Attachment 156 Details for Bug 1737 – 0001737-curs.tex

0001737-curs.tex

0001737-curs.tex (text/x-tex), 3.24 KB, created by inger@altlinux.org on 2002-12-19 15:22:06 MSK

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>% íÁÔÅÒÉÁÌÙ ÄÌÑ ËÕÒÓÏ×ÏÊ
>\documentclass[a4paper]{article}
>\usepackage[koi8-r]{inputenc}
>\usepackage[english,russian]{babel}
>\usepackage{amsmath,amssymb}
>
>\title{òÁÓÞÅÔ ÓÏÓÔÏÑÎÉÊ ÁÔÏÍÁ ÇÅÌÉÑ}
>\author{óÔÁÎÉÓÌÁ× éÅ×ÌÅ×}
>
>%ÏÂÝÉÅ ÏÐÒÅÄÅÌÅÎÉÑ
>\let\vect = \overrightarrow
>\let\f = \varphi
>\begin{document}
>\maketitle
>
>\section{ôÅÏÒÅÔÉÞÅÓËÉÅ ÏÓÎÏ×Ù}
>\paragraph{íÁÔÅÍÁÔÉÞÅÓËÉÊ ÁÐÐÁÒÁÔ Ë×ÁÎÔÏ×ÏÊ ÍÅÈÁÎÉËÉ}
>ó×ÏÊÓÔ×Á Ë×ÁÎÔÏ×ÙÈ ÏÂßÅËÔÏ× ÏÐÉÓÙ×ÁÀÔÓÑ Ó ÐÏÍÏÝØÀ ×ÏÌÎÏ×ÏÊ
>ÆÕÎËÃÉÉ. ÷ÏÌÎÏ×ÁÑ ÆÕÎËÃÉÑ Ñ×ÌÑÅÔÓÑ ËÏÍÐÌÅËÓÎÏÊ, ÏÄÎÏÚÎÁÞÎÏÊ É
>ÎÅÐÒÅÒÙ×ÎÏÊ ÆÕÎËÃÉÅÊ ÒÁÄÉÕÓ-×ÅËÔÏÒÁ ÞÁÓÔÉÃÙ  $r$ É ×ÒÅÍÅÎÉ $t$.
>\begin{gather}
>\label{wave} \Psi=\Psi(\vect{r},t) \\
>\label{vector_func} \vect{r}=\vect{r}(x,y,z)
>\end{gather}
>ðÏÄÓÔÁ×É× × ×ÙÒÁÖÅÎÉÅ ~(\ref{wave}) ×ÅËÔÏÒ-ÆÕÎËÃÉÀ ~(\ref{vector_func}), ×ÏÌÎÏ×ÕÀ ÆÕÎËÃÉÀ ÍÏÖÎÏ
>ÚÁÐÉÓÁÔØ × ×ÉÄÅ $\Psi=\Psi(x,y,z,t)$.
>ë×ÁÄÒÁÔ ÍÏÄÕÌÑ ×ÏÌÎÏ×ÏÊ ÆÕÎËÃÉÉ ×ÚÑÔÙÊ × ÔÏÞËÅ $(x,y,z)$ × ÍÏÍÅÎÔ
>×ÒÅÍÅÎÉ t, ÕÍÎÏÖÅÎÎÙÊ ÎÁ ÜÌÅÍÅÎÔÁÒÎÙÊ ÏÂßÅÍ
>$\triangle x\triangle y\triangle z$, 
>×ËÌÀÞÁÀÝÉÊ ÜÔÕ ÔÏÞËÕ ÄÁÅÔ ×ÅÒÏÑÔÎÏÓÔØ ÔÏÇÏ, ÞÔÏ × ÍÏÍÅÎÔ ×ÒÅÍÅÎÉ t ÍÙ ÏÂÎÁÒÕÖÉÍ ÞÁÓÔÉÃÕ × ÜÔÏÍ ÏÂßÅÍÅ.
>åÓÌÉ ÄÏÐÏÌÎÉÔÅÌØÎÏ ×ÙÐÏÌÎÅÎÏ ÕÓÌÏ×ÉÅ ÎÏÒÍÉÒÏ×ËÉ
>$\int\limits_V|\Psi(x,y,z,t)|^2\,dx\,dy\,dz=1$, ÔÏ Ë×ÁÄÒÁÔ ÍÏÄÕÌÑ
>ÍÏÖÅÔ ÂÙÔØ ÎÁÚ×ÁÎ ÐÌÏÔÎÏÓÔØÀ ×ÅÒÏÑÔÎÏÓÔÉ.
>÷ÓÑ ÒÁÂÏÔÁ ÐÒÏÉÚ×ÏÄÉÔÓÑ × ÌÉÎÅÊÎÏÍ ÐÒÏÓÔÒÁÎÓÔ×Å ÆÕÎËÃÉÊ. óËÁÌÑÒÎÏÅ
>ÐÒÏÉÚ×ÅÄÅÎÉÅ ÏÐÒÅÄÅÌÅÎÏ ÓÌÅÄÕÀÝÉÍ ÏÂÒÁÚÏÍ:
>\begin{equation}
>\label{scal} \langle{f,g}\rangle=\int f(r)^*g(r)\,dr
>\end{equation}
>ÇÄÅ $x^*$ ÏÐÅÒÁÃÉÑ ËÏÍÐÌÅËÓÎÏÇÏ ÓÏÐÒÑÖÅÎÉÑ
>ëÁÖÄÏÊ ÆÉÚÉÞÅÓËÏÊ ×ÅÌÉÞÉÎÅ ÓÏÏÔ×ÅÔÓÔ×ÕÅÔ ÎÅËÏÔÏÒÙÊ ÌÉÎÅÊÎÙÊ
>ÏÐÅÒÁÔÏÒ.
>óÒÅÄÎÅÅ ÚÎÁÞÅÎÉÅ ÆÉÚÉÞÅÓËÏÊ ×ÅÌÉÞÉÎÙ, ÚÁÄÁÎÎÏÊ ÏÐÅÒÁÔÏÒÏÍ L
