Skip to content

fix incorrect formula #49

New issue

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

Sign up for GitHub

By clicking “Sign up for GitHub”, you agree to our terms of service and privacy statement. We’ll occasionally send you account related emails.

Already on GitHub? Sign in to your account

Merged
jeanwsr merged 3 commits into from
Jul 8, 2024
Merged

fix incorrect formula #49

Show file tree
Hide file tree
Changes from all commits
Commits
Show all changes
3 commits
Select commit Hold shift + click to select a range
6b820ba
fix incorrect formula
Usu171 Jul 7, 2024
e79d4b9
update
Usu171 Jul 7, 2024
7530b19
update
Usu171 Jul 7, 2024
File filter

Filter by extension

Filter by extension
Conversations
Failed to load comments.
Loading
Jump to
Jump to file
Failed to load files.
Loading
Diff view
Diff view
20 changes: 10 additions & 10 deletions Chaps/Chap2.tex
View file
Open in desktop
Original file line number Diff line number Diff line change
Expand Up @@ -343,7 +343,7 @@ \subsection{反对称/Pauli不相容原理}
它是我们熟悉的Pauli不相容原理最常见的一种表述.
这个原理是量子力学的基本假设之一.
波函数不仅要满足\sch 方程,
也要满足反对称要求, 即\autoref{2.20}.
也要满足反对称要求, 即\autoref{2.22}.
以后我们就会明白, 反对称性可以用Slater行列式来保证.

\section{轨道, Slater行列式,基函数}
Expand Down Expand Up @@ -1479,7 +1479,7 @@ \subsection{\phrase{极小基 $\mathrm{H}_2$}的矩阵元}
常将所有双电子积分中的傀标都写成电子1和2的坐标。
现引入记号来表示含自旋轨道的双电子积分：
\begin{equation}
\braket{ij|kl}=\braket{\chi_i\chi_j|\chi_k\chi_l}=\int\dd\mathbf{x}_1\dd\mathbf{x}_2\chi_i^*(\mathbf{x}_1)\chi_j^*(\mathbf{x}_1)r_{12}^{-1}\chi_k*(\mathbf{x}_1)\chi_l(\mathbf{x}_2)
\braket{ij|kl}=\braket{\chi_i\chi_j|\chi_k\chi_l}=\int\dd\mathbf{x}_1\dd\mathbf{x}_2\chi_i^*(\mathbf{x}_1)\chi_j^*(\mathbf{x}_1)r_{12}^{-1}\chi_k(\mathbf{x}_1)\chi_l(\mathbf{x}_2)
\label{2.90}
\end{equation}
那么就有
Expand Down Expand Up @@ -2804,11 +2804,11 @@ \subsection{产生、湮灭算符及其反对易关系}
我们取(2.194)的伴随.
由于(见练习1.3)：
\begin{align}
\left( \mathscr{AB} \right)^\dagger = \mathscr{B}\dagger\mathscr{A}^\dagger
\left( \mathscr{AB} \right)^\dagger = \mathscr{B}^\dagger\mathscr{A}^\dagger
\end{align}
那么
\begin{equation}
a_ja_i + a_ia+j = 0 =\{a_j,a_i\}
a_ja_i + a_ia_j = 0 =\{a_j,a_i\}
\end{equation}
因此
\begin{align}
Expand Down Expand Up @@ -3235,9 +3235,9 @@ \subsection{自旋算符}

在通常的非相对论处理下(本书即如此),
哈密顿量中并无自旋坐标,
因此$\ts,\ts_z$都与哈密顿量对易：
因此$\ts^2,\ts_z$都与哈密顿量对易：
\begin{align}
[\hs \ts^2] = 0 = [\hs,\ts_z]
[\hs,\ts^2] = 0 = [\hs,\ts_z]
\end{align}
那么,
哈密顿量的精确本征函数也就是这两个自旋算子的本征函数：
Expand Down Expand Up @@ -3321,7 +3321,7 @@ \subsection{限制性行列式与自旋匹配组态}
\draw (0,0)--++(.5,0)node{$\uparrow$}-- ++(.5,0)node{$\downarrow$}-- ++(.5,0)node[right]{$\psi_1$};
\draw (0,.6)--++(.5,0)node{$\uparrow$}-- ++(.5,0)node{$\downarrow$}-- ++(.5,0)node[right]{$\psi_2$};
\draw (0,1.2)--++(.5,0)--node{} ++(.5,0)node{}-- ++(.5,0)node[right]{$\psi_3$};
\draw (0,1.8)--++(.5,0)node{$\uparrow$}-- ++(.5,0)node{$\uparrow$}-- ++(.5,0)node[right]{$\psi_4$};
\draw (0,1.8)--++(.5,0)node{$\uparrow$}-- ++(.5,0)node{$\downarrow$}-- ++(.5,0)node[right]{$\psi_4$};
\draw (0,2.4)--++(.5,0)node{}-- ++(.5,0)node{}-- ++(.5,0)node[right]{$\psi_5$};
\draw (0,3)--++(.5,0)node{}-- ++(.5,0)node{}-- ++(.5,0)node[right]{$\psi_5$};
\filldraw (0.75,3.4)circle(.5pt) (.75,3.6)circle(.5pt) (.75,3.8)circle(.5pt);
Expand Down Expand Up @@ -3365,7 +3365,7 @@ \subsection{限制性行列式与自旋匹配组态}
}

