Schrödinger Equation for a Dirac Bubble Potential
Schrödinger Equation for a Dirac Bubble Potential
The Schrödinger equation has been solved in closed form for about 20 quantum-mechanical problems. This Demonstration describes one such example published some time ago. A particle moves in a potential that is zero everywhere except on a spherical bubble of radius , drawn as a red circle in the contour plots. This result has been applied to model the buckminsterfullerene molecule and also to approximate the interatomic potential in the helium van der Waals dimer .
r
0
C
60
He
2
The relevant Schrödinger equation is given by , in units with , and in bohrs, and in hartrees. For , the equation has separable continuum solutions , where the are spherical harmonics. The radial function has the form (r)=const(kr) for and for . Here and are spherical Bessel functions and the are phase shifts. For each value of , a single bound state will exist, provided that . The bound-state radial function is (r)=-λκ()(iκ)(iκ), where and are the greater and lesser of and , and is a Hankel function. The energy is given by , with determined by the transcendental equation . Both the bound and continuum wavefunctions are continuous at but have discontinuous first derivatives. The produces a deltafunction in the second derivative.
-ψ+δ(r-)ψ=Eψ
1
2
2
∇
λ
r
0
r
0
ℏ=m=1
r
r
0
E
E=/2>0
2
k
ψ(r,θ,ϕ)=(r)(θ,ϕ)
R
l
Y
lm
Y
lm
R
l
j
l
r⩽
r
0
const[(kr)cos-(kr)sin]
j
l
δ
l
y
l
δ
l
r⩾
r
0
j
l
y
l
δ
l
l
λ⩽-(2l+1)
R
l
r
0
R
l
r
0
j
l
r
<
(1)
h
l
r
>
r
>
r
<
r
r
0
(1)
h
l
E=-/2
2
κ
κ
-λκ(iκ)(iκ)=1
r
0
j
l
r
0
(1)
h
l
r
0
r=
r
0
This Demonstration shows plots of the radial functions (r) and a cross section of the density plots of for . The wavefunction is positive in the blue regions and negative in the white regions. Be cautioned that the density plots might take some time to complete.
R
l
Reψ(r,θ,ϕ)
l=0,1,2