1. 狄拉克符号

1.1 基矢

\(|0 \rang = \binom{1}{0}\) \(|1\rang = \binom{0}{1}\)

\(\lang 0| = (1~0)\) \(\lang 1| = (0~1)\)

1.2 态矢

\(| \psi \rangle = a|0\rangle + b|1\rangle\)

\(\lang \psi| = |\psi \rang ^{\dagger} = \Big( \left(|\psi \rang \right)^* \Big)^T = \Big( \left(|\psi \rang \right)^T \Big)^* = a^* \lang0| + b^*\lang1|\)

共轭，转置顺序可以互换.

2. 内积

\(\lang \varphi | \psi \rang = \lang \varphi || \psi \rang = | \varphi \rang^ \dagger |\psi\rang\) ，也可以表示为 \(( | \varphi \rang, |\psi \rang)\)

内积的性质：

(1) \((\cdot~, \cdot)\) 对第二个变量是线性的，即
\((|v\rang, \sum_{i}\lambda_i|w_i\rang) = \sum_i \lambda_i(|v\rang, |w_i\rang)\)
(2) \(\lang \varphi| \psi \rang = \Big(\lang \psi| \varphi \rang \Big)^*\)
推导如下：
\(\lang \varphi | \psi \rang = \lang \varphi|| \psi \rang = | \varphi\rang^ \dagger |\psi\rang\)
\(\lang \psi| \varphi \rang = \lang \psi|| \varphi \rang = | \psi\rang^ \dagger |\varphi\rang\)
\(\Big(\lang \psi| \varphi \rang \Big)^* = \Big(| \psi\rang^ \dagger |\varphi\rang \Big)^* = |\psi\rang^T |\varphi\rang^* = (|\varphi\rang^*)^T |\psi\rang = | \varphi\rang^ \dagger |\psi\rang\).

(3) 内积非负，当且仅当\(|v\rang=0\) 时取等号.
\(\lang i|j \rang = \delta_{ij} = \left\{\begin{matrix} 1~~~i=j \\ 0~~~~i\neq j \end{matrix}\right.\)

参考文献

1. 马瑞霖. 量子密码通信[M]. 北京：科学出版社，2006.
2. [英]尼尔森，庄著. 量子计算和量子信息(一)--量子计算部分[M]. 赵千川译. 北京：清华大学出版社，2009.

标签：lang,rang,psi,dagger,量子力学,Big,基础,varphi
来源： https://www.cnblogs.com/one2Four/p/15489298.html