- #1
Cogswell
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Homework Statement
[2 Dimensional infinite square well]
Show that you can separate variables such that the solution to the time independent schrodinger equation, ## \hat{H} \psi (x,y) = E \psi (x,y) ## can be written as a product state ## \psi (x,y) = \phi (x) \chi (y) ## where ## \phi (x)## is a function of only the x coordinate and ##\chi(y)## is a function of only the y coordinate.
Homework Equations
[tex]\hat{H} = -\dfrac{\hbar^2}{2m} \left( \dfrac{\partial^2}{\partial x^2} + \dfrac{\partial^2}{\partial y^2} \right) + V(x,y)[/tex]
The Attempt at a Solution
So... putting in ## \psi (x,y) = \phi (x) \chi (y) ## I'll get:
[tex]\hat{H} \phi (x) \chi (y) = -\dfrac{\hbar^2}{2m} \left( \dfrac{\partial^2 \phi (x)}{\partial x^2} \chi (y) + \dfrac{\partial^2 \chi (y)}{\partial y^2} \phi (x) \right) + V(x,y) \phi (x) \chi (y) [/tex]
And then dividing both sides by ## \phi (x) \chi (y) ## I'll get:
[tex]\hat{H} = -\dfrac{\hbar^2}{2m} \left( \dfrac{1}{\phi (x)} \dfrac{\partial^2 \phi (x)}{\partial x^2} + \dfrac{1}{\chi (y)} \dfrac{\partial^2 \chi (y)}{\partial y^2} \right) + V(x,y)[/tex]
And then... I don't really know where to go from here. I don't get what I'm supposed to do next.