In Cylindrical Coordinates, the Scale Factors are
,
,
and the separation functions are
,
,
, so the Stäckel Determinant is 1. Attempt Separation of Variables by writing
![\begin{displaymath}
F(r, \theta, z) = R(r)\Theta(\theta)Z(z),
\end{displaymath}](h_936.gif) |
(1) |
then the Helmholtz Differential Equation becomes
![\begin{displaymath}
{d^2R\over dr^2}\Theta Z + {1\over r}{dR\over dr}\Theta Z + ...
...ver d\theta^2}RZ + {d^2Z\over dz^2}R\Theta +k^2 R\Theta Z = 0.
\end{displaymath}](h_937.gif) |
(2) |
Now divide by
,
![\begin{displaymath}
\left({{r^2\over R}{d^2R\over dr^2}+ {r\over R}{dR\over dr}}...
...ta^2}{1\over \Theta }+ {d^2Z\over dz^2}{r^2\over Z} + k^2 = 0,
\end{displaymath}](h_939.gif) |
(3) |
so the equation has been separated. Since the solution must be periodic in
from the definition of the circular
cylindrical coordinate system, the solution to the second part of (3) must have a Negative separation constant
![\begin{displaymath}
{d^2\Theta\over d\theta^2}{1\over\Theta}= -(k^2+m^2),
\end{displaymath}](h_941.gif) |
(4) |
which has a solution
![\begin{displaymath}
\Theta(\theta) = C_me^{-i\sqrt{k^2+m^2}\,\theta}+D_me^{i\sqrt{k^2+m^2}\,\theta}.
\end{displaymath}](h_942.gif) |
(5) |
Plugging (5) back into (3) gives
![\begin{displaymath}
{r^2\over R}{d^2R\over dr^2}+ {r\over R}{dR\over dr}-m^2 + {d^2Z\over dz^2}{r^2\over Z}= 0
\end{displaymath}](h_943.gif) |
(6) |
![\begin{displaymath}
{1\over R}{d^2R\over dr^2}+ {1\over rR}{dR\over dr}-{m^2\over r^2} + {d^2Z\over dz^2}{1\over Z}= 0.
\end{displaymath}](h_944.gif) |
(7) |
The solution to the second part of (7) must not be sinusoidal at
for a physical solution, so the
differential equation has a Positive separation constant
![\begin{displaymath}
{d^2Z\over dz^2}{1\over Z}= n^2,
\end{displaymath}](h_946.gif) |
(8) |
and the solution is
![\begin{displaymath}
Z(z) = E_ne^{-nz}+F_ne^{nz}.
\end{displaymath}](h_947.gif) |
(9) |
Plugging (9) back into (7) and multiplying through by
yields
![\begin{displaymath}
{d^2R\over dr^2}+ {1\over r}{dR\over dr}+ \left({n^2 - {m^2\over r^2}}\right)R = 0
\end{displaymath}](h_949.gif) |
(10) |
![\begin{displaymath}
{1\over n^2}{d^2R\over dr^2}+ {1\over (nr)}{1\over n}{dR\over dr} + \left[{1 - {m^2\over (nr)^2}}\right]R = 0
\end{displaymath}](h_950.gif) |
(11) |
![\begin{displaymath}
{d^2R\over d(nr)^2}+ {1\over (nr)}{dR\over d(nr)}+\left[{1-{m^2\over(nr)^2}}\right]R = 0.
\end{displaymath}](h_951.gif) |
(12) |
This is the Bessel Differential Equation, which has a solution
![\begin{displaymath}
R(r) = A_{mn}J_m(nr)+B_{mn}Y_m(nr),
\end{displaymath}](h_952.gif) |
(13) |
where
and
are Bessel Functions of the First and
Second Kinds, respectively. The general solution is therefore
|
|
|
(14) |
Actually, the Helmholtz Differential Equation is separable for general
of the form
![\begin{displaymath}
k^2(r,\theta,z)=f(r)+{g(\theta)\over r^2}+h(z)+k'^2.
\end{displaymath}](h_957.gif) |
(15) |
See also Cylindrical Coordinates, Helmholtz Differential Equation
References
Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York:
McGraw-Hill, pp. 514 and 656-657, 1953.
© 1996-9 Eric W. Weisstein
1999-05-25