Orthogonality, normalisation, and ⟨m|n⟩ = δₘₙ

How fast does the partial sum of |n⟩⟨n| over infinite-well eigenkets approach the identity for a chosen state, and where does the missing norm show up in position space? Useful evidence includes coefficients cₙ = ⟨n|ψ⟩ by numerical overlap, the running sum of |cₙ|² against N, truncated reconstructions drawn over ψ(x), the residual norm 1 minus that sum, and three test functions compared - smooth and vanishing at the walls, kinked, and deliberately not vanishing at the walls - with coefficients falling as 1/n³, 1/n² and 1/n and Gibbs overshoot appearing only at the genuine jump, never at the kink, where convergence is merely slower.

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.