for Fireant, etc (math people): Need help with a PDE

[Cartesiantheater]

I need to change the two dimensional time-independent Schrödinger Equation from Cartesian coordinates to Polar Coordinates. Then, if any of you know how to do it, I need help solving it using the separation of variables technique (the Partial Differential Equation seperatio of variables technique, not the Ordinary Differential Equation seperation of variables technique).

Here is the equation:


(∂^2Ψ(x,y)/∂x^2) + (∂^2Ψ(x,y)/∂y^2) = -k^2 Ψ(x,y)

where k^2 is a constant, dependent upon energy and mass, and Ψ(x,y) is a function of x and y.

Now, here is what I tried to do. If you know how, tell me if and where I made mistakes.


First I want to write this in ∂/∂x (∂/∂x (Ψ(x,y))) form, if it is allowed:



∂/∂x (∂/∂x (Ψ(x,y))) + ∂/∂y (∂/∂y (Ψ(x,y))) = -k^2Ψ(x,y)


Next, I want to use the relationship between (x, y) and (r, θ):


x = r cos θ

y = r sin θ

Then, I want to find ∂x/∂r and ∂y/∂θ



∂x/∂r = cos θ

∂y/∂θ = r cos θ

Then divide by ∂x and ∂y


1/∂r = cos θ /∂x

1/∂θ = r cos θ/∂y

Then I want to multiply both sides by ∂ and isolate ∂/∂x and ∂/∂y:


(1/ cos θ) ∂/∂r = ∂/∂x

(1/ r cos θ) ∂/∂θ = ∂/∂y


Next, I want to replace every ∂/∂x and ∂/∂y with their equivalent in polar coordinates according to the above (assuming all of that is right). So plugging in for ∂/∂x and ∂/∂y:


(1/cos θ) ∂/∂r ( (1/cos θ) ∂/∂r (Ψ(r,θ)) ) + (1/r)(1/ cos θ) ∂/∂θ ( (1/r)(1/ cos θ) ∂/∂θ (Ψ(r,θ)) ) = -k^2Ψ(r,θ)

Then returning to the normal notation for second partial derivatives and factoring out the cosine functions:


(1/cos^2 θ) (∂^2/∂r^2 (Ψ(r,θ)) ) + (1/r^2) (1/ cos^2 θ) ( ∂^2/∂θ^2 (Ψ(r,θ)) ) = -k^2Ψ(r,θ)




So, a) have I successfully changed this to polar coordinates?

and b)

does anyone know how to solve this using separation of variables?