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:

(-hbar

^{2}/2I) ∂^{2}/∂φ^{2}ψ = EψThis 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...

## 4 comments:

For the radial part of the wavefunction, we are given two different equations. On the equation sheet from the quiz, we have the Laguerre equation as a function of rho. However, on the handouts in class and notes, it is a function of rho/n. Which is it?

The handouts are correct. Thanks for pointing that out.

When drawing the probability diagram for a particle in a 2D box, (where nx = 2 , ny = 3), do you include the (+) and (-) in the boxes? How would you depict the nodes?

When you're referring to probability, the value can never be negative; thus it can only be zero or positive. For this reason, we often don't even bother writing the +'s and indicate the nodes with a line (just as in the wavefunction plots).

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