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# Finding the Coefficients of a Spherical Harmonic Series

**Keywords:**Central Forces, Quantum Mechanics, Rigid Rotor, Spherical Harmonics, Series Expansion, Small Group Activity

### Highlights of the activity

- This small group activity is designed to help upper division undergraduate students expand functions in terms of spherical harmonics.
- Students work in small groups to find the coefficients of a given function expanded in spherical harmonics.
- The whole class wrap-up discussion includes group presentations focusing on the notion that any function on the unit sphere can be expanded in terms of the orthonormal set of spherical harmonics.

### Reasons to spend class time on the activity

Students typically do not see the bigger picture with regard to the special functions they learn in quantum mechanics. In particular, they often do not see the connection between what they learn when studying continuous and discrete variables in quantum mechanics. This activity is intended to reinforce the idea that one can write any given spin state (or other discrete state) as a linear combination of the spin eigenstates is the same as the idea that one can write any given wavefunction as a linear combination of the eigenstates of the Hamiltonian for that system. This activity helps to reinforce the idea that any function can be written in terms of a set of orthonormal eigenfunctions that span a given space. This activity also gives them practice in how to find the coefficients for the series expansion and draws a parallel with this same process for Fourier series and Legendre polynomial expansions they have done before.