Solve the 2-D time-independent Schrödinger equation with V (x,y) = 0:
-ћ2/2m ( ∂2Ψ(x,y)/∂x2 + ∂2Ψ(x,y)/∂y2 ) = EΨ(x,y)
The Attempt at a Solution
I started by getting -ћ2/2m to one side:
( ∂2Ψ(x,y)/∂x2 + ∂2Ψ(x,y)/∂y2 ) = -2mE/ћ2 Ψ(x,y)
Here is my problem: I'm not sure what to do with everything on the right side of the = sign.
What follows is my misguided attempt at a solution:
I assumed a solution of the form
Ψ(x,y) = X(x)Y(y)
Then I substituted this in for Ψ(x,y), giving:
Y(y) d2X(x)/dx2 + X(x) d2Y(y)/dy2 = -2mE/ћ2 X(x)Y(y)
Then I divided both sides by X(x)Y(y):
1/X(x) d2X(x)/dx2 + 1/Y(y) d2Y(y)/dy2 = -2mE/ћ2
And here is where I hit the road block. If the two 2nd order ODEs are equal to a constant, then it is pretty obvious that since they add together to get -2mE/ћ2
and are equal to eachother, then the constant they are equal to has to be -mE/ћ2. I can't really see how to solve those ODEs. I mean, I can't just use the auxilery equation, because that would be m2 = 0, which would give me two solutions of zero. Obviously I am missing something pretty important here.
Any insight would be much appreciated. Please keep in mind that this is very new to me, and any explanations will have to be pretty elementary if I am to understand.