The 1/2 is for the moment of inertia of a solid disk about its center perpendicular to the plane of the disk, not angular momentum. I used Wikipedia because it's a known result, and I didn't want to do this:
\begin{equation} I = \int dm \ r^2 = \int \left(\frac{M}{\pi R^2} r d\theta dr\right)...
You calculate a change in angular momentum in part 3 after assuming the angular momentum is constant in part 2.
The angular momentum is I * ω, and since the moment of inertia I is constant, the change in angular momentum is I * Δω, where
(...
Your expression for Ψ(x,t) is incorrect. The wavefunction Ψ(x,t) is a sum over the stationary states ψ_1(x) and ψ_2(x), with both states multiplied by their respective exponential time factors, exp(- i E_n t / hbar), where E_n is the energy of the n-th stationary state.
I haven't watched the video, but if you perform a http://mathworld.wolfram.com/RotationMatrix.html" by an angle \theta about the z axis on the vector {\bf r} = \left(x,y,z\right) , you get r' = r + \Delta r = \left(\cos \theta x - \sin \theta y, \sin \theta x + \cos \theta y ,z\right) . For...