University Quantum Mechanics I · Introduction to the Hydrogen Atom · 15.06

Atomic orbitals: the s, p and d shapes

Each (n, l, m) triple names an orbital, written 1s, 2s, 2p or 3d in spectroscopic notation, with n minus l minus one radial nodes and l angular nodes.

Course-map guide · not a complete lesson or simulation

Scope & orientation

What this subsection covers

Each (n, l, m) triple names an orbital, written 1s, 2s, 2p or 3d in spectroscopic notation, with n minus l minus one radial nodes and l angular nodes.

A strong response uses radial probability densities with nodes marked 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 the tabulated radial functions up to n = 3 normalise to one, and where does each radial probability density peak relative to the Bohr radius?

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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..

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.

01 · Physical claimAtomic orbitals: the s, p and d shapes

Each (n, l, m) triple names an orbital, written 1s, 2s, 2p or 3d in spectroscopic notation, with n minus l minus one radial nodes and l angular nodes.

Read the complete note

Each (n, l, m) triple names an orbital, written 1s, 2s, 2p or 3d in spectroscopic notation, with n minus l minus one radial nodes and l angular nodes. The m = 0 members such as p_z are already real; pₓ, p_y and the lobed d shapes are real combinations of the plus-m and minus-m partners. Every such picture is a contour of |ψ|-squared, not an orbit.

02 · RepresentationThe subsection's claim

Model, evidence, and boundary

03 · TestPrediction before measurement

Which response about Atomic orbitals: the s, p and d shapes 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

Interactive diagram for Atomic orbitals: the s, p and d shapes: 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 Atomic orbitals: the s, p and d shapes is most defensible?

Quick check

Test the reasoning, not recall

Which response about Atomic orbitals: the s, p and d shapes is most defensible?

Choose an answer to test the model.