“Helmholtz Differential Equation--Spherical Coordinates”的概念、定义、习题解题方法及翻译-数学辞典

单词

Helmholtz Differential Equation--Spherical Coordinates

释义

Helmholtz Differential Equation--Spherical Coordinates

In Spherical Coordinates, the Scale Factors are , , , andthe separation functions are , , , giving a StäckelDeterminant of . The Laplacian is

(1)

To solve the Helmholtz Differential Equation in SphericalCoordinates, attempt Separation of Variables by writing

(2)

Then the Helmholtz Differential Equation becomes

(3)

Now divide by



	(4)

(5)

The solution to the second part of (5) must be sinusoidal, so the differential equation is

(6)

which has solutions which may be defined either as a Complex function with

, ...,

(7)

or as a sum of Real sine and cosine functions with

, ...,

(8)

Plugging (6) back into (7),

(9)

The radial part must be equal to a constant

(10)

(11)

But this is the Euler Differential Equation, so we try a series solution of the form

(12)

Then


	(13)


	(14)

(15)

This must hold true for all Powers of

. For the

term (with

(16)

which is true only if

and all other terms vanish. So

for

. Therefore, thesolution of the

component is given by

(17)

Plugging (17) back into (9),

(18)

(19)

which is the associated Legendre Differential Equation for

and

, ...,

. The generalComplex solution is therefore


	(20)

where

(21)

are the (Complex) Spherical Harmonics. The general Real solution is

(22)

Some of the normalization constants of

can be absorbed by

and

, so this equation may appear in theform


	(23)

where

(24)

(25)

are the Even and Odd (real) Spherical Harmonics. If azimuthal symmetry is present, then

is constant and the solution of the

component is a Legendre Polynomial

. Thegeneral solution is then

(26)

Actually, the equation is separable under the more general condition that is of the form

(27)

References

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 514 and 658, 1953.

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