What are kets?

TL;DR

This paper explores the dual interpretation of kets in Dirac notation, proposing a flexible view that treats them as vectors or functions depending on context, without requiring quantum mechanics background.

Contribution

It introduces a perspective that allows viewing kets as both vectors and functions, enhancing the understanding of bra-ket notation in a more flexible and context-dependent way.

Findings

Kets can be interpreted as vectors or functions.

This dual view simplifies the understanding of outer products.

The approach does not require prior quantum mechanics knowledge.

Abstract

According to Dirac's bra-ket notation, in an inner-product space, the inner product $\langle x\,|\,y\rangle$ of vectors $x,y$ can be viewed as an application of the bra $\langle x|$ to the ket $|y\rangle$. Here $\langle x|$ is the linear functional $|y\rangle \mapsto \langle x\,|\,y\rangle$ and $|y\rangle$ is the vector $y$. But often -- though not always -- there are advantages in seeing $|y\rangle$ as the function $a \mapsto a\cdot y$ where $a$ ranges over the scalars. For example, the outer product $|y\rangle\langle x|$ becomes simply the composition $|y\rangle \circ \langle x|$. It would be most convenient to view kets sometimes as vectors and sometimes as functions, depending on the context. This turns out to be possible. While the bra-ket notation arose in quantum mechanics, this note presupposes no familiarity with quantum mechanics.