University Quantum Mechanics I · Angular Momentum · 12.07
The quantum numbers l and m
Requiring ⟨ψ|ψ⟩ ≥ 0 at both ends of the ladder terminates it, forcing L²|l, m⟩ = ℏ² l(l+1)|l, m⟩ and Lz|l, m⟩ = m ℏ |l, m⟩ with m from −l to l in 2l+1 steps.
Course-map guide · not a complete lesson or simulationScope & orientation
What this subsection covers
Requiring ⟨ψ|ψ⟩ ≥ 0 at both ends of the ladder terminates it, forcing L²|l, m⟩ = ℏ² l(l+1)|l, m⟩ and Lz|l, m⟩ = m ℏ |l, m⟩ with m from −l to l in 2l+1 steps.A strong response uses polar plots of |yₗm(θ, φ)|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
Do matrices built only from the Lₚₘ matrix elements reproduce [Lₓ, Ly] = i ℏ Lz, and does L² come out as ℏ² l(l+1) times the identity for each l?
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Do matrices built only from the Lₚₘ matrix elements reproduce [Lₓ, Ly] = i ℏ Lz, and does L² come out as ℏ² l(l+1) times the identity for each l? Useful evidence includes matrices for l = 1, 2 and 3 built from the Lₚₘ elements, commutator residuals near machine ε, L² equal to ℏ² l(l+1) times I, Lₚₗᵤₛ annihilating m = l, and the algebra fixing the spectrum, not which l occurs..
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.
Requiring ⟨ψ|ψ⟩ ≥ 0 at both ends of the ladder terminates it, forcing L²|l, m⟩ = ℏ² l(l+1)|l, m⟩ and Lz|l, m⟩ = m ℏ |l, m⟩ with m from −l to l in 2l+1 steps. The algebra permits half-integer l; single-valued e(i m φ) rules it out for orbital L.
Model, evidence, and boundary
What is the strongest test of a claim about 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 is the strongest test of a claim about The quantum numbers l and m?
Quick check
Test the reasoning, not recall
What is the strongest test of a claim about The quantum numbers l and m?
Choose an answer to test the model.