Activity: Time Evolution of a Spin-1/2 System

Quantum Fundamentals 2022 (3 years)
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.

Prerequisite Knowledge

  • Spin 1/2 systems
  • Familiarity with how to calculate measurement probabilities
  • Solutions to the Schrodinger equation for a time independent Hamiltonian
  • Dirac notation

Activity: Introduction

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.

Student Task

Time Evolution of a Spin-1/2 System

Two particles are under the influence of an interaction with a Hamiltonian that is proportional to \(\hat{S}_{z}\):

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

At \(t=0\):

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


Activity: Wrap-up

The main points of this activity are addressed in the last question. Students should recognize:
  • a pattern of how these calculations proceed,
  • to recognize a stationary state
  • that measurement probabilities of non-stationary states will be time-dependent UNLESS you measure a quantity that commutes with the Hamiltonian.
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Author Information
Elizabeth Gire
Learning Outcomes