# 純態

## 量子力學

${\displaystyle S=|\Psi \rangle }$

${\displaystyle S=\rho =|\Psi \rangle \langle \Psi |}$

${\displaystyle S=\rho =\sum _{i=1}^{N}c_{i}|\Psi _{i}\rangle \langle \Psi _{i}|}$

### 區分純態與混態

#### 舉例

${\displaystyle \rho _{1}={\begin{pmatrix}{\frac {1}{2}}&{\frac {1}{2}}\\{\frac {1}{2}}&{\frac {1}{2}}\end{pmatrix}}}$為純態，${\displaystyle \rho _{2}={\begin{pmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{pmatrix}}}$為混態

${\displaystyle \Rightarrow tr(\rho _{1})=tr(\rho _{2})={\frac {1}{2}}+{\frac {1}{2}}=1}$

${\displaystyle \rho _{1}^{2}=\rho _{1}*\rho _{1}={\begin{pmatrix}{\frac {1}{2}}&{\frac {1}{2}}\\{\frac {1}{2}}&{\frac {1}{2}}\end{pmatrix}}}$${\displaystyle \rho _{2}^{2}=\rho _{2}*\rho _{2}={\begin{pmatrix}{\frac {1}{4}}&0\\0&{\frac {1}{4}}\end{pmatrix}}}$

${\displaystyle \Rightarrow tr(\rho _{1}^{2})=tr(\rho _{1})={\frac {1}{2}}+{\frac {1}{2}}=1}$${\displaystyle tr(\rho _{2}^{2})={\frac {1}{4}}+{\frac {1}{4}}={\frac {1}{2}}\neq tr(\rho _{2})=1}$

${\displaystyle \rho _{1}={\begin{pmatrix}{\frac {1}{2}}&{\frac {1}{2}}e^{-{\frac {t}{T_{2}}}}\\{\frac {1}{2}}e^{-{\frac {t}{T_{2}}}}&{\frac {1}{2}}\end{pmatrix}}\Rightarrow \rho _{1}(t=0)={\begin{pmatrix}{\frac {1}{2}}&{\frac {1}{2}}\\{\frac {1}{2}}&{\frac {1}{2}}\end{pmatrix}}}$

${\displaystyle {\overset {t\rightarrow \infty }{\to }}\rho _{2}={\begin{pmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{pmatrix}}}$

## 參考资料

1. ^ {S. VanEnk, "Mixed states and pure states," [Online Note]. University of Oregon. Available: https://pages.uoregon.edu/svanenk/solutions/Mixed_states.pdf [Accessed: September 25, 2023]}