## The Overall Movement

The angle \(\theta\) has two meanings in our inertial frame. It is both coordinate and angle of the **molecular axis** to the electric field. Using the expression for the conserved **angular momentum** \(p_{\varphi}\), we find a relation between the angular velocity \(\dot{\varphi}\) and the angle \(\theta\):\[\begin{eqnarray*}\dot{\varphi} & = & \frac{p_{\varphi}}{mR^{2}\sin^{2}\theta}\end{eqnarray*}\] Note that neither \(\theta\) nor \(\dot{\varphi}\) is constant in general!

So, how can we employ this relationship to understand the movement? We may replace \(\dot{\varphi}\) entirely with terms in \(\theta\). Next we can use the second conserved quantity, namely the **energy** \(\mathrm{E}\). Remembering that \(\mathrm{E}=T+V\) we find \[\begin{eqnarray*}\mathrm{E} & = & \frac{mR^{2}}{2}\left(\dot{\theta}^{2}+\sin^{2}\theta\dot{\varphi}^{2}\right)-qRE_{0}\cos\theta\\& = & \frac{mR^{2}}{2}\left(\dot{\theta}^{2}+\frac{p_{\varphi}^{2}}{m^{2}R^{4}\sin^{2}\theta}\right)-qRE_{0}\cos\theta\ .\end{eqnarray*}\]Now this formula is extremely useful. Using it we can entirely understand the movement of the molecule. Let us for example set \(p_{\varphi}=0\) for the moment. We find that \(\dot{\varphi}=0\) and\[\begin{eqnarray*}\mathrm{E}\left(p_{\varphi}=0\right) & = & \frac{mR^{2}}{2}\dot{\theta}^{2}-qRE_{0}\cos\theta

\end{eqnarray*}\]which is an **oscillator equation** for small \(\theta\). So, for small energies the molecule oscillates back and forth in \(\theta\).

But what about a **non-vanishing** **angular momentum** \(p_{\varphi}\neq0\)? Here one can look at the "**effective potential**''. This concept originates from **celestial mechanics**. There, one is for example regarding the orbital motion of two bodies under the mutual influence of gravitation. If this motion has a certain angular momentum, the bodies may never collide.

In our case, the situation is just like that. Rearranging terms we find\[\begin{eqnarray*}\frac{mR^{2}}{2}\dot{\theta}^{2} & = & \mathrm{E}-\frac{p_{\varphi}^{2}}{2mR^{2}\sin^{2}\theta}+qRE_{0}\cos\theta\\& \equiv & \mathrm{E}-V_{\mathrm{eff}}\left(\theta\right)\ .\end{eqnarray*}\]Let us take some **characteristic quantities** for molecular systems to look at \(V_{\mathrm{eff}}\): \(p_{\varphi}=\hbar=6.63\times10^{-34}\text{Js}\) (**Planck** constant), \(m=m_{p}=1.67\times10^{-27}\text{kg}\) (proton mass), \(R=a_{0}=5.29\times10^{-10}\text{m}\) (Bohr radius) and \(q=e=1.60\times10^{-19}\text{C}\) (elementary charge). Then,\[\begin{eqnarray*}V_{\mathrm{eff}}\left(\theta\right) & = & \frac{\hbar^{2}}{2m_{p}a_{0}^{2}\sin^{2}\theta}-ea_{0}E_{0}\cos\theta\\& \approx & \frac{1.19\times10^{-21}\text{J}}{\sin^{2}\theta}-8.48\times10^{-30}\text{A s m}\cdot E_{0}\cos\theta\end{eqnarray*}\] and we can see that an electric field has to be as big as \(10^{9\dots10} \text{V/m}\) to actually have an effect!

In the figure you can see how \(V_{\mathrm{eff}}\) changes for different \(E_{0}\) from \(10^{10}\text{V/m}\) (blue curve) to \(6\times10^{10}\text{V/m}\) (green). At different energies \(\mathrm{E}\) the molecule oscillates in a larger \(\theta\)-range. But similarly to the case of orbital motion, the points at \(\theta=0\) or \(\pi\) are never reached since \(p_{\varphi}\neq0\). Along with the angular movement in \(\varphi\), we can conclude that the molecule is performing a kind of **precession**.

We have seen that the details of the motion depend on the balance between the conserved quantities and the strength of the applied electric field.