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← University Quantum Mechanics I

University Quantum Mechanics I · Operators and Observables · 3.08

Expectation values, variance, and standard deviation

⟨A⟩ = integral ψ* A-hat ψ dx, the operator between the factors so it acts on ψ first, equal to sum aₙ |cₙ|² in an eigenbasis; σ² = ⟨A²⟩ - ⟨A⟩² needs A-hat applied twice, not ⟨A⟩ squared.

Course-map guide · not a complete lesson or simulation

Scope & orientation

What this subsection covers

⟨A⟩ = integral ψ* A-hat ψ dx, the operator between the factors so it acts on ψ first, equal to sum aₙ |cₙ|² in an eigenbasis; σ² = ⟨A²⟩ - ⟨A⟩² needs A-hat applied twice, not ⟨A⟩ squared.

A strong response uses ⟨a⟩ and sigma marked on the density |ψ|² and states where the model stops being reliable.

Reasoning checklist

Evidence, assumptions and limits

01

Assumptions to state

State the system, observable, approximation, and conditions held fixed before using a model.

02

Evidence to collect

Does inserting a filter at 45 degrees between crossed polarisers restore transmission, and does applying the same two filters in the opposite order change the result?

Read the complete note

Does inserting a filter at 45 degrees between crossed polarisers restore transmission, and does applying the same two filters in the opposite order change the result? Useful evidence includes transmitted intensity for the crossed pair and with the middle filter at several angles, fitted to cos squared, plus the same filters in reversed order; a statement that Malus's classical law fits the data equally well, so this is an analogy for non-commuting projections rather than evidence for quantum mechanics; and an explicit note that the order-dependence belongs to the operators, not to a filter mechanically disturbing photons.

03

Limits to state

This GioPhysics course map is an adaptable learning sequence, not academic credit, accreditation, or a universal university syllabus.

Read the complete note

This GioPhysics course map is an adaptable learning sequence, not academic credit, accreditation, or a universal university syllabus. It is a first dedicated quantum-mechanics course, distinct from University Physics V, which covers quantum mechanics alongside atomic, nuclear and particle physics in twenty units: this course is narrower, slower, and teaches the linear algebra it needs rather than assuming it. Hydrogen appears here as an introduction, with the full radial derivation belonging to a second course. A midterm examination is assumed around week 8. Follow your institution's published scope, notation, laboratory programme, and assessment rules. A result should be checked against units, signs, limiting cases, and the conditions under which its model was derived.

Diagram & examples

Work the claim before choosing an equation

Interactive concept map

Follow the model from claim to evidence.

01 · Physical claimExpectation values, variance, and standard deviation

⟨A⟩ = integral ψ* A-hat ψ dx, the operator between the factors so it acts on ψ first, equal to sum aₙ |cₙ|² in an eigenbasis; σ² = ⟨A²⟩ - ⟨A⟩² needs A-hat applied twice, not ⟨A⟩ squared.

Read the complete note

⟨A⟩ = integral ψ* A-hat ψ dx, the operator between the factors so it acts on ψ first, equal to sum aₙ |cₙ|² in an eigenbasis; σ² = ⟨A²⟩ - ⟨A⟩² needs A-hat applied twice, not ⟨A⟩ squared. Both are means over an ensemble of identically prepared systems, never the outcome of one measurement, and sigma is the spread of that ensemble rather than an instrument error.

02 · RepresentationThe subsection's claim

Model, evidence, and boundary

03 · TestPrediction before measurement

When should a model used for Expectation values, variance, and standard deviation be revised?

04 · Evidence & boundaryDecide, then qualify

Does inserting a filter at 45 degrees between crossed polarisers restore transmission, and does applying the same two filters in the opposite order change the result?

Read the complete note

Does inserting a filter at 45 degrees between crossed polarisers restore transmission, and does applying the same two filters in the opposite order change the result? Useful evidence includes transmitted intensity for the crossed pair and with the middle filter at several angles, fitted to cos squared, plus the same filters in reversed order; a statement that Malus's classical law fits the data equally well, so this is an analogy for non-commuting projections rather than evidence for quantum mechanics; and an explicit note that the order-dependence belongs to the operators, not to a filter mechanically disturbing photons.

Interactive diagram for Expectation values, variance, and standard deviation: follow the physical claim through its representation, proposed test, evidence, and model boundary.

modelModel, evidence, and boundary turns the stated idea into a representation that can make a prediction.

Example questions

Try the reasoning before revealing the structure.

Diagram check

When should a model used for Expectation values, variance, and standard deviation be revised?

Quick check

Test the reasoning, not recall

When should a model used for Expectation values, variance, and standard deviation be revised?

Choose an answer to test the model.