130_notes.dvi

15.4 Particle in a Sphere*

This islike the particle in a boxexcept now the particle is confined to the inside of a sphere of
radiusa. Inside the sphere, the solution is a Bessel function. Outside the sphere, the wavefunction
is zero. The boundary condition is that the wave function go to zeroon the sphere.

jℓ(ka) = 0

There are an infinite number of solutions for eachℓ. We only need to find the zeros of the Bessel

functions. The table below gives the lowest values ofka=

√

2 ma^2 E
̄h^2 which satisfy the boundary
condition.

ℓ n= 1 n= 2 n= 3

0 3.14 6.28 9.42

1 4.49 7.73

2 5.72 9.10

3 6.99 10.42

4 8.18

5 9.32

We can see bothangular and radial excitations.

15.5 Bound States in a Spherical Potential Well*

We now wish to find theenergy eigenstatesfor a spherical potential well of radiusaand potential
−V 0.

Spherical Potential Well (Bound States)

V(r)

r

a

-V 0

E

We must use the Bessel function near the origin.

Rnℓ(r) =Ajℓ(kr)

k=

√

2 μ(E+V 0 )

̄h^2