Orthogonality of eigenfunctions belonging to different eigenvalues

Does inserting a filter at 45 degrees between crossed polarisers restore transmission, and does applying the same two filters in the opposite order change the result? Useful evidence includes transmitted intensity for the crossed pair and with the middle filter at several angles, fitted to cos squared, plus the same filters in reversed order; a statement that Malus's classical law fits the data equally well, so this is an analogy for non-commuting projections rather than evidence for quantum mechanics; and an explicit note that the order-dependence belongs to the operators, not to a filter mechanically disturbing photons.

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.