The number operator and stepping the ladder

Do a finite-difference diagonalisation of the quadratic well and a truncated ladder-operator matrix return the same Eₙ, and where does each route fail first? Useful evidence includes grid eigenvalues against (n + 1/2) ℏ ω, the same spectrum from truncated a and a-dagger matrices, node count and parity per eigenvector, convergence in spacing and box size, and the top-of-basis truncation error..

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.