Rigid Rotation in 2d and 3d

After having studied translation and vibration in the particle-in-a-box and harmonic oscillator quantum systems, respectively, it is natural to turn to rotation. The simplest such system is the particle on a ring (also called rigid rotation in 2d) which, after replacing the mass m with the reduced mass μ can be used for two-body problems (like a diatomic molecule).

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 e^imφ

Continuity conditions demands that cyclic boundary conditions hold; that is, that ψ(φ + 2π) = ψ(φ). This restriction leads directly to quantization, and the quantum number m_l can only take integral values. This restriction on m_l requires that both energy and position/displacement be quantized.

Further, after constructing the appropriate operator, we see that we have a new quantized property arises naturally in angular momentum, called L_z because the vector lies perpendicular to the plane of rotation.

Rotation in 3d is very similar but we need to use spherical coordinates...