Normalisation, the ground state, and zero-point energy

Do the resonances of a string clamped at both ends fall at fₙ = n f₁ with n - 1 interior nodes, and where does the analogy with Eₙ break? Useful evidence includes resonances for n = 1 to 6 with node positions measured, fₙ fitted against n, the fitted f₁ against v/2L, and fₙ proportional to n set beside Eₙ proportional to n² with the differing dispersion named.

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.