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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, d^{0}P/dz^{0} 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 Σa_{k}z^{k} where k runs from 0 up, then
the coefficients a_{k} must be such that

(k+2)(k+1)a_{k+2} − [k(k+1) − Λ]a_{k} = 0
which reduces to
a_{k+2} = a_{k}[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.