Seperation of variables, infinite cubic well

[student111]

Suppose one is to find the stationary states of a particle in an infinite cubic well. Inside the box the time independent SE is:

$$- \frac{\hbar}{2m} \big( \frac{\partial ^2 \psi}{\partial x ^2 } + \frac{\partial ^2 \psi}{\partial z ^2 } + \frac{\partial ^2 \psi}{\partial z ^2 } \big)= E\psi$$

Using separation of variables: $$\psi = X(x)Y(x)Z(z)$$ we get:

$$YZ\frac{\partial ^2 X}{\partial x^2} + XZ\frac{\partial ^2 Y}{\partial y^2} + XY\frac{\partial ^2 Z}{\partial z^2} = \frac{-2mE}{\hbar ^2} XYZ$$

After this one divides both sides by XYZ. My question is the following:
When dividing by XYZ one must assume that XYZ is different from zero. However the solution we obtain has lots of zeroes. Is this not a problem?

Thanks in advance

 

[Vanadium 50]

And?

 

[Meir Achuz]

You just have to rule out XYZ being zero everhywhere.
At an isolated zero, the limit of the ratio X"/X is well behaved for an eigenfunction.