University Quantum Mechanics I · The Infinite Square Well · 6.05
Orthogonality and completeness of the sine basis
A product-to-sum integral gives the integral of ψₘ ψₙ over [0, L] as δₘₙ, and Fourier's theorem makes the set complete for every square-integrable function on [0, L].
Course-map guide · not a complete lesson or simulationScope & orientation
What this subsection covers
A product-to-sum integral gives the integral of ψₘ ψₙ over [0, L] as δₘₙ, and Fourier's theorem makes the set complete for every square-integrable function on [0, L].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
Assumptions to state
State the system, observable, approximation, and conditions held fixed before using a model.
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?
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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.
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.
Model, evidence, and boundary
When should a model used for Orthogonality and completeness of the sine basis 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 Orthogonality and completeness of the sine basis be revised?
Quick check
Test the reasoning, not recall
When should a model used for Orthogonality and completeness of the sine basis be revised?
Choose an answer to test the model.