# 狄拉克符号

（重定向自狄拉克括号

## 矩陣表示

${\displaystyle |\psi \rangle ={\begin{pmatrix}\psi _{1}\\\psi _{2}\\\psi _{3}\\\psi _{4}\\\vdots \\\psi _{N}\\\end{pmatrix}}}$
${\displaystyle \langle \psi |={\begin{pmatrix}\psi _{1}^{*},&\psi _{2}^{*},&\psi _{3}^{*},&\psi _{4}^{*},&\cdots ,&\psi _{N}^{*}\end{pmatrix}}}$

## 性質

• 給定任何左矢${\displaystyle \langle \phi |}$ 、右矢${\displaystyle |\psi _{1}\rangle }$ 以及${\displaystyle |\psi _{2}\rangle }$ ，還有複數c1c2，則既然左矢是線性泛函，根據線性泛函的加法與純量乘法的定義，
${\displaystyle \langle \phi |\;{\bigg (}c_{1}|\psi _{1}\rangle +c_{2}|\psi _{2}\rangle {\bigg )}=c_{1}\langle \phi |\psi _{1}\rangle +c_{2}\langle \phi |\psi _{2}\rangle }$
• 給定任何右矢${\displaystyle |\psi \rangle }$ 、左矢${\displaystyle \langle \phi _{1}|}$ 以及${\displaystyle \langle \phi _{2}|}$ ，還有複數c1c2，則既然右矢是線性泛函
${\displaystyle {\bigg (}c_{1}\langle \phi _{1}|+c_{2}\langle \phi _{2}|{\bigg )}\;|\psi \rangle =c_{1}\langle \phi _{1}|\psi \rangle +c_{2}\langle \phi _{2}|\psi \rangle }$
• 給定任何右矢${\displaystyle |\psi _{1}\rangle }$ ${\displaystyle |\psi _{2}\rangle }$ ，還有複數c1c2，根據內積的性質（其中c*代表c的複數共軛），
${\displaystyle c_{1}|\psi _{1}\rangle +c_{2}|\psi _{2}\rangle }$ ${\displaystyle c_{1}^{*}\langle \psi _{1}|+c_{2}^{*}\langle \psi _{2}|}$ 對偶。
• 給定任何左矢${\displaystyle \langle \phi |}$ 及右矢${\displaystyle |\psi \rangle }$ ，內積的一個公理性質指出
${\displaystyle \langle \phi |\psi \rangle =\langle \psi |\phi \rangle ^{*}}$
• 給定任何算符${\displaystyle X}$ 、左矢${\displaystyle \langle \phi |}$ 及右矢${\displaystyle |\psi \rangle }$ ，它們之間的合法相乘滿足乘法結合公理，例如，[2]:16-17
${\displaystyle (|\omega \rangle \langle \phi |)\ |\psi \rangle =|\omega \rangle \ (\langle \phi |\psi \rangle )}$
${\displaystyle \langle \phi |\ (X|\psi \rangle )=(\langle \phi |X)\ |\psi \rangle }$

## 參考文獻

1. ^ PAM Dirac. A new notation for quantum mechanics. Mathematical Proceedings of the Cambridge Philosophical Society 35 (3). 1939: 416–418. doi:10.1017/S0305004100021162.
2. ^ Sakukrai, J. J.; Napolitano, Jim, Modern Quantum Mechanics 2nd, Addison-Wesley, 2010, ISBN 978-0805382914