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

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?

Activity: Wrap-up

• 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.