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SphericalY

The Spherical Harmonics function

 

Calling Sequence

Parameters

Description

Examples

References

Calling Sequence

SphericalY(λ, μ, θ, φ)

Parameters

λ

-

algebraic expression

μ

-

algebraic expression

θ

-

algebraic expression

φ

-

algebraic expression

Description

  

SphericalY(λ, μ, θ, φ) represents spherical harmonics, that is, the angular part of the solution to Laplace's equation in spherical coordinates (r,θ,φ).

Diff(r^2*Diff(f(r,theta,phi),r),r) + 1/sin(theta)*Diff(sin(theta)*Diff(f(r,theta,phi),theta),theta) + 1/sin(theta)^2*Diff(f(r,theta,phi),phi,phi) = 0;

rr2rfr,θ,φ+θsinθθfr,θ,φsinθ+2φ2fr,θ,φsinθ2=0

(1)
  

The SphericalY functions are particularly relevant in quantum mechanics, where they are eigenfunctions of observable operators associated with angular momentum - see Abramowitz and Stegun, Chapter VI. SphericalY is normalized such that

Int(Int(abs(SphericalY(lambda,lambda,theta,phi))^2*sin(theta),theta=0..Pi),phi=0..2*Pi) = 1;

02π0πSphericalYλ,λ,θ,φ2sinθⅆθⅆφ=1

(2)
  

so that when written in terms of the associated LegendreP function of the first kind, SphericalY is given by

FunctionAdvisor( definition, SphericalY );

SphericalYλ,μ,θ,φ=−1μ2λ+1πλμ!ⅇIφμLegendrePλ,μ,cosθ2λ+μ!,¬λ+μ::?¬λμ::?

(3)
  

Attention should be paid to the normalization conventions adopted. The requirement that the double integral mentioned is equal to one does not fix a phase, which can then be chosen in different ways; following the definitions given by references 2 and 3 (at the bottom), thus, in Maple the right-hand side of the definition above includes the multiplicative factor −1μ. In second place, the Maple choice for the branch cuts of LegendrePλ,μ,z follow conventions which, for λ and μ not integers and outside a unit circle around z=0, are slightly different than those presented for instance in the first reference below. Finally, noting that SphericalY is more frequently used with λ and μ integers, λ positive and μλ, in this case the three square roots entering the definition above,

((2*lambda+1)/Pi)^(1/2)*(lambda-mu)!^(1/2)/(lambda+mu)!^(1/2);

2λ+1πλμ!λ+μ!

(4)
  

can be combined,

combine((4)) assuming posint;

2λ+1λμ!πλ+μ!

(5)
  

resulting into a form of the definition usually presented in textbooks - this combination of the radicals, however, is not valid for arbitrary complex values of λ or μ.

  

The SphericalY functions constitute a complete set of orthonormal functions satisfying

Int(Int(SphericalY(lambda,mu,theta,phi)*conjugate(SphericalY(rho,nu,theta,phi))*sin(theta),theta=0..Pi),phi=0..2*Pi) = delta[lambda,rho]*delta[mu,nu];

02π0πSphericalYλ,μ,θ,φSphericalYρ,ν,θ,φ&conjugate0;sinθⅆθⅆφ=δλ,ρδμ,ν

(6)
  

where in the right-hand side we have Kronecker deltas. Due to the rich structure of these functions, including periodicity with respect to both θ and φ and reflection properties regarding each of its four arguments, the number of identities they satisfy is rather large. Some important ones are

FunctionAdvisor( identities, SphericalY );

SphericalYλ,μ,θ,φ=SphericalYλ,μ,θ,φ,SphericalYλ,μ,θ,φ=SphericalYλ,μ,θ,φⅇ2Iμφ,SphericalYλ,μ,θ,φ=SphericalY1λ,μ,θ,φ2λ+1Γλ+μΓλμ+112λΓλμΓμ+λ+1,λ+μ::¬?λμ::¬?μ+λ+1::¬?λμ+1::¬?,SphericalYλ,μ,θ,φ=SphericalYλ,μ,θ,φⅇ2Iμφ,λ::?μ::?μλλμ,SphericalYλ,μ,θ,φ=−1μSphericalYλ,μ,θ,φ&conjugate0;,λ::?μ::?μλλμ,SphericalYλ,μ,θ,φ=SphericalYλ,μ,2nπ+θ,φ,n::?,SphericalYλ,μ,θ,φ=SphericalYλ,μ,θ,φ+2πnμ,n::?μ0

(7)

Examples

Expressing SphericalY in terms of LegendreP

convertSphericalYλ,μ,θ,φ,LegendreP

−1μ2λ+1πλμ!ⅇIφμLegendrePλ,μ,cosθ2λ+μ!

(8)

In the typical case where λ is a positive integer, μ is an integer and μλ the square roots are automatically combined resulting in the form frequently found in textbooks

convertSphericalYλ,μ,θ,φ,LegendrePassumingλ::posint,μ::integer,μλ

−1μ2λ+1λμ!πλ+μ!ⅇIφμLegendrePλ,μ,cosθ2

(9)

Special values

FunctionAdvisorspecial_values,SphericalY

SphericalYλ&comma;μ&comma;θ&comma;φ=0&comma;2λ+1=0&comma;SphericalYλ&comma;μ&comma;θ&comma;φ=0&comma;μ::?θπ::even&comma;SphericalYλ&comma;μ&comma;θ&comma;φ=0&comma;μ<0θπ::even&comma;SphericalYλ&comma;μ&comma;θ&comma;φ=0&comma;λ::?μ::?λ<μ&comma;SphericalYλ&comma;μ&comma;θ&comma;φ=0&comma;λ::?θπ::?μ::?&comma;SphericalYλ&comma;μ&comma;θ&comma;φ=−1λ2θ2ππθπ2λ+1π2&comma;λ::?θπ::?μ=0

(10)

Hypergeometric representation

FunctionAdvisorspecialize&comma;SphericalY&comma;hypergeom

SphericalYλ&comma;μ&comma;θ&comma;φ=−1μ2λ+1πλμ!&ExponentialE;Iφμcosθ+1μ2hypergeomλ&comma;λ+1&comma;1μ&comma;12cosθ22λ+μ!cosθ1μ2Γ1μ&comma;¬λ+μ::?¬λμ::?¬1μ::?

(11)

References

  

Abramowitz, M., and Stegun, I., eds. Handbook of Mathematical Functions. New York: Dover Publications.

  

Arfken, G., and Weber, H.J. Mathematical Methods for Physicists. 3rd ed. Academic Press, 1985.

  

Cohen-Tannoudji, C.; Diu, B.; and Laloe, F. Quantum Mechanics. Paris: Hermann, 1977. Vol. 1, Complement A-VI.

See Also

FunctionAdvisor

hypergeom

JacobiP

LegendreP