Townsend Quantum Mechanics Solutions

Assume kR = (u(r)/r)Ylm(θ,φ). In classical mechanics, a free particle with angular momentum L about the origin and momentum p moves in a straight line with closest distance to the origin b = L/p; b is called the impact parameter relative to the origin. If L = ħ(l(l + 1))½ and p = ħk, then bl(k) = (1/k)(l(l + 1))½.bl(k) is the radius where the angular momentum barrier potential, Ueff(r) = l(l+1)ħ2/(2mr2) is equal to the total energy of the free particle.l(l+1)ħ2/(2m(bl(k))2) = ħ2k2/(2m), bl(k) = (1/k)(l(l + 1))½.In Quantum Mechanics, for r < bl(k), in the classically forbidden region, the eigenfunctions |klm> can only have an approximately exponentially decaying tail.In a finite range potential, if the partial wave with angular momentum quantum number l only has an approximately exponentially decaying tail in the region where the potential is non-zero, then it is minimally affected by the potential and the phase shiftδl is approximately zero.]For a finite range potential U(r) of range R the phase shift δ0 is negative if U(r)r 0, and it is positive if U(r)r 2b1af7f3a8