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Spherical Bessel Differential Equation

Take the Helmholtz differential equation

del ^2F+k^2F=0
(1)

in spherical coordinates. This is just Laplace's equation in spherical coordinates with an additional term,

(d^2R)/(dr^2)PhiTheta+2/r(dR)/(dr)PhiTheta+1/(r^2sin^2phi)(d^2Theta)/(dtheta^2)PhiR+(cosphi)/(r^2sinphi)(dPhi)/(dphi)ThetaR+1/(r^2)(d^2Phi)/(dphi^2)ThetaR+k^2RPhiTheta=0.
(2)

Multiply through by r^2/RPhiTheta,

(r^2)/R(d^2R)/(dr^2)+(2r)/R(dR)/(dr)+k^2r^2+1/(Thetasin^2phi)(d^2Theta)/(dtheta^2)+(cosphi)/(Phisinphi)(dPhi)/(dphi)+1/Phi(d^2Phi)/(dphi^2)=0.
(3)

This equation is separable in R. Call the separation constant n(n+1),

(r^2)/R(d^2R)/(dr^2)+(2r)/R(dR)/(dr)+k^2r^2=n(n+1).
(4)

Now multiply through by R,

r^2(d^2R)/(dr^2)+2r(dR)/(dr)+[k^2r^2-n(n+1)]R=0.
(5)

This is the spherical Bessel differential equation. It can be transformed by letting x=kr, then

r(dR(r))/(dr)=kr(dR(r))/(kdr)=kr(dR(r))/(d(kr))=x(dR(r))/(dx).
(6)

Similarly,

r^2(d^2R(r))/(dr^2)=x^2(d^2R(r))/(dx^2),
(7)

so the equation becomes

x^2(d^2R)/(dx^2)+2x(dR)/(dx)+[x^2-n(n+1)]R=0.
(8)

Now look for a solution of the form R(r)=Z(x)x^(-1/2), denoting a derivative with respect to x by a prime,

R^'=Z^'x^(-1/2)-1/2Zx^(-3/2)
(9)
R^('')=Z^('')x^(-1/2)-1/2Z^'x^(-3/2)-1/2Z^'x^(-3/2)-1/2(-3/2)Zx^(-5/2)
(10)
=Z^('')x^(-1/2)-Z^'x^(-3/2)+3/4Zx^(-5/2),
(11)

so

x^2(Z^('')x^(-1/2)-Z^'x^(-3/2)+3/4Zx^(-5/2))+2x(Z^'x^(-1/2)-1/2Zx^(-3/2))+[x^2-n(n+1)]Zx^(-1/2)=0
(12)
x^2(Z^('')-Z^'x^(-1)+3/4Zx^(-2))+2x(Z^'-1/2Zx^(-1))+[x^2-n(n+1)]Z=0
(13)
x^2Z^('')+(-x+2x)Z^'+[3/4-1+x^2-n(n+1)]Z=0
(14)
x^2Z^('')+xZ^'+[x^2-(n^2+n+1/4)]Z=0
(15)
x^2Z^('')+xZ^'+[x^2-(n+1/2)^2]Z=0.
(16)

But the solutions to this equation are Bessel functions of half integral order, so the normalized solutions to the original equation are

R(r)=A(J_(n+1/2)(kr))/(sqrt(kr))+B(Y_(n+1/2)(kr))/(sqrt(kr))
(17)

which are known as spherical Bessel functions. The two types of solutions are denoted j_n(x) (spherical Bessel function of the first kind) or n_n(x) (spherical Bessel function of the second kind), and the general solution is written

R(r)=A^'j_n(kr)+B^'n_n(kr),
(18)

where

j_n(z)=sqrt(pi/2)(J_(n+1/2)(z))/(sqrt(z))
(19)
n_n(z)=sqrt(pi/2)(Y_(n+1/2)(z))/(sqrt(z)).
(20)

See also

Spherical Bessel Function, Spherical Bessel Function of the First Kind, Spherical Bessel Function of the Second Kind

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References

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 437, 1972.Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 121, 1997.

Referenced on Wolfram|Alpha

Spherical Bessel Differential Equation

Cite this as:

Weisstein, Eric W. "Spherical Bessel Differential Equation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SphericalBesselDifferentialEquation.html

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