For an electron in the infinite spherical well of radius nm treated in Chapter 11
PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Central potentials and canonical three-dimensional systems · Central potentials and canonical three-dimensional systems
Quantum Mechanics: From the Photon to the Hydrogen Atom, Ch The hydrogen atom
Problem
For an electron in the infinite spherical well of radius 1.00 nm treated in Chapter 11, the three lowest levels are 0.376 eV (with l=0), 0.769 eV (with l=1) and 1.27 eV (with l=2). In hydrogen, by contrast, the n=2 level contains both the 2s and the 2p states at −3.40 eV, both with orbital angular momentum L=l(l+1)ℏ. In both problems each l carries 2l+1 values of ml and the electron has two spin states; V(r) enters only the radial equation, so the angular factor Ylml is identical in the two problems for the same l (Chapter 11). The proton is a fixed point charge.
((a)) How many independent states, counting spin, does each of the three lowest well levels hold, in order of increasing energy?
((b)) How many independent states, counting spin, lie at hydrogen's n=2 energy?
((c)) The well's l=1 level and hydrogen's 2p state both carry L=2ℏ. Are their full three-dimensional wave functions therefore identical? Referring to how V(r) enters the separated equations, say in one or two sentences what is shared between them and what is not.