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.
Little introduction is needed, although you may want to review how to find a time-evolved state for a time-independent Hamiltonian.
Students work in groups to solve for the time dependence of two quantum particles under the influence of a Hamiltonian. Students find the time dependence of the particles' states and some measurement probabilities.
Time Evolution of a Spin-1/2 SystemConsider a spin-1/2 system with a Hamiltonain that is proportional to \(\hat{S}_{z}\):
\[\hat{H} = \omega_0 \hat{S}_{z}\]
At \(t=0\):
- one particle is in the state \(\vert + \rangle _{x}\).
- another particle is in the state \(\vert + \rangle\)
For each particle:
- What values of energy could you measure?
- What are the energy eigenstates?
- What state is each particle in at a later time t?
- What are the probabilities for each energy measurement?
- What is the probability that you would measure \(S_x = {\hbar \over 2}\) state at time \(t\)? Does this probability change with time?
- What is the probability that you would measure \(S_z = {\hbar \over 2}\) at time \(t\)? Does this probability change with time?
- Given a Hamiltonian, how would you determine which states are stationary states (states where no probabilities change with time)? Under what circumstances do measurement probabilities change with time?