---
title: "Observables as Hermitian Operators and Their Spectra"
description: "Every measurable quantity is a Hermitian operator whose eigenvalues are the possible results and whose eigenvectors span the state space. Continuous spectra need a different normalisation, which is wh"
canonical: https://lightmysky.com/learn/science/observables-as-hermitian-operators-and-their-spectra-mt_TXCN8uiWqs
source: https://lightmysky.com/learn/science/observables-as-hermitian-operators-and-their-spectra-mt_TXCN8uiWqs.md
retrieved: 2026-09-12
---

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# Observables as Hermitian Operators and Their Spectra

Every measurable quantity is a Hermitian operator whose eigenvalues are the possible results and whose eigenvectors span the state space. Continuous spectra need a different normalisation, which is why a momentum eigenstate is not an ordinary state.

Subject: Science · Area: Quantum & Modern Physics · Ages 22 to 23
Page: https://lightmysky.com/learn/science/observables-as-hermitian-operators-and-their-spectra-mt_TXCN8uiWqs

## Ready when they can

- Shows that a Hermitian operator has real eigenvalues and orthogonal eigenvectors
- Expands an arbitrary state in an operator's eigenbasis and predicts the measurement probabilities
- Explains why a continuous spectrum forces a different normalisation convention

## Lesson: Measuring devices as Hermitian operators

Every measurable quantity, like position, momentum, or energy, is represented by an operator. The spectrum of an operator is the menu of results a measurement can return: only its eigenvalues ever appear on the dial. Observables must be Hermitian because Hermitian operators always have real eigenvalues, exactly what detectors report. Their eigenvectors are mutually orthogonal, so distinct outcomes match distinct, non overlapping states.

**Example.** The hydrogen atom shows the machinery at work. Solving its equation yields states labeled by quantum numbers, each with a definite energy eigenvalue. A general electron state is a combination of these eigenstates. Expand an arbitrary state in the eigenbasis and the squared weights give the probability of measuring each energy. For amplitudes 0.6 and 0.8, the odds are 0.36 and 0.64.

Measurement is probabilistic at its root: the state fixes only the odds of each eigenvalue. Continuous spectra, like position, need extra care: probabilities come from densities rather than plain lists, and normalization works differently than for discrete lines. Bound and free states therefore use different conventions. Atomic spectra are the experimental face of this: each sharp line marks a jump between two definite energies.

**Tip.** To predict any measurement, expand the state in that observable's eigenbasis and square the weights. Orthogonality guarantees the outcomes do not overlap. When comparing theory with a table of lines, quote which lines you used and the uncertainty on each. A claim that the model matches the data holds only if prediction gaps sit inside the stated errors.

**Recap.** Hermitian operators give real eigenvalues as outcomes, and squared expansion weights give the odds.

## Practice

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

## Needs first

- [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)
- [Compact Self-Adjoint Operators and the Spectral Theorem](https://lightmysky.com/learn/mathematics/compact-self-adjoint-operators-and-the-spectral-theorem-mt_P_Jr9LWFQj)
- [Symmetric Matrices and the Spectral Theorem](https://lightmysky.com/learn/mathematics/symmetric-matrices-and-the-spectral-theorem-mt_sdQv4m7Nbk)

## Opens up

- [Crystal Lattices, the Reciprocal Lattice and Bloch's Theorem](https://lightmysky.com/learn/science/crystal-lattices-the-reciprocal-lattice-and-blochs-theorem-mt_nbNnHpGnZI)
- [Commutators, Compatible Observables and the General Uncertainty Relation](https://lightmysky.com/learn/science/commutators-compatible-observables-and-the-general-uncertainty-relation-mt_TKoUwu74K9)