内積
$A$体系での計算
$A$体系の波動関数 \begin{align} \Phi(\boldsymbol{Z}_{1},\cdots,\boldsymbol{Z}_{A}) & = \frac{1}{\sqrt{A!}}\begin{vmatrix}\langle\boldsymbol{r}_{1}|\varphi_{1}\rangle & \cdots & \langle\boldsymbol{r}_{1}|\varphi_{A}\rangle\\ \vdots & \ddots & \vdots\\ \langle\boldsymbol{r}_{A}|\varphi_{1}\rangle & \cdots & \langle\boldsymbol{r}_{A}|\varphi_{A}\rangle\end{vmatrix}\\ & = \frac{1}{\sqrt{A!}}\det\{\varphi_{1}(\boldsymbol{r}_{1})\cdots\varphi_{A}(\boldsymbol{r}_{A})\}\\ \varphi & = \phi\chi\eta\\ \phi_{i}(\boldsymbol{r};\boldsymbol{Z}) & =\left(\frac{2\nu}{\pi}\right)^{3/4}\exp\left\{ -\nu\left(\boldsymbol{r}-\frac{\boldsymbol{Z}_{i}}{\sqrt{\nu}}\right)^{2}+\frac{\boldsymbol{Z}_{i}^{2}}{2}\right\} \label{1nucleon} \end{align} このように表わされるのだった。 mtkblog.hatenablog.com
この内積を計算しよう。 \begin{align} \bra{\Phi}|\ket{\Psi} & =\frac{1}{A!}\bra{\det\{\varphi_{1}(\boldsymbol{r}_{1})\cdots\varphi_{A}(\boldsymbol{r}_{A})\}}|\ket{\det\{\psi_{1}(\boldsymbol{r}_{1})\cdots\psi_{A}(\boldsymbol{r}_{A})\}}\\ & =\bra{\varphi_{1}(\boldsymbol{r}_{1})\cdots\varphi_{A}(\boldsymbol{r}_{A})}|\ket{\det\{\psi_{1}(\boldsymbol{r}_{1})\cdots\psi_{A}(\boldsymbol{r}_{A})\}}\\ & =\sum_{P_{1}\cdots P_{A}}\varepsilon(P_{1}\cdots P_{A})\bra{\varphi_{1}}|\ket{\psi_{P_{1}}}\cdots\bra{\varphi_{A}}|\ket{\psi_{P_{A}}}\\ & =\det\{\bra{\varphi_{i}}|\ket{\psi_{j}}\}\\ & =\det\{B_{ij}\} \end{align}
- 2体系で確認してみよう
$N\alpha$系での例
$\alpha$粒子はスピン$\chi=\{\uparrow,\ \downarrow\}$, アイソスピン$\eta=${proton, neutron}。 \begin{align} \bra{\Phi}|\ket{\Psi} & = \det\{B_{ij}\},\ (i,\ j=1,\cdots,A)\\ & =\begin{vmatrix}\boxed{b_{ij}}\\ & \boxed{b_{ij}}\\ & & \boxed{b_{ij}}\\ & & & \boxed{b_{ij}}\end{vmatrix}\\ & = \left|\ \boxed{b_{ij}}\ \right|^{4},\ (i,\ j = 1,\cdots,N) \end{align}
参考文献
- D.$\ $M. Brink, in Proceedings of the International School of Physics Enrico Fermi, Course 36, Varenna, edited by C. Bloch (AcademicPress, New York, 1966)
- Chapter III. Kernels of GCM, RGM and OCM and Their Calculation Methods | Progress of Theoretical Physics Supplement | Oxford Academic
*1:適宜、加筆します。
