University Quantum Mechanics I · Introduction to the Hydrogen Atom · 15.03
The angular solutions: the quantum numbers l and m
Spherical harmonics quoted from tables rather than derived: single-valuedness in phi forces integer m, regularity at the poles forces integer l with |m| at most l, giving L-squared = l(l+1) ℏ-squared and Lz = m ℏ.
Course-map guide · not a complete lesson or simulationScope & orientation
What this subsection covers
Spherical harmonics quoted from tables rather than derived: single-valuedness in phi forces integer m, regularity at the poles forces integer l with |m| at most l, giving L-squared = l(l+1) ℏ-squared and Lz = m ℏ.A strong response uses energy-level diagrams with series arrows 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 measured Balmer wavelengths fit one over lambda equal to R times one-quarter minus one over n-squared for n of three and above, and does the fitted R match the value implied by 13.6 eV?
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Do measured Balmer wavelengths fit one over lambda equal to R times one-quarter minus one over n-squared for n of three and above, and does the fitted R match the value implied by 13.6 eV? Useful evidence includes calibrated wavelengths for at least four Balmer lines from a hydrogen lamp, a fitted Rydberg constant with residuals, propagated angular uncertainty, and the grating calibration that bounds the result..
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.
Spherical harmonics quoted from tables rather than derived: single-valuedness in phi forces integer m, regularity at the poles forces integer l with |m| at most l, giving L-squared = l(l+1) ℏ-squared and Lz = m ℏ. Every central potential shares them.
Model, evidence, and boundary
What must be stated before selecting an equation for The angular solutions: the quantum numbers l and m?
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.
What must be stated before selecting an equation for The angular solutions: the quantum numbers l and m?
Quick check
Test the reasoning, not recall
What must be stated before selecting an equation for The angular solutions: the quantum numbers l and m?
Choose an answer to test the model.