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for a Particle in a Central Field
Consider a particle in a central field with a potential energy function V(r), where r is the distance from the center of the field. For a spherical coordinate system of (r, θ, φ) the method of separation of variables applied to the Schröinger Equationfor the system leads to the following equation
where m is is an integer and Λ is a constant.
This equation may be somewhat simplified by transforming the independent variable to z=cos(θ). This results in the equation
Now replace Θ with
(where if m=0, d0P/dz0 is just P.)
This leads to the equation
The function P(z) is known as a Legendre polynomial.
If P(z) is a polynomial of the form Σakzk where k runs from 0 up, then the coefficients ak must be such that
The series solution will terminate only if there is some integer l such that l(l+1)−Λ=0. This means that
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