Solving the $\Theta$ Equation using the Series Solution Method (90 minutes)

FIXME See 2012 video for series solution lecture with sequence of SWBQs.

Central Forces Notes Section 20-23

  • To help improve students understanding of the geometric constraints on the system, we use a physical sphere (a volleyball) when discussing this problem. The use of a physical prop helps students quickly grasp the geometry so they can focus their intellectual energy on other aspects of the solution.
  • The Schrodinger Equation is written down for the rigid rotor (aka motion of a particle on a sphere and separated into the $\Theta$ and $\Phi$ parts. By inspection it is shown that the $\Phi$ part of the equation was already solved in the previous section (The Ring Problem).
  • A change of variables is applied to cast the $\Theta$ equation in dimensionless terms. We strongly recommend using the variable $z=\cos\theta$ rather than some other letter (many texts use $w$) since $\cos\theta$ is literally the physical variable $z$ from rectangular coordinates.
  • The $\Theta$ equation is then solved using the Series Solution Method (for the case in which m=0).
  • The Legendre Polynomials are defined and shown to be the solutions of the $\Theta$ equation.
  • Notice that the $\Theta$ equation is a second order linear differential equation. As such, it should have two solutions for all values of the separation constant. However, there are only a few values of the separation constant for which the solutions will be regular (i.e. not blow up) at both the north and south poles. These special solutions are the physical ones and all other solutions are thrown out, i.e two solutions are thrown out for most values of the separation constant and one solution is thrown out for the special values. Regularity at the poles is boundary condition just like having a solution go to zero at a boundary or fall off appropriately at infinity. The discreteness of the quantum number $\ell$ is a direct consequence of this boundary condition.

Small White Board Questions

  • Write down the following Legendre Polynomial, $P_2(z)$. Students sometimes have difficulty with this the first time they do it because they are not used to using the recurrence relation in this way. In addition, they do not realize that they need to set the values of $a_0$ and $a_1$.

Common Student Confusions/Questions

  • Q: What is the difference between a Legendre Polynomial and a Taylor Series? A: A Legendre Polynomial is a specific example of a (short!) Taylor Series.
  • Trying to find $a_0$ and $a_1$. Students don't realize they have to set these values, either $a_0=a_0$ and $a_1=0$ (even), or $a_0=0$ and $a_1=a_1$ (odd)
  • Students also are often confused about how you set $a_0$. They don't realize that this is set by the initial/boundary conditions in most physical situations. In quantum examples, the constant is determined by the normalization condition on the eigenstates.
  • It is common for students to wonder how you can set the odd or even solution to be zero. It is helpful to remind them that we expect two solutions for a second order differential equation. One of which is odd and one is even. If the value of $\ell$ is even then the odd series does not terminate and, in fact, blows up at either the north or south pole, whereas if the value of $\ell$ is odd, the even series does not terminate and thus blows up. The recurrence relation shows that only one of the two solutions will terminate and thus be well behaved at the poles for a given value of $\ell$.

Personal Tools