Activity: Quantum Expectation Values

Expectation Values and Uncertainty

You have a system that consists of quantum particles with spin. On this system, you will perform a Stern-Gerlach experiment with an analyzer oriented in the \(z\)-direction.

Consider one of the different initial spin states described below:

A spin 1/2 particle described by:

  1. \(\left|{+}\right\rangle \)
  2. \(\frac{i}{2}\left|{+}\right\rangle -\frac{\sqrt{3}}{2}\left|{-}\right\rangle \)
  3. \(\left|{+}\right\rangle _x\)

    A spin 1 particle described by:

  4. \(\left|{0}\right\rangle \)
  5. \(\left|{-1}\right\rangle _x\)
  6. \(\frac{2}{3}\left|{1}\right\rangle +\frac{i}{3}\left|{0}\right\rangle -\frac{2}{3}\left|{-1}\right\rangle \)
  • List the possible values of spin you could measure and determine the probability associated with each value of the z-component of spin.


  • Plot a histogram of the probabilities.


  • Find the expectation value and uncertainty of the z-component of spin.



Learning Outcomes