\documentclass[a4paper,10pt]{report} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} % Title Page \title{Phys 261} \author{The class} \begin{document} \maketitle \begin{abstract} Here are the notes from the class. They are written i realtime. \end{abstract} \section{Third lecture 29.08.06} \subsection{Atomic world construction} $a_0$ - radius, length $\Delta x \approx a_0$ $\Delta k \approx 1/a_0$ ($\hbar \Delta k = \Delta p$ wavenumber $\Delta E \approx \frac{(\Delta p )^2}{2m} = \frac{\hbar ^2}{2ma_0^2} \approx \frac{\hbar ^2}{ma_0^2}$ (kinetic energy T$_0$. ) Potential energy V$_0$ $|-\frac{e^2}{a_0}$ $a_0 = 0,529$ \AA = $\frac{\hbar^2}{e^2m}$ Lenght unit Energy unit $\frac{e^2}{a_0} = \frac{\hbar}{ma_0^2} = 27,2$ eV $$ = $$ should be in the same order. Atomic unit of time Atomic unit of velocity $p_0 = \hbar k_0 = \frac{\hbar}{a_0} v_0 = \frac{p_0}{m} = \frac{\hbar}{m}\frac{1}{a_0} = \frac{me^2}{\hbar ^2}\frac{\hbar}{m} = \frac{e^2}{\hbar} \frac{v_0}{c} = \frac{e^2}{\hbar c} = \alpha = \frac{1}{137}$ $t_0 =\frac{a_0}{v_0} = (\frac{a_0}{e^2}) \hbar = \frac{\hbar}{E_0}$ Alternative postulate $t_0 = \frac{\hbar}{E-O}$ (Statement a.u. $\longleftrightarrow e = m \hbar = 1$) $\hbar = 0,66 \cdot 10^{-15}$ eVs $t_0 = \frac{0,66 \cdot 10^{-15}}{27,2}$s=$0,24 \cdot 10^{-15}$ (2 $\pi$ for angular freq.) $\nu$ frequency | $\omega$ - angular frequency $k_0 T$ is the ``physical temperature''. Room temperature is thus $\frac{1}{40}$eV or 25 meV Atomic unit of energy $\longrightarrow$ VERY HOT More about bound states in H Ground state $-\frac{1}{2}$ a.u. States characterized by $n, l$ ($m$ - magnetic) In H energies given by $\frac{1}{n^2}(-\frac{1}{2})$a.u **Sett inn bilde av matrise og brønnpotensiale har bilder** Ladi says its nonsens to talk about m as a quantum number. Angular momentum $L = \omega \varphi$ $T_{rot} = \frac{1}{2} \frac{L^2}{g}$ $E=t+V$ is negativ ** bane bilde ** $I_n$ QM it looks different 3-dim Schr.Eq. $\rightarrow$ Seperation of variables $x, y, z, \rightarrow r, \nu , \varphi | \frac{\delta ^2}{\delta x^2}+\frac{\delta ^2}{\delta y^2}+\frac{\delta ^2}{\delta z^2} \longrightarrow T_r +\frac{L^2(\nu , \varphi )}{r^2}$ $T_r \longrightarrow \frac{1}{r}\frac{\delta}{\delta r}\frac{1}{r}\frac{\delta}{\delta r}$ $L^2$ is ugly (can be made very elegant) This is generally used in many fields. Exercise: Look on the separation and how it's done. (0)s-states, (1)p-states, (2)d-states, (3)f-states, ... there are more, but it is irrelevant here \end{document}