时间反演对称和空间反演对称性

哈密顿量:

\[H(r)=\sum_ke^{ikr}H(k)e^{-ikr} \]

一，时间反演对称性 \(\hat{T}\):
\([\hat{T},H(r)]=\hat{T}H(r)-H(r)\hat{T}=0\) 得到: \(\hat{T}H(r)\hat{T}^{-1}=H(r)\)

\[\hat{T}\sum_{k}e^{ikr}H(k)e^{-ikr}\hat{T}^{-1} = H(r) \\ =\sum_{k}e^{-ikr}\hat{T}H(k)\hat{T}^{-1}e^{ikr} =\sum_ke^{ikr}H(k)e^{-ikr} \\ \hat{T}H(k)\hat{T}^{-1} = H(-k) \]

时间反演算符 \(T=UK\).

\[\hat{U}\hat{K}H(k)\hat{K}^{-1}\hat{U}^{-1} = H(-k) \]

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二，空间反演对称性 \(\hat{P}\):
\([\hat{P},H(r)]=\hat{P}H(r)-H(r)\hat{P}=0\) 得到: \(\hat{P}H(r)\hat{P}^{-1}=H(r)\)

\[\hat{P}\sum_{k}e^{ikr}H(k)e^{-ikr}\hat{P}^{-1} = H(r) \\ =\sum_{k}e^{-ikr}\hat{P}H(k)\hat{P}^{-1}e^{ikr} =\sum_ke^{ikr}H(k)e^{-ikr} \\ \hat{P}H(k)\hat{P}^{-1} = H(-k) \]

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三，旋转对称性 \(\hat{R}\):
\([\hat{R},H(r)]=\hat{R}H(r)-H(r)\hat{R}=0\) 得到: \(\hat{R}H(r)\hat{R}^{-1}=H(r)\)

\[\hat{R}\sum_{k}e^{ikr}H(k)e^{-ikr}\hat{R}^{-1} = H(r) \\ =\sum_{k}e^{ik\hat{R}r}\hat{R}H(k)\hat{R}^{-1}e^{-ik\hat{R}r} =\sum_ke^{ikr}H(k)e^{-ikr} \\ \hat{R}H(k)\hat{R}^{-1} = H(\hat{R}k) \]

标签：ikr,ke,sum,反演,对称性,空间反演,hat
来源： https://www.cnblogs.com/ghzhan/p/16684816.html