QM Rigid Rotor Intro, Associated Legendre Polynomials

Students should be able to:

1. (FirstNine)

   Write out the first 9 terms in the sum: $$\sum_{\ell=0}^{\infty} \sum_{m=-\ell}^{\ell} c_{\ell, m} Y_{\ell, m}$$ Describe the energy degeneracy of the rigid rotor system, i.e. give the number of eigenstates that all have the same energy.

2. (Separation)

   Use the separation of variables procedure on the angular equation

   $$\mathbf{L}^2 Y\left(\theta,\phi\right)=A\hbar^2Y\left(\theta,\phi\right)$$

   $$\mathrm{where}\,\,\,\mathbf{L}^2= -\hbar^2\left[{1\over \sin\theta}{\partial\over\partial\theta}\left( \sin\theta{\partial\over\partial\theta}\right)+{1\over\sin^2\theta} {\partial^2\over \partial \phi^2}\right]$$

   to obtain the following two equations for the polar and azimuthal angles:

   $$\left[\frac{1}{\sin\theta}\frac{d}{d\theta}\left(\sin\theta\frac{d}{d\theta}\right)-B\frac{1}{\sin^2\theta}\right]\Theta(\theta)=-A\Theta(\theta)$$

   $$\frac{d^2 \Phi(\phi)}{d\phi^2}=-B\Phi(\phi)$$