Spherical Harmonic Wave Function =? 3D Wave Function

  • #1

Homework Statement


Prove that the spherical harmonic wave function [tex] \frac{1}{r}e^{i(kr-{\omega}t)} [/tex] is a solution of the three-dimensional wave equation, where [tex] r = (x^2+y^2+z^2)^{\frac{1}{2}} [/tex]. The proof is easier if spherical coordinates are used.

Homework Equations



Wave function: [tex] \frac{{\partial}^2U}{\partial x^2} + \frac{{\partial}^2U}{\partial y^2} + \frac{{\partial}^2U}{\partial z^2} = \frac{1}{u^2} \frac{{\partial}^2U}{\partial t^2}[/tex]

The Attempt at a Solution



I really just don't even know where to start. Do I first convert the x,y,z into polar coordinates? or do I just substitue what's above in for r? But then what's up with imaginary part?
 
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Answers and Replies

  • #2
vela
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You wrote the wave equation using Cartesian coordinates. More generally, you can write it as[tex]\nabla^2 U = \frac{1}{u^2}\frac{\partial^2 U}{\partial t^2}[/tex]
In your textbook, you can probably find how to write the Laplacian [itex]\nabla^2[/itex] using spherical coordinates. (Or just Google it.)
 

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