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In Electrostatics, if we consider a region without charges the electrostatic potential $V$ obeys Laplace's Equation $\nabla^2 V = 0$. We can tackle this with separation of variables. In cartesian coordinates we have $V(x,y,z) = X(x)Y(y)Z(z)$ and so the equation is:

$$\nabla^2 V = 0 \Longleftrightarrow \dfrac{X''(x)}{X(x)}+\dfrac{Y''(y)}{Y(y)}+\dfrac{Z''(z)}{Z(z)} = 0.$$

For this to happen there should be three constants $C_1,C_2,C_3$ such that

$$\begin{cases}X''(x) &= C_1 X(x), \\ Y''(y) &= C_2 Y(y), \\ Z''(z) &= C_3Z(z).\end{cases}$$

Now, another example of a problem in Physics tackled with separation of variables is Schrödinger's equation

$$i\hbar \dfrac{\partial \Psi}{\partial t} = H\Psi,$$

separation of variables yields two equations one of which is $H\psi = E\psi$ for the spatial part and the separation constant has one physical meaning as expected value of energy.

Now, the constants $C_1,C_2,C_3$ appearing when we separate variables on Laplace's equation for electrostatic potential has some physical meaning? If they do, what is it?

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Now, the constants C1,C2,C3 appearing when we separate variables on Laplace's equation for electrostatic potential has some physical meaning? If they do, what is it?

The constants are the related to the square of the spatial (angular) frequency or a spatial growth/decay constant.

For an example of spatial frequency, let

$$X(x) = A \sin (k_xx) + B \cos(k_xx)$$

Then

$$X''(x) = -k^2_x X(x)$$

Or, for an example of a spatial growth/decay constant, let

$$Y(y) = A e^{k_yy} + B e^{-k_yy}$$

Then

$$Y''(y) = k^2_y Y(y)$$

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