Quantum numbers: which condition produces each label

Does a spherical finite well always bind a state the way a one-dimensional well does, and how much deeper must it be before an l = 1 state appears? Useful evidence includes shot solutions u(r) started from u(0) = 0 with the small-r behaviour u ~ r^(l+1) respected, eigenvalues located by node counting and sign changes, grid-spacing and cutoff convergence, binding thresholds for l = 0 and l = 1, and the 1D well for contrast.

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.