# Calculate energy of wavefunctions for a particle in infinite spherical well

1. May 6, 2012

### h2obrain

1. The problem statement, all variables and given/known data
Consider a particle in a 2nm sphere with infinite potential energy outside and zero potential energy inside the sphere. Calculate the energy of the following wavefunctions: 1s, 2p, 3d

2. Relevant equations
H(hat) = p(hat)^2/2m(sub zero) + V(r)
V(r) = ∞ when r ≥ 2 nm
V(r) = 0 when r< 2 nm

3. The attempt at a solution
for 1s l=0
spherical Bessel fxn is j(sub zero)(x)=(sin x)/x
E(sub k0) = [(k(pi))^2*h(bar)^2]/[2m(sub 0)r(sub 0)^2]

2. May 7, 2012

### CFede

This problems are all solved by the same logic. First you should solve the free particle inside the sphere, wich will give a solution on functions of the possition, that also depend on l (like the spherical Bessel functions).

Then apply the boundary condition that the wave function vanishes at the surface of the sphere (since you have infinite potential outside it), from this condition you'll find and equation something like j_l(alpha*R)=0. This is an equation for alpha, wich will depend on the energy, so, summing up:

1: Solve the free particle for an arbitrary l.
2: Apply boundary condition for the wave function to in the surface of the sphere.
3: You should get a trascendental equation for E.

Hope this helps. (sorry for the sloppy english)