Why Can the Hamiltonian Be Split for a Stationary State?

[Niles]

Homework Statement

Hi all.

I have a Hamiltonian given by:

$$H = H_x + H_y = -\frac{\hbar^2}{2m}(d^2/dx^2 + d^2/dy^2).$$

Now I have a stationary state on the form $\psi(x,y)=f(x)g(y)$. According to my teacher, then the Hamiltonian can be split up, i.e. we have the two equations:

$$H_x f(x) = E_xf(x) \qquad \text{and}\qquad H_y g(y)=E_yg(y).$$

I can't see why this must be true. Inserting in the time-independent Schrödinger-equation doesn't give me these expressions. What am I missing here?

Thanks in advance.


Niles

 

[xboy]

If you put it in the time-indep Schrödinger equation, and then if you divide both sides by f(x)g(x), and then if you take one of the two terms on the left hand side to the right hand side, you get : a LHS that is a function of x only, and RHS that's function of y only. This can only be true if both sides are equal to some constant. Try to take it from there.

 

[Niles]

Ahh, yes. I see.

Thanks for that.