现来看开壳层限制性行列式.
开壳层行列式\emph{不是}$\ts_2$的本征函数,
开壳层行列式\emph{不是}$\ts^2$的本征函数,
除非所有开壳层电子的自旋都平行,
如图2.12中那样.
举个例子,
Expand All @@ -3375,7 +3375,7 @@ \subsection{限制性行列式与自旋匹配组态}
\begin{subequations}
\begin{align}
\ket{\Psi_1^{\bar{2}}} & = \ket{\bar{2}\bar{1}} = -2^{-1/2}[\psi_1(1)\psi_2(2) - \psi_2(1)\psi_1(2)]\beta(1)\beta(2)\\
\ket{\Psi_1^{{2}}} & = \ket{12} = -2^{-1/2}[\psi_1(1)\psi_2(2) - \psi_2(1)\psi_1(2)]\alpha(1)\alpha(2)\\
\ket{\Psi_{\bar{1}}^2} & = \ket{12} = -2^{-1/2}[\psi_1(1)\psi_2(2) - \psi_2(1)\psi_1(2)]\alpha(1)\alpha(2)
\end{align}
\end{subequations}
是$\ts^2$的本征函数,
Expand All @@ -3386,7 +3386,7 @@ \subsection{限制性行列式与自旋匹配组态}
\begin{subequations}
\begin{align}
\ket{\Psi_1^2} & = \ket{2\bar{1}}\\
\ket{\Psi_{\bar{1}}^{2}} & = \ket{1\bar{2}}
\ket{\Psi_{\bar{1}}^{\bar{2}}} & = \ket{1\bar{2}}
\end{align}
\end{subequations}
不是纯自旋态.
Expand Down
6 changes: 3 additions & 3 deletions Chaps/Chap3.tex
View file
Open in desktop
Original file line number Diff line number Diff line change
Expand Up @@ -2735,17 +2735,17 @@ \subsection{STO-3G下的H$_2$}
正则分子轨道构成分子点群的一个表示.
也就是说,
对于同核双原子分子,
轨道可以标为$\sigma_g$、$\sigma_\mu$、$\pi_g$、$\pi_\mu$等等.
轨道可以标为$\sigma_g$、$\sigma_u$、$\pi_g$、$\pi_u$等等.
在极小基下只有两个分子轨道.
能量低的那个就是占据分子轨道,
是具有$\sigma_g$对称的成键轨道
\begin{align}
\psi_1 = [2(1+S_{12})]^{-1/2}(\phi_1+\phi_2)
\end{align}
虚分子轨道就是对应的反键轨道,
对称性为$\sigma_\mu$:
对称性为$\sigma_u$:
\begin{align}
\psi_1 = [2(1-S_{12})]^{-1/2}(\phi_1-\phi_2)
\psi_2 = [2(1-S_{12})]^{-1/2}(\phi_1-\phi_2)
\end{align}
该问题的最终系数矩阵成为
\begin{align}
Expand Down
2 changes: 1 addition & 1 deletion Chaps/Chap4.tex
View file
Open in desktop
Original file line number Diff line number Diff line change
Expand Up @@ -729,7 +729,7 @@ \section{自然轨道与单粒子约化密度矩阵}
\end{align*}
其中$(\mathbf{d})_{ij}=d_i\delta_{ij}$. 请证明
\begin{align*}
\mathbf{U^\dagger CC\dagger U} = \mathbf{d}^2
\mathbf{U^\dagger CC^\dagger U} = \mathbf{d}^2
\end{align*}
\item 请证明
\begin{align*}
Expand Down