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University Quantum Mechanics I · Angular Momentum · 12.02

The components in spherical coordinates

Rewriting the Cartesian operators in r, theta and phi gives Lz = −i ℏ d/dphi and makes L² equal to −ℏ² times the angular part of the Laplacian, which is how L² enters the central-potential Hamiltonian.

Course-map guide · not a complete lesson or simulation

Scope & orientation

What this subsection covers

Rewriting the Cartesian operators in r, theta and phi gives Lz = −i ℏ d/dphi and makes L² equal to −ℏ² times the angular part of the Laplacian, which is how L² enters the central-potential Hamiltonian.

A strong response uses cone diagrams for l at fixed lz and states where the model stops being reliable.

Reasoning checklist

Evidence, assumptions and limits

01

Assumptions to state

State the system, observable, approximation, and conditions held fixed before using a model.

02

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?

Read the complete note

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..

03

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.

01 · Physical claimThe components in spherical coordinates

Rewriting the Cartesian operators in r, theta and phi gives Lz = −i ℏ d/dphi and makes L² equal to −ℏ² times the angular part of the Laplacian, which is how L² enters the central-potential Hamiltonian.

Read the complete note

Rewriting the Cartesian operators in r, theta and phi gives Lz = −i ℏ d/dphi and makes L² equal to −ℏ² times the angular part of the Laplacian, which is how L² enters the central-potential Hamiltonian. Both act on angles alone. This form realises orbital L only, and regularity at the poles is an added demand.

02 · RepresentationThe subsection's claim

Model, evidence, and boundary

03 · TestPrediction before measurement

What must be stated before selecting an equation for The components in spherical coordinates?

04 · Evidence & boundaryDecide, then qualify

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?

Read the complete note

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..

Interactive diagram for The components in spherical coordinates: follow the physical claim through its representation, proposed test, evidence, and model boundary.

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.

Diagram check

What must be stated before selecting an equation for The components in spherical coordinates?

Quick check

Test the reasoning, not recall

What must be stated before selecting an equation for The components in spherical coordinates?

Choose an answer to test the model.