Talk:Rigid rotor

Untitled
The rigid rotor article is too limited (only linear rotors). Does anybody object to me extending it? (P.wormer)

I finished extending it P.wormer 13:38, 19 November 2006 (UTC)

Units
A previous author wrote \bar B in wavenumbers by dividing B by c (speed of light). An author with IP from UCLA changed this. Now the bar on top of B and the sentence about wavenumbers don't make sense. Please, anonymous from LA look at it more deeply.--86.81.145.23 09:50, 12 December 2006 (UTC) (= P.wormer)

Waited 10 days and put units straight. --P.wormer 13:42, 22 December 2006 (UTC) (P.wormer)

Outside link
A link to http://www.teraproofs.com/rigid_rotor.html has been removed from this page by "Jouster." Since the owner of the website posted it, he has taken it upon himself to remove the link. It has a comprehensive fully worked out solution to the rigid rotor problem. Please check it out, and if any of you think it is worth looking at, please link to it on the main pg.


 * I had no problems with the link to teraproofs.com. Contents are OK. My only point of criticism is the number of ads on the teraprooofs.com site. For me that wasn't enough reason to remove this outside link.--P.wormer 14:02, 24 May 2007 (UTC)

Conversion of Angular velocity
Following the explanations in Goldstein 3rd Edition page 174 (end of section 4.9), to write the Angular velocity form, I obtained the general structure of the conversion matrix

\begin{pmatrix} \omega_x \\ \omega_y \\ \omega_z \\ \end{pmatrix} = M \begin{pmatrix} \dot{\alpha} \\ \dot{\beta} \\ \dot{\gamma} \\ \end{pmatrix}. $$ If the rotation is described in the form $$\boldsymbol{R}(\alpha,\beta,\gamma)=R_1(\hat{e}_1, \alpha)R_2(\hat{e}_2, \beta)R_3(\hat{e}_3, \gamma)$$ where $$\hat{e}_1,\hat{e}_2,\hat{e}_3$$ are the axes of the convention (ZXZ means $$\hat{e}_1=\hat{e}_3=\hat{Z}, \hat{e}_2=\hat{X}$$) and $$(\hat{b}_1,\hat{b}_2,\hat{b}_3)$$ are the body-axes, the general form of the conversion matrix that I obtain is

\begin{pmatrix} \hat{e}_1\cdot(\boldsymbol{R}\hat{X}) & (R_1\hat{e}_2)\cdot(\boldsymbol{R}\hat{X}) & 0 \\ \hat{e}_1\cdot(\boldsymbol{R}\hat{Y}) & (R_1\hat{e}_2)\cdot(\boldsymbol{R}\hat{Y}) & 0\\ \hat{e}_1\cdot(\boldsymbol{R}\hat{Z}) & (R_1\hat{e}_2)\cdot(\boldsymbol{R}\hat{Z}) & 1\\ \end{pmatrix} $$

This is obtained by writing the angular velocities of each angle in its vectorial form
 * $$ \dot{\boldsymbol{\alpha}} = \dot{\alpha}\hat{e}_1 $$
 * $$ \dot{\boldsymbol{\beta}} = \dot{\beta} \hat{N} = \dot{\beta} R_1\hat{e}_2 $$
 * $$ \dot{\boldsymbol{\gamma}} = \dot{\gamma}\hat{b}_3 = \dot{\gamma} R\hat{e}_3 $$

Where $$\hat{N}$$ is the line of nodes (the new position of the second axis of the convention, after applying the first rotation). the last step is projecting over the body-axes
 * $$ \hat{b}_1 = \boldsymbol{R}\hat{X} $$
 * $$ \hat{b}_2 = \boldsymbol{R}\hat{Y} $$
 * $$ \hat{b}_3 = \boldsymbol{R}\hat{Z} $$

With this setup, when using the convention ZXZ (which is the one used in Goldstein) we get

\begin{pmatrix} \hat{Z}\cdot(\boldsymbol{R}\hat{X}) & (R_1\hat{X})\cdot(\boldsymbol{R}\hat{X}) & 0 \\ \hat{Z}\cdot(\boldsymbol{R}\hat{Y}) & (R_1\hat{X})\cdot(\boldsymbol{R}\hat{Y}) & 0\\ \hat{Z}\cdot(\boldsymbol{R}\hat{Z}) & (R_1\hat{X})\cdot(\boldsymbol{R}\hat{Z}) & 1\\ \end{pmatrix} = \begin{pmatrix} \sin\beta\sin\gamma & \cos\gamma & 0 \\ \sin\beta\cos\gamma & -\sin\gamma & 0 \\ \cos\beta & 0 & 1\\ \end{pmatrix} $$ Which gives equations (4.87) of Goldstein
 * $$ \omega_{x} = \dot{\alpha}\sin\beta\sin\gamma + \dot{\beta}\cos\gamma $$
 * $$ \omega_{y} = \dot{\alpha}\sin\beta\cos\gamma - \dot{\beta}\sin\gamma $$
 * $$ \omega_{z} = \dot{\alpha}\cos\beta + \dot{\gamma} $$

