What is the final expression for the energy in separation of variables?

[aaaa202]

Suppose you have some partiel DE describing a physical system with 2 degrees of freedom (e.g. the SE). If you try separation of variables you get something like:

Hg(x)h(y) = Eg(x)h(y)

now you can separate this to two equations, but the energy has to go in one of them. Is the final expression for the energy dependent on which one you choose to put it in?

 

[vela]

E typically doesn't go with one or the other. When separation works, what happens is you get
$$\hat{H}[g(x)h(y)] = G(x)h(y) + g(x)H(y).$$ The Hamiltonian acts on each piece separately. Then you can divide both sides by g(x)h(y) to get
$$\frac{G(x)}{g(x)} + \frac{H(y)}{h(y)} = E.$$ The only way this can be satisfied for all x and y is if the two terms on the left are each constants. The energy is the sum of those constants.