Many-electron atoms and the exclusion principle

Where do numerically integrated hydrogen radial functions place the most probable radius, and how does it compare with the Bohr radius for each n and l? Useful evidence includes radial solutions, normalisation checks, plots of the radial probability density formed as r-squared times the modulus-squared radial function, most probable and mean radii, grid-convergence tests, and a stated non-relativistic fixed-nucleus limit.

This GioPhysics course map is an adaptable learning sequence, not academic credit, accreditation, or a universal university syllabus. Third-semester content is the least standardised of the introductory sequence: departments place oscillations, waves, optics, and thermal physics differently, and the modern-physics units here are a bounded survey rather than a complete course in relativity, quantum mechanics, atomic, nuclear, or particle physics. 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.