スピントロニクス理論の基礎/8-8 のバックアップソース(No.1)

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[[スピントロニクス理論の基礎]]

* 8-8 相互作用の摂動論的扱い [#te6bf6a7]

&math(H=H_0+V);

に対して、

(8.96), (8.97)

&math(U=(U_0U_0^\dagger) U=U_0(U_0^\dagger U)\equiv U_0 U_V);

として、&math(U); を &math(U_0); と &math(U_V); に分けて書く。

(8.98), (8.99)

&math(
i\hbar\frac{\PD}{\PD t}U_V&=i\hbar\frac{\PD U_0^\dagger}{\PD t} U+i\hbar U_0^\dagger\frac{\PD U}{\PD t}\\
&=i\hbar\left(\frac{H_0 U_0}{i\hbar}\right)^\dagger U+i\hbar U_0^\dagger \left(\frac{H U}{i\hbar}\right)\\
&=-U_0^\dagger H_0 U + U_0^\dagger H U\\
&=U_0^\dagger (H-H_0) U\\
&=U_0^\dagger V U\\
&=U_0^\dagger V (U_0 U_0^\dagger) U\\
&=V_{H_0}U_V
);

したがって (8.7) と同様にして、

(8.100)

&math(U_V(t,t_0)=Te^{-\frac{i}{\hbar}\int_{t_0}^tdt'V_{H_0}(t')});

これと &math(O_\mathrm H=U^\dagger O U=U_V^\dagger U_0^\dagger O U_0 U_V=U_V^\dagger O_{\mathrm H_0}U_V); を用いて、

(8.101)

&math(
\overline O(t)&=
\frac{1}{Z_0}\trace[U(-i\beta/\hbar+t_0,t_0)U_V^\dagger(t,t_0)O_{\mathrm H_0}(t)U_V(t,t_0)]\\
&=\frac{1}{Z_0}\trace[T_C\,e^{-\frac{i}{\hbar}\int_Cd\tau'V_{\mathrm H_0}(\tau')}O_{\mathrm H_0}(\tau)]\\
&=\big\langle T_C\,e^{-\frac{i}{\hbar}\int_Cd\tau'V_{\mathrm H_0}(\tau')}O_{\mathrm H_0}(\tau)\big\rangle
);

同様に、

(8.102)

&math(
G(\bm r,t,\bm r',t)=
&=-i\big\langle T_C\,e^{-\frac{i}{\hbar}\int_Cd\tau'V_{\mathrm H_0}(\tau')}
c_{\mathrm H_0}^\dagger(\bm r',\tau')c_{\mathrm H_0}(\bm r,\tau)\big\rangle
);

(8.63) の &math(H); を &math(H_0+V); に置き換え、(8.24A), (8.30A) を用いれば、

(8.103)

&math(
&i\hbar\frac{\PD}{\PD \tau}G=\hbar\delta(\tau-\tau')\delta^3(\bm r-\bm r')
+i\big\langle T_C\,e^{-\frac{i}{\hbar}\int_Cd\tau'H(\tau')}
[H_0+V,c(\bm r,\tau)]c^\dagger(\bm r',\tau')\big\rangle\\
&=\hbar\delta(\tau-\tau')\delta^3(\bm r-\bm r')
-\left(\frac{\hbar^2}{2m}\nabla^2_{\bm r}+\varepsilon_F\right)G
+i\big\langle T_C\,e^{-\frac{i}{\hbar}\int_Cd\tau''H(\tau'')}
[V,c(\bm r,\tau)]c^\dagger(\bm r',\tau')\big\rangle\\
);

&math(
&\left(i\hbar\frac{\PD}{\PD \tau}+\frac{\hbar^2}{2m}\nabla^2_{\bm r}+\varepsilon_F\right)G
=\hbar\delta(\tau-\tau')\delta^3(\bm r-\bm r')
+i\big\langle T_C\,e^{-\frac{i}{\hbar}\int_Cd\tau''H(\tau'')}
[V,c(\bm r,\tau)]c^\dagger(\bm r',\tau')\big\rangle\\
);

この &math(\delta); 関数を &math(g_0); で書き換えると、

(8.104)

&math(
&\left(i\hbar\frac{\PD}{\PD \tau}+\frac{\hbar^2}{2m}\nabla^2_{\bm r}+\varepsilon_F\right)G(\bm r,t,\bm r',t')\\
&=
\left(i\hbar\frac{\PD}{\PD \tau}+\frac{\hbar^2}{2m}\nabla^2_{\bm r}+\varepsilon_F\right)g_0(\bm r,t,\bm r',t')\\
&+\int_Cd\tau_1\int d^3r_1\hbar\delta(\tau-\tau_1)\delta^3(\bm r-\bm r_1)\times
\frac{i}{\textcolor{red}{\hbar}}\big\langle T_C\,e^{-\frac{i}{\hbar}\int_Cd\tau''H(\tau'')}
[V,c(\bm r_1,\tau_1)]c^\dagger(\bm r',\tau')\big\rangle\\
&=
\left(i\hbar\frac{\PD}{\PD \tau}+\frac{\hbar^2}{2m}\nabla^2_{\bm r}+\varepsilon_F\right)g_0(\bm r,t,\bm r',t')\\
&+\left(i\hbar\frac{\PD}{\PD \tau}+\frac{\hbar^2}{2m}\nabla^2_{\bm r}+\varepsilon_F\right)
\int_Cd\tau_1\int d^3r_1g_0(\bm r,t,\bm r',t')\times
\frac{i}{\textcolor{red}{\hbar}}\big\langle T_C\,e^{-\frac{i}{\hbar}\int_Cd\tau''H(\tau'')}
[V,c(\bm r_1,\tau_1)]c^\dagger(\bm r',\tau')\big\rangle\\
);

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