Thermodynamics, Statistical Physics, and Quantum Mechanics

QUANTUM MECHANICS 323

5.71 Attractive Delta Function in 3D (Princeton)

a) The amplitude of thewavefunction is continuous at the point of

the deltafunction. For the derivative wefirst note that the eigenfunctions

are written interms of aradialfunction andangular functions:

Since thedeltafunction isonly for theradial variable only thefunction

has a discontinuousslope. From the radialpart of thekineticenergy

operator we integratefrom to

Thisformula isused tomatch theslopes at

b) In order to findboundstates, we assume that theparticle has an energy

given by where needs to bedetermined by aneigenvalue

equation. Theeigenfunctions arecombinations of exp In order to

be zero at and to vanish atinfinity, wemust choose theform

We match thevalues of at Wematch the derivative, using the
results of part (a):

We eliminatethe constants A and B and obtain theeigenvalue equation
for whichwe proceed tosimplify: