Commutators and the canonical relation [x-hat, p-hat] = i ℏ

For a Gaussian, a triangular state and a half-cosine lobe, does the computed σ-x σ-p sit at ℏ/2 or above it - and what happens to σ-p for a discontinuous top-hat as the grid is refined? Useful evidence includes psi on a stated box and spacing, σ-x and σ-p from FFT momentum amplitudes, the product against ℏ/2 with only the Gaussian saturating, a spacing-refinement check showing the first three settle while the top-hat's σ-p grows without bound because ⟨p²⟩ diverges for a discontinuous ψ, and the boundary rows named as where the discrete [x-hat, p-hat] stops equalling i ℏ..

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.