# Computing Inner Products

## The Prompt

“Compute the inner product $_{x}\langle -\vert +\rangle_{x}$.”

## Context

This SWBQ could be used in two different ways. For students that are in the process of finding defining $\vert+ \rangle_{x}$ and $\vert- \rangle_{x}$ in terms of the $z$-basis, this small white board question can be given to the students to help them find the relative phase between the $\vert -\rangle_z$ components of the two states. (See section ?.? in McIntyre). If students have already found what $\vert+ \rangle_{x}$ and $\vert- \rangle_{x}$ are in terms of the $z$-basis, this SWBQ could also be used as a quick check to see if they know that the spin-up and spin-down states in the x-direction are orthogonal.

## Wrap Up

Some students may quickly recognize the orthogonality of the given states, while others may be unsure how to progress. Inviting the latter students to change $\vert+ \rangle_{x}$ and $\vert- \rangle_{x}$ to their corresponding representations in the $z$-basis may help them justify why the given bra and ket must be orthogonal.

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