# Properties of Outer Products

## The Prompt

Compute the outer product of the general state vector

$$\vert \psi\rangle=\left[\begin{array}{c} \sin(\frac{\theta}{2})\\ \cos(\frac{\theta}{2})e^{i\phi}\\ \end{array}\right]$$

Take the determinant of the computed 2 x 2 matrix.

## Context

After being introduced to the density matrix, students are commonly skeptical about how useful it can be in finding probabilities and describing mixed states. This SWBQ introduces several of the properties of the density matrix and helps students transition into making states described by density matrices. The activity also shows that for any density matrix formed by the outer product of a general state vector with itself must have $\text{Tr}(\vert \psi\rangle \langle\psi\vert) = 1$ and $\text{Det}(\vert \psi\rangle \langle\psi\vert) = 0$.

## Wrap Up

- Extra Information “After being introduced to the density matrix, students are commonly skeptical about how useful it can be in finding probabilities and describing mixed states.”

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