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University Quantum Mechanics I · Introduction to the Hydrogen Atom · 15.02

Separating the radial and angular equations

Writing ψ = R(r) Y(θ, φ) splits the eigenvalue equation into a radial and an angular problem joined by the separation constant l(l+1), with a second constant m-squared appearing when Y is split further.

Course-map guide · not a complete lesson or simulation

Scope & orientation

What this subsection covers

Writing ψ = R(r) Y(θ, φ) splits the eigenvalue equation into a radial and an angular problem joined by the separation constant l(l+1), with a second constant m-squared appearing when Y is split further.

A strong response uses radial probability densities with nodes marked 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

Do the tabulated radial functions up to n = 3 normalise to one, and where does each radial probability density peak relative to the Bohr radius?

Read the complete note

Do the tabulated radial functions up to n = 3 normalise to one, and where does each radial probability density peak relative to the Bohr radius? Useful evidence includes normalisation integrals, radial probability curves for 1s, 2s and 2p, most probable and mean radii in units of a-nought, node counts checked against n minus l minus one, and the integration cutoff stated..

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 claimSeparating the radial and angular equations

Writing ψ = R(r) Y(θ, φ) splits the eigenvalue equation into a radial and an angular problem joined by the separation constant l(l+1), with a second constant m-squared appearing when Y is split further.

Read the complete note

Writing ψ = R(r) Y(θ, φ) splits the eigenvalue equation into a radial and an angular problem joined by the separation constant l(l+1), with a second constant m-squared appearing when Y is split further. Separation needs only that V depend on r alone, so by itself it fixes no energy.

02 · RepresentationThe subsection's claim

Model, evidence, and boundary

03 · TestPrediction before measurement

Which response about Separating the radial and angular equations is most defensible?

04 · Evidence & boundaryDecide, then qualify

Do the tabulated radial functions up to n = 3 normalise to one, and where does each radial probability density peak relative to the Bohr radius?

Read the complete note

Do the tabulated radial functions up to n = 3 normalise to one, and where does each radial probability density peak relative to the Bohr radius? Useful evidence includes normalisation integrals, radial probability curves for 1s, 2s and 2p, most probable and mean radii in units of a-nought, node counts checked against n minus l minus one, and the integration cutoff stated..

Interactive diagram for Separating the radial and angular equations: 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

Which response about Separating the radial and angular equations is most defensible?

Quick check

Test the reasoning, not recall

Which response about Separating the radial and angular equations is most defensible?

Choose an answer to test the model.