# 狄拉克符号

## 矩阵表示

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

## 性质

• 给定任何左矢${\displaystyle \langle \phi |}$、右矢${\displaystyle |\psi _{1}\rangle }$以及${\displaystyle |\psi _{2}\rangle }$，还有复数c1c2，则既然左矢是线性泛函，根据线性泛函的加法与标量乘法的定义，
${\displaystyle \langle \phi |\;{\bigg (}c_{1}|\psi _{1}\rangle +c_{2}|\psi _{2}\rangle {\bigg )}=c_{1}\langle \phi |\psi _{1}\rangle +c_{2}\langle \phi |\psi _{2}\rangle }$
• 给定任何右矢${\displaystyle |\psi \rangle }$、左矢${\displaystyle \langle \phi _{1}|}$以及${\displaystyle \langle \phi _{2}|}$，还有复数c1c2，则既然右矢是线性泛函
${\displaystyle {\bigg (}c_{1}\langle \phi _{1}|+c_{2}\langle \phi _{2}|{\bigg )}\;|\psi \rangle =c_{1}\langle \phi _{1}|\psi \rangle +c_{2}\langle \phi _{2}|\psi \rangle }$
• 给定任何右矢${\displaystyle |\psi _{1}\rangle }$${\displaystyle |\psi _{2}\rangle }$，还有复数c1c2，根据内积的性质（其中c*代表c的复数共轭），
${\displaystyle c_{1}|\psi _{1}\rangle +c_{2}|\psi _{2}\rangle }$${\displaystyle c_{1}^{*}\langle \psi _{1}|+c_{2}^{*}\langle \psi _{2}|}$对偶。
• 给定任何左矢${\displaystyle \langle \phi |}$及右矢${\displaystyle |\psi \rangle }$，内积的一个公理性质指出
${\displaystyle \langle \phi |\psi \rangle =\langle \psi |\phi \rangle ^{*))$
• 给定任何算符${\displaystyle X}$、左矢${\displaystyle \langle \phi |}$及右矢${\displaystyle |\psi \rangle }$，它们之间的合法相乘满足乘法结合公理，例如，[2]:16-17
${\displaystyle (|\omega \rangle \langle \phi |)\ |\psi \rangle =|\omega \rangle \ (\langle \phi |\psi \rangle )}$
${\displaystyle \langle \phi |\ (X|\psi \rangle )=(\langle \phi |X)\ |\psi \rangle }$

## 参考文献

1. ^ PAM Dirac. A new notation for quantum mechanics 35 (3). 1939: 416–418 [2014-01-31]. doi:10.1017/S0305004100021162. （原始内容存档于2013-12-03）. |journal=被忽略 (帮助)
2. ^ Sakukrai, J. J.; Napolitano, Jim, Modern Quantum Mechanics 2nd, Addison-Wesley, 2010, ISBN 978-0805382914

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