Since the potential is a constant we can define it to be zero. The natural coordinate system to use is polar coordinates which greatly simplifies the Schrödinger Equation:
(-hbar2/2I) ∂2/∂φ2ψ = Eψ
This is identical to the very first problem we solved, expect we have traded translational properties (x, m) with rotational analogs (φ, I), a standard practice seen even in classical physics. This leads to, after normalizing, the following equation:
ψ(φ) = 1/√2π eimφ
Further, after constructing the appropriate operator, we see that we have a new quantized property arises naturally in angular momentum, called Lz because the vector lies perpendicular to the plane of rotation.
Rotation in 3d is very similar but we need to use spherical coordinates...