Mechanika kwantowa/Zasada wariacyjna Schwingera: Różnice pomiędzy wersjami
Usunięta treść Dodana treść
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{{IndexWzór|<MATH>[\hat{b}_k^-,\hat{b}^-_{k^{'}}]={{\hbar}\over{4m_0c^2L^3}}\left(\omega_k\omega_{k^'}\right)^{-{{1}\over{2}}}\int\int d^3\vec{r} d^3\vec{r}^'\exp(-i(\vec{k}\vec{r}-i\vec{k}^{'}\vec{r}^')\exp(i(\omega_k+\omega_{k^'})t)\cdot[\omega_k\hat{\Phi}(\vec{r},t)+i\hat{\Pi},\omega_{k^'}\hat{\Phi}(\vec{r}^',t)+i\hat{\Pi}(\vec{r}^',t)]=\;</MATH><BR>
<MATH>={{\hbar}\over{4m_0c^2L^3}}\left(\omega_k\omega_{k^'}\right)^{-{{1}\over{2}}}\int\int d^3\vec{r} d^3\vec{r}^'\exp(-i\vec{k}\vec{r}-i\vec{k}^{'}\vec{r}^')\exp(i\omega_kt+i\omega_{k^'}t)\cdot\;</MATH><BR>
<MATH>\cdot\Bigg\{\omega_k\omega_{k^'}[\hat{\Phi}(
<MATH>={{\hbar}\over{4m_0c^2L^3}}\left(\omega_k\omega_{k^'}\right)^{-{{1}\over{2}}}\int\int d^3\vec{r} d^3\vec{r}^'\exp(-i\vec{k}\vec{r}-i\vec{k}^{'}\vec{r}
<MATH>={{\hbar}\over{4m_0c^2L^3}}\left(\omega_k\omega_{k^'}\right)^{-{{1}\over{2}}}\int\int d^3\vec{r} d^3\vec{r}^'\exp(-i\vec{k}\vec{r}-i\vec{k}^{'}\vec{r}^')\exp(i\omega_kt+i\omega_{k^'}t)\cdot\left\{\omega_k ii{{2m_0c^2}\over{\hbar}}\delta^3(\vec{r}-\vec{r}^')-ii{{2m_0c^2}\over{\hbar}}\omega_{k^'}\delta^3(\vec{r}-\vec{r}^') \right\}=\;</MATH><BR>
<MATH>={{1}\over{2L^3}}\left(\omega_k\omega_{k^'}\right)^{-{{1}\over{2}}}\int d^3\vec{r}\exp(-i(\vec{k}+\vec{k}^')\vec{r})\exp(i(\omega_k +\omega_{k^'})t)(\omega_{k^'}-\omega_k)=\;</MATH><BR><MATH>={{1}\over{2L^3}}\left(\omega_k\omega_{k^'}\right)^{-{{1}\over{2}}}\exp(i(\omega_k +\omega_{k^'})t)(\omega_{k^'}-\omega_k)\int\exp(-i(\vec{k}+\vec{k}^')\vec{r})d^3\vec r=0\;</MATH>|31.80}}
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{{IndexWzór|<MATH>[\hat{b}_k^-,\hat{b}^-_{k^{'}}]=0\;</MATH>|31.81}}
Następnie krokiem jest wyznaczenie wyrażenia oparte tylko na operatorach kreacji:
{{IndexWzór|<MATH>[\hat{b}^+_k,\hat{b}^+_{k^{'}}]=\;</MATH><BR><MATH>={{\hbar}\over{4m_0c^2L^3}}\left(\omega_k\omega_{k^'}\right)^{-{{1}\over{2}}}\int\int d^3\vec{r} d^3\vec{r}^'\exp(i\vec{k}\vec{r}+i\vec{k}^{'}\vec{r}^')\exp(-i(\omega_k+\omega_{k^'})t)\cdot[\omega_k\hat{\Phi}(\vec{r},t)-i\hat{\Pi}(\vec{r},t),\omega_{k^'}\hat{\Phi}(\vec{r}
<MATH>={{\hbar}\over{4m_0c^2L^3}}\left(\omega_k\omega_{k^'}\right)^{-{{1}\over{2}}}\int\int d^3\vec{r}d^3\vec{r}^'\exp(i\vec{k}\vec{r}+i\vec{k}^{'}\vec{r}^')\exp(-i\omega_kt-i\omega_{k^'}t)\cdot\Bigg\{\omega_k\omega_{k^'}[\hat{\Phi}(\vec{r},t),\hat{\Phi}(\vec{r}^',t)]-\omega_k i[\hat{\Phi}(\vec{r},t),\hat{\Pi}(\vec{r}^',t)]+\;</MATH><BR><MATH>-i\omega_{k^'}[\hat{\Pi}(\vec{r},t),\hat{\Phi}(\vec{r}^',t)]+i[\hat{\Phi}(\vec{r},t),\hat{\Phi}(\vec{r}^',t)]\Bigg\}={{\hbar}\over{4m_0c^2L^3}}\left(\omega_k\omega_{k^'}\right)^{-{{1}\over{2}}}\int\int d^3\vec{r} d^3\vec{r}^'\exp(i\vec{k}\vec{r}+i\vec{k}^{'}\vec{r}^')\exp(-i\omega_kt-i\omega_{k^'}t)\cdot\;</MATH><BR>
<MATH>\cdot\left\{-\omega_k i[\hat{\Phi}(\vec{r},t),\hat{\Pi}(\vec{r}^',t)]-i\omega_{k^'}[\hat{\Pi}(\vec{r},t),\hat{\Phi}(\vec{r}^',t)]\right\}={{\hbar}\over{4m_0c^2L^3}}\left(\omega_k\omega_{k^'}\right)^{-{{1}\over{2}}}\int\int d^3\vec{r}d^3\vec{r}^'\exp(i\vec{k}\vec{r}+i\vec{k}^{'}\vec{r}^')\exp(-i\omega_kt-i\omega_{k^'}t)\cdot</MATH><BR><MATH>\cdot\left\{-\omega_k ii{{2m_0c^2}\over{\hbar}}\delta(\vec{r}-\vec{r}^')+ii{{2m_0c^2}\over{\hbar}}\omega_{k^'}\delta^3(\vec{r}-\vec{r}^') \right\}={{1}\over{2L^3}}\left(\omega_k\omega_{k^'}\right)^{-{{1}\over{2}}}\int d^3\vec{r}\exp(i(\vec{k}+\vec{k}^')\vec r)\exp(-i(\omega_k +\omega_{k^'})t)(\omega_k-\omega_{k^'})=\;</MATH><BR>
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