>×ÙÞÉÓÌÑÅÔÓÑ ÐÏ ÆÏÒÍÕÌÅ
>\begin{equation}
>\label{avg} \langle{f}\rangle=\int\psi^*(r)L\psi(r)\,dr
>\end{equation}
>, ÇÄÅ $\psi$ - ÎÏÒÍÉÒÏ×ÁÎÎÁÑ ×ÏÌÎÏ×ÁÑ ÆÕÎËÃÉÑ
>%TODO: ÎÁÐÉÓÁÔØ ÅÝÅ ÐÒÏ ÎÏÒÍÉÒÏ×ËÕ ÎÁ ÂÅÓËÏÎÅÞÎÏÓÔÉ ÐÏ ÄÅÌØÔÁ-ÆÕÎËÃÉÉ
> á ÓÏÓÔÏÑÎÉÅ ÞÁÓÔÉÃÙ ÏÐÉÓÙ×ÁÅÔÓÑ ÕÒÁ×ÎÅÎÉÑÍÉ, ÓÏÄÅÒÖÁÝÉÍÉ
>ÏÐÅÒÁÔÏÒÙ. òÅÛÅÎÉÅ ÔÁËÏÇÏ ÕÒÁ×ÎÅÎÉÑ - ×ÏÌÎÏ×ÁÑ ÆÕÎËÃÉÑ. ÷ÏÔ ÎÅÓËÏÌØËÏ
>ÏÓÎÏ×ÎÙÈ ÏÐÅÒÁÔÏÒÏ× ÆÉÚÉÞÅÓËÉÈ ×ÅÌÉÞÉÎ:\\
>\begin{tabular}
> {|l|l|l|}
>  \hline æÉÚ. ×ÅÌÉÞÉÎÁ & æÏÒÍÕÌÁ & ïÐÅÒÁÔÏÒ \\
>  \hline ëÏÏÒÄÉÎÁÔÁ & 
>   $\begin{cases}
>    r \\
>    x,y,z \\
>   \end{cases}$ &
>   $\begin{cases}
>    r  \\
>    x,y,z
>   \end{cases}$ \\
>  \hline éÍÐÕÌØÓ &
>   $\begin{cases}
>    {\bf p} \\
>     p_x, p_y, p_z \\
>   \end{cases}$ &
>   $\begin{cases}
>    -i\hbar\nabla \\
>    -i\hbar\frac{\partial }{\partial x},-i\hbar\frac{\partial}{\partial y},-i\hbar\frac{\partial }{\partial z} 
>   \end{cases}$ \\
>   \hline õÇÌÏ×ÏÊ ÍÏÍÅÎÔ&
>   $\begin{cases}
>    L=[r\times p] \\
>    L_x=yp_z-zp_y\\
>    L_y=zp_x-xp_z \\
>    L_z=xp_y-yp_x
>   \end{cases}$ &
>   $\begin{cases}
>    \hat{L}=-i\hbar[r\times\nabla] \\
>    \hat{L}_x=-i\hbar(y\frac{\partial }{\partial
>    z}-z\frac{\partial}{\partial y}) \\
>    \hat{L}_y=-i\hbar(z\frac{\partial }{\partial
>    x}-x\frac{\partial}{\partial z}) \\
>    \hat{L}_z=-i\hbar(x\frac{\partial }{\partial y}-y\frac{\partial}{\partial z})
>   \end{cases}$ \\
>   \hline ðÏÌÎÁÑ ÜÎÅÒÇÉÑ &
>    $E=\frac{p^2}{2m}+U(r)$ &
>    $\hat{H}=-\frac{\hbar^2}{2m}\triangle+U(r)$ \\
>   \hline
>\end{tabular}\\
>÷ ÔÁÂÌÉÃÅ $\triangle=\nabla^2=\frac{\partial^2}{\partial
>  x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial
>  x^2}$ - ÏÐÅÒÁÔÏÒ ìÁÐÌÁÓÁ (ÌÁÐÌÁÓÉÁÎ). ïÐÅÒÁÔÏÒ ÜÎÅÒÇÉÉ $\hat{H}$
>  ÎÁÚÙ×ÁÅÔÓÑ ÔÁËÖÅ ÇÁÍÉÌØÔÏÎÉÁÎ.
>\end{document}

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