Bras, the dual space, and the inner product

Do two- and three-sheet transmissions follow from |⟨a|b⟩|² projection probabilities, and does the middle sheet restore light because a single projector is not the identity? Useful evidence includes intensity against angle for two- and three-sheet stacks with cos² fits; each sheet's own transmission efficiency measured on its own first and divided out, because real sheets pass only about eighty percent of the aligned component and the completeness test I(θ) + I(θ+90) = Iᵢₙ fails by construction without that normalisation; stray light and detector nonlinearity quantified alongside; and Malus's classical law recorded as fitting the same data equally well, so the experiment illustrates the projector algebra rather than proving quantum mechanics.

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.