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{\footnotesize This note was written in "realtime" by Ingjald Pilskog.
Edited by L. K.}

{\bf \Large Third lecture 29.08.06}

{\it 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$_0$
$|-\frac{e^2}{a_0}$

{\bf Atomic unit of length }


$a_0 = 0,529$ \AA = $\frac{\hbar^2}{e^2m}$ Lenght unit

{\it Atomic unit of energy }

Energy unit $\frac{e^2}{a_0} = \frac{\hbar}{ma_0^2} = 27,2$ eV

$<T_0>$ = $<V_0>$ should be of the same order.

{\it 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} = $  \ \ \ \ \ \ \ \ \
$\frac{me^2}{\hbar ^2}\frac{\hbar}{m} = \frac{e^2}{\hbar}$  \ \ \ \ \ \ \ \ \
$\frac{v_0}{c} = \frac{e^2}{\hbar c} = \alpha = \frac{1}{137}$


{\it Atomic unit of time }


$t_0 =\frac{a_0}{v_0} = (\frac{a_0}{e^2}) \hbar = \frac{\hbar}{E_0}$

Alternative postulate $t_0 = \frac{\hbar}{E_0}$ 

(Statement a.u. $\longleftrightarrow e = m \hbar = 1$ is useless)

$\hbar = 0,66 \cdot 10^{-15}$ 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
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{\bf More about bound states in H }
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Ground state $-\frac{1}{2}$ a.u.

\includegraphics[height=4.5cm]{klasse1.pdf}   

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States characterized by $n, l$ ($m$ - magnetic)

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In H energies given by $\frac{1}{n^2}(-\frac{1}{2})$a.u.

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"Ladi says its nonsens to talk about m as a quantum number",
wrote Ingjald. He did not write: "unless we have magnetic field on".

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Angular momentum

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$L = \omega {\cal I} $ \ \ \ \ \ \ \ \ \
$T_{rot} = \frac{1}{2} \frac{L^2}{{\cal I}}$  \ \ \ \ \ \ \ \ \
$T_{rot} = \frac{1}{2}   {\cal I} \omega^2 $  \ \ \ \ \ \ \ \ \
$ {\cal I} $  is the {\it moment of inertia }

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$E=T+V$ is negativ   .... $T$  is kinetic energy



\includegraphics[height=5.0cm]{klasse2.pdf}   



\includegraphics[height=2.5cm]{klasse3.pdf}   


$I_n$ QM it looks different

3-dim Schr.Eq. $\rightarrow$ Seperation of variables

$x, y, z, \rightarrow r, \nu , \varphi $

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$ \frac{\delta ^2}{\delta x^2}+\frac{\delta ^2}{\delta y^2}+\frac{\delta ^2}{\delta z^2} \longrightarrow T_r +\frac{L^2(\theta , \varphi )}{r^2}$

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$T_r \longrightarrow \frac{1}{r^2}\frac{\partial}{\partial r}{r^2}\frac{\partial}{\partial r}$ 

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$L^2$ is ugly (but can be made very elegant)

This is generally used in many fields.

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$$
\frac{L^2(\theta , \varphi )}{r^2} \longrightarrow 
  \frac{1}{r^2} \frac{1}{ sin \theta} \frac{\partial}{\partial \theta}
    \left( sin \theta \frac{\partial \psi}{\partial \theta} \right)
  + \frac{1}{r^2 sin^2 \theta} \frac{\partial^2 \psi}{\partial \phi^2}
$$

Solutions of the above are $Y_l^m(\theta, \phi)$,
known as the spherical harmonics. (The constants were not taken care of)

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From the web ("stolen latex code")

{\tt http://vergil.chemistry.gatech.edu/notes/quantrev/node25.html }


 \begin{equation}
- \frac{\hbar^2}{2 \mu} \left[ \frac{1}{r^2} \frac{\partial}{\partial r}
  \left( r^2 \frac{\partial \psi}{\partial r} \right)
  \frac{1}{r^2 sin \theta} \frac{\partial}{\partial \theta}
    \left( sin \theta \frac{\partial \psi}{\partial \theta} \right)
  + \frac{1}{r^2 sin^2 \theta} \frac{\partial^2 \psi}{\partial \phi^2}
  \right] - \frac{e^2}{4 \pi \epsilon_0 r} \psi(r, \theta, \phi) =
  E \psi(r, \theta, \phi)
\end{equation}

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Exercise:
Look on the separation of variables and how it's done.

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(l=0) s-states, 

(l=1) p-states, 

(l=2) d-states, 

(l=3) f-states, ... there are more, but it is not relevant in typical atoms



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