Using the convention ZYZ, which is the one of this article we obtain the matrix reported here

\begin{pmatrix} \hat{Z}\cdot(\boldsymbol{R}\hat{X}) & (R_1\hat{Y})\cdot(\boldsymbol{R}\hat{X}) & 0 \\ \hat{Z}\cdot(\boldsymbol{R}\hat{Y}) & (R_1\hat{Y})\cdot(\boldsymbol{R}\hat{Y}) & 0\\ \hat{Z}\cdot(\boldsymbol{R}\hat{Z}) & (R_1\hat{Y})\cdot(\boldsymbol{R}\hat{Z}) & 1\\ \end{pmatrix} = \begin{pmatrix} -\sin\beta\cos\gamma & \sin\gamma & 0 \\ \sin\beta\sin\gamma & \cos\gamma & 0 \\ \cos\beta & 0 & 1\\ \end{pmatrix} $$ Kakila (talk) 18:21, 23 October 2011 (UTC)


 * I'm 10 years late to your comment, but this is actually nice and formulas for some other Euler angle convention could be helpful in the article. The Z-Y-Z one used is common but not at all universal. 178.165.175.226 (talk) 22:01, 1 August 2021 (UTC)

Some suggestions
1. Should the article be broken into two, for the classical rigid rotor and the quantum rigid rotor? One page for classical mechanics and one for quantum? The current form goes back and forth and can be somewhat confusing.

2. I think the section on the nonlinear quantum rotor should be expanded. First, it stops in the body-fixed frame, which is confusing because it mentions the 2J+1 degeneracy of EJ0 but does not explain it's origin. Emphasizing this case obscures that the the +/-k levels are also degenerate. Non-zero k has a total degeneracy of 2(2J+1). The section on linear rotors gives the analytical form of the eigenfunctions, i.e. the spherical harmonics (Ylm). Why not include similar information for the nonlinear (symmetric) rotors, for which the eigenfunctions can also be expressed analytically? A have not yet looked up a detailed treatment of the nonlinear rotor to reference. Herzberg (vol II p. 26) cites Dennison (Rev Mod Phys 3 280 1931) and Mulliken (Phys Rev 59 873 1941). These might work, but there must be more modern treatments too.

Zolot (talk) 00:10, 13 May 2010 (UTC)

Coordinates
An anonymous with an IP adress from BRIGHAM-YOUNG-UNIVERSITY-IDAHO changed in the lede  "three Euler angles" to "two angles and a radius". In other words, according to our Mormon friend the coordinates of an arbitrarily shaped rigid rotor would be the same as those of a point particle. Nothing was changed in other sections, so that the lede is now in flagrant contradiction to the body of the article. I wonder, is this change in the lede subtle vandalism, or stupidity? In either case, it brings back to me the reasons why I stopped contributing to WP.--P.wormer (talk) 05:55, 21 September 2012 (UTC)
 * I have now corrected this error. However please be more patient with other editors. This was clearly a "good-faith edit", meaning a case of honest confusion rather than vandalism. After all, the bond length of a diatomic molecule is mentioned further in the article, so I think the problem was that the editor just read the sentence too quickly before modifying it. In any case ad hominem attacks such as "our Mormon friend" are clearly out of line. Dirac66 (talk) 21:59, 31 January 2013 (UTC)

linear vs diatomic
"The linear rigid rotor model can be used in quantum mechanics to predict the rotational energy of a diatomic molecule." Can't it be used on any linear molecule? For example, CO2 is linear but no diatomic. Mverleg (talk) 02:36, 4 October 2013 (UTC)

Dependence of energy on m
The article states: "Note that the energy [of a linear rotor] does depend on m through I." This is nonsense, the whole article is about rigid (linear) rotors, which by definition have a constant geometry, independent of their quantum state. Consequently, the moment of inertia I, by definition depending on geometry only, is independent of the quantum state and does not depend on the quantum number m.

Even if the dependence of the geometry of a non-rigid linear rotor on its quantum state were to be considered, the inertia moment I would still not dependent on m. The geometry dependence would be through the centrifugal force that would stretch the rotor depending on angular velocity/momentum. The centrifugal force is proportional to the square of the total angular momentum  and is independent of m. Stated equivalently, the rotational energy and velocity of a linear rotor depend on the quantum number $$\ell$$ only and not on the quantum number m. P.wormer (talk) 14:27, 16 April 2024 (UTC)