• Eigenvalues and eigenstates
• Measurements of energy, angular momentum, and position in quantum systems
• Quantum probabilities
• Superposition of states
• Quantum calculations in multiple representations
• Degeneracy
• Finding coefficients for a wave function as a linear combination of eigenstates

Estimated Time: 30-45 minutes

Students calculate probability for energy, angular momentum and position for a wavefunction that is not easily separated into eigenstates for a particle on a ring.

• Eigenstates & Eigenvalues
• Familiarity with the postulates of quantum mechanics, particularly those having to do with measurement
• The energy and angular momentum eigenstates and eigenvalues of a particle confined to a ring
• Calculating probabilities using Dirac “bra-ket” and wavefunction notation

• Tabletop Whiteboard with markers
• A handout for each student

If the previous activities (Energy and Angular Momentum for a Particle on a Ring and Time Dependence for a Particle on a Ring) have been done, little introduction is needed. It might be helpful to ask a small whiteboard question to help them remember what the eigenfunctions for a particle on a ring are.

In many cases, students will not think to rewrite the function as a linear combination of eigenstates and if they do know to do this, many will have forgotten how. Thus, it is sometimes useful to start this activity in class and have them finish the calculations for homework.

• Finding coefficients:
  • Students readily grasp the strategy of finding probability amplitudes “by inspection” when they are given an initial state written as a sum of eigenstates. We find that students then find it extremely difficult to find probability amplitudes of wavefunctions that are not written this way (i.e. using an integral to find the expansion coefficients of a function).
  • If they write the cosine in terms of exponentials, they can find the coefficients without using an integral. It is fantastic if a group recognizes this and these groups will often finish early, so having them go back and do the integral is a good way to give them extra practice and keep them engaged with the material.

• Degeneracy: students may experience some difficulty due to the degeneracy of some states, in particular, that you have to include all the states that share that particular eigenvalue.

$$P_{E={m^2\,\hbar^2\over 2I}}=\vert \langle m\vert \psi\rangle\vert^2+\vert \langle -m\vert \psi\rangle\vert^2$$

Use their work to demonstrate how finding all of the probabilities allows you to rewrite the wavefunction as a linear combination of eigenstates. $$P_{L_z=m\hbar}=\vert\langle m\vert\Psi\rangle\vert^2=\left|\int_{-\infty}^{\infty} \Phi_m^*(\phi)\Psi(\phi)\,d\phi\right|^2=\vert c_m \vert^2$$ $$\vert\Psi\rangle=\sum_m c_m \vert m\rangle \doteq \sum_m c_m e^{im\phi}$$

• Quantum Ring Sequence: This is a part of a sequence of activities and homework problems that use a particle confined to a ring as a touchstone example.