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← University Quantum Mechanics I

University Quantum Mechanics I · The Infinite Square Well · 6.09

Expectation values that move, and ones that do not

For a two-term state ⟨x⟩(t) oscillates at angular frequency ω = (E₂ - E₁)/ℏ, period 2 π ℏ/(E₂ - E₁), because the cross term survives, while ⟨H⟩ and each |cₙ|² stay constant.

Course-map guide · not a complete lesson or simulation

Scope & orientation

What this subsection covers

For a two-term state ⟨x⟩(t) oscillates at angular frequency ω = (E₂ - E₁)/ℏ, period 2 π ℏ/(E₂ - E₁), because the cross term survives, while ⟨H⟩ and each |cₙ|² stay constant.

A strong response uses energy ladder with levels spaced as n² 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

Do the resonances of a string clamped at both ends fall at fₙ = n f₁ with n - 1 interior nodes, and where does the analogy with Eₙ break?

Read the complete note

Do the resonances of a string clamped at both ends fall at fₙ = n f₁ with n - 1 interior nodes, and where does the analogy with Eₙ break? Useful evidence includes resonances for n = 1 to 6 with node positions measured, fₙ fitted against n, the fitted f₁ against v/2L, and fₙ proportional to n set beside Eₙ proportional to n² with the differing dispersion named.

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 that move, and ones that do not

For a two-term state ⟨x⟩(t) oscillates at angular frequency ω = (E₂ - E₁)/ℏ, period 2 π ℏ/(E₂ - E₁), because the cross term survives, while ⟨H⟩ and each |cₙ|² stay constant.

Read the complete note

For a two-term state ⟨x⟩(t) oscillates at angular frequency ω = (E₂ - E₁)/ℏ, period 2 π ℏ/(E₂ - E₁), because the cross term survives, while ⟨H⟩ and each |cₙ|² stay constant. All of these are averages over identically prepared systems, not the path of one particle, and the σₓ σₚ ≥ ℏ/2 floor computed here follows from x and p failing to commute, not from a probe disturbing the particle.

02 · RepresentationThe subsection's claim

Model, evidence, and boundary

03 · TestPrediction before measurement

When should a model used for Expectation values that move, and ones that do not be revised?

04 · Evidence & boundaryDecide, then qualify

Do the resonances of a string clamped at both ends fall at fₙ = n f₁ with n - 1 interior nodes, and where does the analogy with Eₙ break?

Read the complete note

Do the resonances of a string clamped at both ends fall at fₙ = n f₁ with n - 1 interior nodes, and where does the analogy with Eₙ break? Useful evidence includes resonances for n = 1 to 6 with node positions measured, fₙ fitted against n, the fitted f₁ against v/2L, and fₙ proportional to n set beside Eₙ proportional to n² with the differing dispersion named.

Interactive diagram for Expectation values that move, and ones that do not: 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 that move, and ones that do not be revised?

Quick check

Test the reasoning, not recall

When should a model used for Expectation values that move, and ones that do not be revised?

Choose an answer to test the model.