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Spherical Bessel Differential Equation

The spherical Bessel differential equation arises by separating 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)

Their Wronskian is

j_n(z)n_n^'(z)-j_n^'(z)n_n(z)=1/(z^2)
(21)

for z!=0 (Abramowitz and Stegun 1972, p. 437). The identity also holds for complex number order when the defining Bessel functions use consistent branches.

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 Resource. https://mathworld.wolfram.com/SphericalBesselDifferentialEquation.html

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