## 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