University Quantum Mechanics I · Operators and Observables · 3.04
Eigenvalue equations: solving A-hat ψ = a ψ
Act on a trial function and ask whether it returns multiplied: exp(i k x) is an eigenfunction of p-hat with eigenvalue ℏ k, and sin(n π x / a), handed over here as a function to test rather than as a system already solved, is one of H-hat inside a box.
Course-map guide · not a complete lesson or simulationScope & orientation
What this subsection covers
Act on a trial function and ask whether it returns multiplied: exp(i k x) is an eigenfunction of p-hat with eigenvalue ℏ k, and sin(n π x / a), handed over here as a function to test rather than as a system already solved, is one of H-hat inside a box.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
Assumptions to state
State the system, observable, approximation, and conditions held fixed before using a model.
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?
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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.
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.
Model, evidence, and boundary
When should a model used for Eigenvalue equations: solving A-hat ψ = a ψ 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 Eigenvalue equations: solving A-hat ψ = a ψ be revised?
Quick check
Test the reasoning, not recall
When should a model used for Eigenvalue equations: solving A-hat ψ = a ψ be revised?
Choose an answer to test the model.