Hermitian operators, and why their eigenvalues are real

On a discretised line, is a central-difference momentum matrix Hermitian while a forward-difference one is not, and do a discretised Hamiltonian's eigenvalues come out real with orthogonal eigenvectors? Useful evidence includes both momentum matrices with their Hermiticity residual max|M - M-dagger|; eigenvalues from numpy.linalg.eig, not eigh, with imaginary parts reported, because eigh assumes Hermiticity, reads only one triangle and returns real eigenvalues by construction, so it cannot test the claim; a pairwise overlap table for the lowest six Hamiltonian eigenvectors; ⟨x⟩ and σₓ for one of them; and the grid spacing, with discretisation named as the source of any residual.

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.