Student handout: Time Evolution of a Spin-1/2 System

Quantum Fundamentals 2021
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.

Time Evolution of a Spin-1/2 System

Consider a spin-1/2 system with a Hamiltonain that is proportional to \(\hat{S}_{z}\):

\[\hat{H} = \omega_0 \hat{S}_{z}\]

At \(t=0\):

  1. one particle is in the state \(\vert + \rangle _{x}\).
  2. another particle is in the state \(\vert + \rangle\)

For each particle:

  1. What values of energy could you measure?
  2. What are the energy eigenstates?
  3. What state is each particle in at a later time t?
  4. What are the probabilities for each energy measurement?
  5. What is the probability that you would measure \(S_x = {\hbar \over 2}\) state at time \(t\)? Does this probability change with time?
  6. What is the probability that you would measure \(S_z = {\hbar \over 2}\) at time \(t\)? Does this probability change with time?
  7. 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?


Author Information
Elizabeth Gire
Keywords
quantum mechanics spin precession time evolution
Learning Outcomes