Difference between revisions of "Rovibrational Transitions"

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\usepackage{eufrak}
 
\usepackage{eufrak}
 
\begin{document}
 
\begin{document}
\subsection*{ Ro-Vibrational Transitions }
+
\section*{ Ro-Vibrational Transitions }
  
 +
\subsection{Vibrational Transitions in Molecules}
 +
 +
Our general technique with vibrational transitions is to model them as
 +
harmonic oscillators.  Thus, they should have the characteristic harmonic
 +
energy series:
 +
$$E_n=(n+\frac12)\hbar\omega_0$$
 +
For a harmonic oscillator, $\omega_0=\sqrt{k\over m}$.  We estimate that since
 +
the force for a spring is $k\cdot x$, and that force should be about the
 +
Coulomb force on $e^-$'s.  If we say that atoms stretch with respect to
 +
each other about a Bohr radius:
 +
$$ka_0\sim{e^2\over a_0^2}$$
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$$\Delta E\big|_{vib\atop trans}\sim Ryd\cdot\sqrt{m_e\over A\cdot m_p}$$
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where A is the atomic mass \# of our atoms.
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 +
\subsection{Rotational Transitions in Molecules}
 +
 +
The thing to remember is that angular momentum comes in units of $\hbar$.
  
 
\end{document}
 
\end{document}
 
<\latex>
 
<\latex>

Revision as of 16:16, 9 November 2015

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<latex> \documentclass[11pt]{article} \usepackage{fullpage} \usepackage{amsmath} \usepackage{eufrak} \begin{document} \section*{ Ro-Vibrational Transitions }

\subsection{Vibrational Transitions in Molecules}

Our general technique with vibrational transitions is to model them as harmonic oscillators. Thus, they should have the characteristic harmonic energy series: $$E_n=(n+\frac12)\hbar\omega_0$$ For a harmonic oscillator, $\omega_0=\sqrt{k\over m}$. We estimate that since the force for a spring is $k\cdot x$, and that force should be about the Coulomb force on $e^-$'s. If we say that atoms stretch with respect to each other about a Bohr radius: $$ka_0\sim{e^2\over a_0^2}$$ $$\Delta E\big|_{vib\atop trans}\sim Ryd\cdot\sqrt{m_e\over A\cdot m_p}$$ where A is the atomic mass \# of our atoms.

\subsection{Rotational Transitions in Molecules}

The thing to remember is that angular momentum comes in units of $\hbar$.

\end{document} <\latex>