University Quantum Mechanics I · Measurement and the Uncertainty Principle · 4.03
Expansion on an eigenbasis and the probability of an eigenvalue
Project with cₙ = integral ψₙ* ψ dx, then P(qₙ) = |cₙ|², sum |cₙ|² = 1, and ⟨Q⟩ = sum qₙ |cₙ|2.
Course-map guide · not a complete lesson or simulationScope & orientation
What this subsection covers
Project with cₙ = integral ψₙ* ψ dx, then P(qₙ) = |cₙ|², sum |cₙ|² = 1, and ⟨Q⟩ = sum qₙ |cₙ|2.A strong response uses phase-space uncertainty ellipses and states where the model stops being reliable.
Reasoning checklist
Evidence, assumptions and limits
Assumptions to state
State the system, observable, approximation, and conditions held fixed before using a model.
Evidence to collect
For a Gaussian, a triangular state and a half-cosine lobe, does the computed σ-x σ-p sit at ℏ/2 or above it - and what happens to σ-p for a discontinuous top-hat as the grid is refined?
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For a Gaussian, a triangular state and a half-cosine lobe, does the computed σ-x σ-p sit at ℏ/2 or above it - and what happens to σ-p for a discontinuous top-hat as the grid is refined? Useful evidence includes psi on a stated box and spacing, σ-x and σ-p from FFT momentum amplitudes, the product against ℏ/2 with only the Gaussian saturating, a spacing-refinement check showing the first three settle while the top-hat's σ-p grows without bound because ⟨p²⟩ diverges for a discontinuous ψ, and the boundary rows named as where the discrete [x-hat, p-hat] stops equalling i ℏ..
Limits to state
This GioPhysics course map is an adaptable learning sequence, not academic credit, accreditation, or a universal university syllabus.
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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.
Project with cₙ = integral ψₙ* ψ dx, then P(qₙ) = |cₙ|², sum |cₙ|² = 1, and ⟨Q⟩ = sum qₙ |cₙ|2. Completeness is assumed rather than proved; a degenerate eigenvalue needs its probabilities summed, and a continuous spectrum gives a density.
Model, evidence, and boundary
When should a model used for Expansion on an eigenbasis and the probability of an eigenvalue be revised?
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.
When should a model used for Expansion on an eigenbasis and the probability of an eigenvalue be revised?
Quick check
Test the reasoning, not recall
When should a model used for Expansion on an eigenbasis and the probability of an eigenvalue be revised?
Choose an answer to test the model.