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Unformatted text preview: , but it can be on one side
or the other, and one expects moves between the two sides. It
crosses the plane, but it is never found there!
Monday, April 1, 2013 A 3D spherical box
A complication, not covered in the textbook, is to have a 3D
spherical box. Now using x, y, z makes the problem more difﬁcult
since the, e.g., x limits depend on the y and z values. Instead we
use spherical coordinates, r, θ, φ. The 3D Laplacian operator
o perating on ψ makes S.E., slightly rearranged, of the form: ∂
1 ∂ 2 ∂ψ
∇ ψ ( ) = 2
r sin θ ∂θ
r sin θ ∂φ
It is natural to guess the solution is of the form ψ(r,θ,φ) =
R(r)Θ(θ)Φ(φ). Since each term differentiates only one of the three
functions, we come up with 3 independent equations, like in the
3D cubical box case. The radial function R needs to go to 0 at the
e dge of the box, r = r0, but not at the center, r=0. The angular
dependences are usually written as "spherical harmonics" Ylm(θ,φ...
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