Commutation relations: [Lₓ, L_y] = i ℏ L_z

Do matrices built only from the Lₚₘ matrix elements reproduce [Lₓ, L_y] = i ℏ L_z, 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..

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.