavefunctions, eigenstates return eigenvalues, and superpositions yield one random eigenvalue per trial with odds from squared amplitudes.

## Practice

8 questions on this page, each with its working shown.

## Needs first

- [Standard Deviation and Variance](https://lightmysky.com/learn/mathematics/standard-deviation-and-variance-mt_eNpow5tIX2)
- [The Quantum Harmonic Oscillator and Zero-Point Energy](https://lightmysky.com/learn/science/the-quantum-harmonic-oscillator-and-zero-point-energy-mt_HyDxmqyReX)
- [Eigenvalues and Eigenvectors](https://lightmysky.com/learn/mathematics/eigenvalues-and-eigenvectors-mt_TVqqaw11qa)

## Opens up

- [States as Vectors: Dirac Notation and the State Space](https://lightmysky.com/learn/science/states-as-vectors-dirac-notation-and-the-state-space-mt_9g0aNMxQ8M)
- [Separating the Schrödinger Equation for Hydrogen](https://lightmysky.com/learn/science/separating-the-schrodinger-equation-for-hydrogen-mt_Yk2qHB0_v2)