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The Eigenvalue of the Angular Equation
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

(1/sin(θ))d(sin(θ(dΘ/dθ))/dθ −m²Θ/sin²(θ) = −ΛΘ

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

d((1-z²)(dΘ/dz)) + [Λ −m²/(1-z²)]Θ = 0

Now replace Θ with

Θ(z) = (1−z²)|m|/2(d|m|P/dz|m|)

(where if m=0, d0P/dz0 is just P.)

This leads to the equation

(1−z²)(d²P/dz²) − 2z(dP/dz) + ΛP = 0

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

(k+2)(k+1)ak+2 − [k(k+1) − Λ]ak = 0
which reduces to
ak+2 = ak[k(k+1)−Λ]/[(k+2)(k+1)]

The series solution will terminate only if there is some integer l such that l(l+1)−Λ=0. This means that

the eigenvalue Λ must be equal to l(l+1) for some integer l.

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