Central Forces Notes Section 24

• The orthogonality of the Legendre Polynomials is established
• The normalization constant for the Legendre Polynomials is found.
• These properties are then used to define the Legendre Polynomial series $f(z) = \sum_l a_l P_l(z)$.
• Then the properties of the Legendre Polynomials are used to find a method for determining the coefficients of the series in analogy to how the coefficients of the Fourier Series are found.

$$a_k = (k + \frac{1}{2})\int_{-1}^1 P_k(z)^* f(z) dz $$

• How do you find the coefficients $a_n$ in a Legendre Polynomial Series, $f(z) = \sum_l a_l P_l(z)$? Students often struggle with this, but it is helpful to remind them of the way you find coefficients for a vector (by taking the dot product, $x = \hat {i}\cdot \Vec {r}$), and coefficients for a Fourier series (the inner product for functions, $a_n = \int_0^{2 \pi} sin(x)f(x)dx$) This also serves as a good lead in to discussing the orthogonality relation for Legendre Polynomials.

• A questions that has arisen is “How/why would you write something like $\sin(nx)$ in terms of Legendre Polynomials?” This is a particular rich question from students since it opens the door to discussing why we choose to write our wave functions in terms of one set of functions rather than another. At this juncture, it is very useful to remind the students that it is useful to write any wave function in terms of the eigenstates of the hamiltonian (whatever those eigenstates be, $\sin(nx)$, $P_\ell(z)$, $|+\rangle$) because we know the energy and the time dependence associated with these energy eigenstates.