---
title: "Operators, Eigenvalues and Measurement"
description: "An observable is represented by a self-adjoint operator. Ideal measurement outcomes lie in its spectrum, which may be discrete or continuous. The state determines their probabilities; an eigenstate ha"
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retrieved: 2026-11-12
---

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# Operators, Eigenvalues and Measurement

An observable is represented by a self-adjoint operator. Ideal measurement outcomes lie in its spectrum, which may be discrete or continuous. The state determines their probabilities; an eigenstate has a definite outcome.

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

- States how position and momentum operators act on a wavefunction in the position representation
- Identifies eigenstates and reads off the eigenvalue
- Explains measurement probabilities and distinguishes a single outcome from an ensemble mean

## Lesson: Operators, eigenvalues and quantum measurement

An observable is represented by a self-adjoint operator. In the position representation, position sends psi(x) to x psi(x). The Hamiltonian H is the energy operator. The time-dependent Schrodinger equation, iℏ ∂psi/∂t = H psi, governs evolution. For time-independent H, stationary energy states solve H psi = E psi, as with the oscillator levels you met. Their probability densities stay fixed while their phases evolve.

**Example.** An eigenstate is a nonzero state that an operator returns multiplied by one constant: A psi = a psi. The constant a is its eigenvalue. Apply A and compare the result with psi. A factor that changes with position does not pass this test. An eigenvalue may be zero; that is different from using the zero state, which never counts as an eigenstate.

*(drawing: H returns the same nonzero state times 5 eV. This is an energy eigenstate, and an ideal energy measurement gives 5 eV.)*

An ideal projective measurement returns a value in the operator's spectrum, which may be discrete or continuous. For a normalized superposition of orthonormal eigenstates with distinct discrete eigenvalues, each probability is its coefficient's squared magnitude. When finite, the expectation value is the probability-weighted mean. For continuous outcomes, use a probability density instead. A mean need not be an allowed result.

In one dimension, canonical momentum acts as -iℏ ∂psi/∂x in the position representation. Here i² = -1 and ℏ is the reduced Planck constant. A plane wave exp(ikx) returns ℏk times itself. On the full line, it is a generalized eigenfunction, not a normalizable state.

Position and momentum satisfy Δx Δp >= ℏ/2, where the deltas are standard deviations in the prepared state. Thus both spreads cannot be zero. This bound comes from [x,p] = iℏ I, not detector imperfections. Here [x,p] = xp - px compares the two operator orders; I leaves every state unchanged.

An ideal discrete projective measurement projects the state into the measured eigenspace. An immediate repeat gives the same result if the state is unchanged. Fresh preparations sample the original probabilities. A discrete result excludes preparations assigning it zero probability, but does not identify the earlier state uniquely: both an eigenstate and a superposition can yield the energy eigenvalue E1.

**Recap.** Operators act on states. Eigenstates have definite outcomes. Spectra may be discrete or continuous. Fresh preparations sample probabilities; unchanged immediate repeats preserve a discrete result.

*Lesson corrections, questions and revised flashcards written for this topic.*

## Practice

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

## Needs first

- [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)

## Optional connections

- [Standard deviation and variance](https://lightmysky.com/learn/mathematics/standard-deviation-and-variance-mt_eNpow5tIX2)

## 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)