A particle of reduced mass is attracted by a thin spherical shell
PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Central potentials and canonical three-dimensional systems · Central potentials and canonical three-dimensional systems
The Structure of Quantum Mechanics, Ch Central potentials as a rotation problem
Problem
A particle of reduced mass μ is attracted by a thin spherical shell, V(r)=−gδ(r−a) with g>0, a three-dimensional version of the delta-function well of the first course.
((a)) Integrate the radial equation for u across r=a to find the condition that joins u′ on the two sides.
((b)) Find the smallest value of the dimensionless strength λ=2μga/ℏ2 at which an l=0 bound state appears.
((c)) Repeat for l=1, give the threshold for general l, and compare the ratio of the l=1 and l=0 thresholds with the spherical well's.