American Journal of Physics

Antilinear operators in Dirac’s bra–ket notation

Antoine Royer

Published Aug 1, 1994• 1 citation

Abstract

A notation is introduced which makes the manipulation of antilinear operators within Dirac’s bra–ket notation as efficient as that of linear operators.

DOI: 10.1119/1.17506

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American Journal of Physics

Antilinear operators in Dirac’s bra–ket notation

Antoine Royer

Published Aug 1, 1994• 1 citation

Publisher

Abstract

A notation is introduced which makes the manipulation of antilinear operators within Dirac’s bra–ket notation as efficient as that of linear operators.

DOI: 10.1119/1.17506

Related papers

Dirac’s Bra-ket Notation and Hermitian Operators

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Springer eBooksJan 1, 20212 citations

On the coordinate representation in the Dirac bra-ket notation

Mehdi Jafari Matehkolaee

Aug 1, 20250 citations

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F.J. Duarte, T S Taylor

Institute of Physics eBooksJul 1, 20210 citations

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Apr 27, 20160 citations

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Yuri G. Gurevich, Andreas R. Blass

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.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.

arXiv:2405.10055May 16, 20240 citations

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.