Where does the 1/X(x) term come from in 3D Schrödinger equation separation?

[Master J]

The Schrödinger equation in 3D (time independent).

Letting Phi = X(x).Y(y).Z(z), and solving as a PDE...

The equation looks pretty much the same except there is a separate Hamiltonian for each of the Cartesian coordinates x y z. However, the 1/X(x) term etc. really confuses me, I don't know where it comes from. Could someone perhaps explain??

ie. H_x = [-(h^2)/2m].[1/X(x)].[(d^2)X(x)/d(X(x))^2] + V(x)
^^^^

where h is representing h-bar, and d the partial derivative.

Cheers guys!

 

[Feldoh]

It occurs because you divide through by 1/XYZ to isolate the equations.

But note that using a separation in Cartesian coordinates is not always a viable solution, and will only work for some potentials.

 

[Master J]

Can you perhaps outline the derivation from the start? It's just clearing it up for me...

 

[Feldoh]

It goes something like assume the potential is an infinite square potential

$$V(x,y,z) = \left(\begin{array}{cc}0 if x,y,z < a \\ \infty else$$

We can assume a separable solution $$\Psi (x,y,z) = X(x)Y(y)Z(z)$$

$$\frac{-\hbar^2}{2m} [Y(y)Z(z) \frac{d^2 X}{dx^2}+X(x)Z(z) \frac{d^2 Y}{dy^2}+X(x)Y(y) \frac{d^2 Z}{dz^2}] + V(x,y,z)XYZ = E(XYZ)$$

Then just divide everything by 1/XYZ.