Rotational Transitions

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

\subsection{Order of Magnitude Energies}

Molecules can rotate, and have rotational energies that depend on their angular momentum. If $J$ is the quantum angular momentum number, $$E_{rot} \sim \frac{\hbar^2 J (J+1)}{2I}$$ $$I \sim M L^2$$ For small $J$, $$E_{rot} \sim \frac{m}{M} E_{elec}$$

So the electronic, vibrational, and rotational energy states have contributions that scale with the electron-to-nucleus mass ratio: $$E_{elec} : E_{vib} : E_{rot} \sim 1: \left(\frac{m}{M}\right)^{1/2} : \left(\frac{m}{M}\right)$$

Rotational energies can be described using the angular momentum number $J$: $$E_{rot} = \frac{\hbar^2}{2I} J (J+1)$$

where $J = 0,\ 1,\ 2,\ ...$, $I$ is the moment of inertia, and for diatomic molecules, we can use the reduced mass $\mu$: $$I = \mu L^2$$ $$\mu = \frac{m_1 m_2}{m_1 + m_2}$$

\end{document} <\latex>