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Matrix Representation of Spin Operators: Instructor's Guide

Main Ideas

  • Examining the matrix representation of spin operators for a spin 1/2 system written in the $S_z$ basis
  • Showing that the spin operators (like any operator) can be written as a linear combination of projection operators.
  • Examining the transformation caused by the spin operators

Students' Task

Estimated Time: 45 min Students work in small groups to:

  • write down matrices with given eigenvalues and eigenvectors (spin operators)
  • write down the spin operators as a linear combination of projection operators
  • explore the transformation a vector undergoes when acted on with the spin operators

Prerequisite Knowledge

  • Matrix representations of $|+\rangle$, $|-\rangle$, $|+\rangle_x$, $|-\rangle_x$, $|+\rangle_y$, and $|-\rangle_y$.
  • Matrix multiplication and vector algebra
  • Eigenvalue equations and the geometric interpretation of eigenvectors and eigenvalues


Activity: Introduction

No introduction is needed

Activity: Student Conversations

Activity: Wrap-up

  • It is important to emphasize that the spin operators do NOT correspond to the transformation that occurs when a quantum measurement is made. Emphasize that operators that correspond to observable quantities encode information about the possible values that could be measured (eigenvalues) and the states that the system will collapse into after a measurement is made (eigenstates).
  • Students should be aware that writing these matrices in the $|+\rangle$, $|-\rangle$ basis is a convention, but these operators (and their eigenvalues) could be written in any basis, and if done, the matrix elements would change.
  • This activity could be easily made into a compare and contrast activity by assigning different groups different spin operators. The wrap-up can then consist of different groups reporting their results


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