University Quantum Mechanics I · Angular Momentum · 12.04
L² and Lz: a compatible pair
L² = Lₓ² + Ly² + Lz² commutes with every component, shown by expanding the double commutator, so L² with one component - by convention Lz - is the largest commuting set the algebra allows.
Course-map guide · not a complete lesson or simulationScope & orientation
What this subsection covers
L² = Lₓ² + Ly² + Lz² commutes with every component, shown by expanding the double commutator, so L² with one component - by convention Lz - is the largest commuting set the algebra allows.A strong response uses matrix arrays of lₓ, ly and lz 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
Viewed transverse to B, does the cadmium 643.8 nm line split into three components spaced by μB B / h, as 2l+1 sublevels with δ-m of 0 and plus or minus 1 require?
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Viewed transverse to B, does the cadmium 643.8 nm line split into three components spaced by μB B / h, as 2l+1 sublevels with δ-m of 0 and plus or minus 1 require? Useful evidence includes ring diameters against magnet current, splitting in wavenumber from the etalon free spectral range, a fitted μB with field-calibration uncertainty, the S = 0 assignment as the reason spin plays no part, and the missing central component when the line is instead viewed along B..
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.
L² = Lₓ² + Ly² + Lz² commutes with every component, shown by expanding the double commutator, so L² with one component - by convention Lz - is the largest commuting set the algebra allows. The z axis is a label, not a direction the system prefers.
Model, evidence, and boundary
When should a model used for L² and Lz: a compatible pair 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 L² and Lz: a compatible pair be revised?
Quick check
Test the reasoning, not recall
When should a model used for L² and Lz: a compatible pair be revised?
Choose an answer to test the model.