• Students find the coefficients of a given function expanded in a Spherical Harmonic Series

Estimated Time: 25 minutes

Students are given a function, for example,

$$Y(\theta, \phi) = \left(\frac{15}{16 \pi}\right)^{\frac{1}{2}} \sin(2 \theta) \sin(\phi)$$

Then each group is asked to find one of the coefficients, e.g., $c_{0,0}$, $c_{1,0}$, $c_{1,1}$, $c_{1,-1}$, etc.

• Students have found the coefficients for a given state written as a linear combination of eigenstates.
• Students have seen and used the correspondence between discrete and continuous variable representations of quantum states using bra-ket notation.
• Students are familiar with the solutions to the $\phi$ and $\theta$ solutions to the rigid rotor problem.
• Students have seen the definition of the spherical harmonics and the orthonormality relation for the spherical harmonics.

• Tabletop Whiteboard with markers

The activity is introduced with a discussion of what a spherical harmonic series will look like, i.e.,

$$Y(\theta,\phi) = \sum^\infty_{l=0} \sum^l_{m=-l} c_{l,m} Y_{l,m}(\theta, \phi)$$

Then students are reminded how they found the $n$th coefficient for the expansion of a given discrete state $|\Phi\rangle = \sum_m a_m |m\rangle$ by applying the ket $|n\rangle$ to the left side of the state $|\Phi\rangle$

$$\langle n|\Phi\rangle = \langle n|\sum_m a_m |m\rangle $$ $$\langle n|\Phi\rangle = \sum_m a_m \langle n|m\rangle $$ $$\langle n|\Phi\rangle = \sum_m a_m \delta_{m,n} = a_m $$

They are then reminded that the scalar product $\langle n|\Phi\rangle$ in the discrete representation is equivalent to $\int_{all \phi} \Phi_n^*(\phi)^* \Phi(\phi) d\phi$ in the functional representation.

Students often begin this calculation by trying to work out the expression for their $Y_{l,m}$ from the general formula. This is a good time to tell them that they can often find the first few $Y_{l,m}$'s in quantum mechanics or mathematical physics texts.

This activity can be used as an introduction to the ylmcombo.mw Maple worksheet activity in which students are able to visualize combinations of the spherical harmonics. This worksheet could be used to verify their calculations and to display both the original function and the linear combination they find in class.

This activity follows a thread throughout the Paradigms in which students are given the opportunity to practice writing arbitrary functions as linear combinations of the eigenstates for that system. Some of the other activities where this is done are

• Operators and Functions (Small Group Activity)
• Quantum Calculations of Energy and Angular Momentum for a Particle Confined to a Ring (Small Group Activity, 30-90 minutes)
• Finding Legendre Coefficients (Maple Activity, 10-15 minutes)