How is the Schrödinger Equation Derived for a Particle in a Ring?

[nokure]

Homework Statement

Write down the Schrödinger equation for a particle inside a circle of radius R

Homework Equations

- Gradientψ(r,θ)=Eψ(r,θ)

The Attempt at a Solution

Take the gradient in spherical form and set r = R; θ=90;∅ as 0

Then you are left with the first two terms of the gradient where sin(90) = 1 and r = R.

I thought I was approaching this correctly but the solutions show this

(d/dr)^2 + 1/r (d/dr) + 1/r^2 (d/dθ)^2

I am not sure how are why you should get three three when the way I am thinking of it "phi" term will be zero.

Thanks for any help in advance!

Anand

 

[dextercioby]

1. Do you know how the SE looks like in cartesian coordinates in a plane ? (2 dimensions)
2. Can you use the symmetry of the problem to simplify the equation at 1. ?