In this small group activity, students solve for the time dependence of two quantum spin 1/2 particles under the influence of a Hamiltonian. Students determine, given a Hamiltonian, which states are stationary and under what circumstances measurement probabilities do change with time.

Students see probability density for eigenstates and linear combinations of eigenstates for a particle on a ring. The three visual representations: standard position vs probability density plot, a ring with colormapping, and cylindrical plot with height and colormapping, are also animated to visualize time-evolution.

Students calculate the expectation value of energy and angular momentum as a function of time for an initial state for a particle on a ring. This state is a linear combination of energy/angular momentum eigenstates written in bra-ket notation.

Quantum MechanicsTime EvolutionSpin PrecessionExpectation ValueBohr FrequencyQuantum Fundamentals 2021
Consider a two-state quantum system with a Hamiltonian
\begin{equation}
\hat{H}\doteq
\begin{pmatrix}
E_1&0\\ 0&E_2
\end{pmatrix}
\end{equation}
Another physical observable \(M\) is described by the operator
\begin{equation}
\hat{M}\doteq
\begin{pmatrix}
0&c\\ c&0
\end{pmatrix}
\end{equation}
where \(c\) is real and positive. Let the initial state of the system be \(\left|{\psi(0)}\right\rangle
=\left|{m_1}\right\rangle \), where \(\left|{m_1}\right\rangle \) is the eigenstate corresponding to the larger of the two possible eigenvalues of \(\hat{M}\). What is the frequency of oscillation of the expectation value of \(M\)? This frequency is the Bohr frequency.

Students calculate probabilities for energy, angular momentum, and position as a function of time for an initial state that is a linear combination of energy/angular momentum eigenstates for a particle confined to a ring written in bra-ket notation. This activity helps students build an understanding of when they can expect a quantity to depend on time and to give them more practice moving between representations.