## Student handout: Superposition States for a Particle on a Ring

Students calculate probabilities for a particle on a ring whose wavefunction is not easily separated into eigenstates by inspection. To find the energy, angular momentum, and position probabilities, students perform integrations with the wavefunction or decompose the wavefunction into a superposition of eigenfunctions.
What students learn
• How to find expansion coefficients of wavefunctions using inner products (integration)
• How to write a compact wavefunction as a superposition (sum) of eigenstates
Consider the following normalized abstract quantum state on a ring: $$\Phi(\phi)= \sqrt{\frac{8}{5\pi r_0}}\cos^3{(2\phi)}$$
1. If you measured the $z$-component of angular momentum, what is the probability that you would measure $2\hbar$? $-3\hbar$?
2. If you measured the $z$-component of angular momentum, what other possible values could you have obtained with non-zero probability?
3. If you measured the energy, what possible values could you have obtained with non-zero probability?
4. What is the probability that the particle can be found in the region $0<\phi<\frac{\pi}{2}$?

Author Information
Corinne Manogue, Kerry Browne, Elizabeth Gire, Mary Bridget Kustusch, David McIntyre
Keywords
central forces quantum mechanics eigenstates eigenvalues quantum measurements angular momentum hermitian operators probability superposition
Learning Outcomes