Angular Systems and Rotors

A particle constrained to a ring or a sphere has angular motion without a radial degree of freedom. These models make periodicity, angular momentum, probability measure, and rotational spectra concrete. The same mathematics describes an ideal rigid molecule whose internal separation is fixed.

Quantized rotation and angular probability

For a particle on a ring with moment of inertia $I$, periodicity requires

$$\psi_m(\phi)=\frac{e^{im\phi}}{\sqrt{2\pi}},\qquad m\in\mathbb Z,\qquad E_m=\frac{\hbar^2m^2}{2I}.$$

The wavefunctions are normalized with $d\phi$. States $m$ and $-m$ have the same energy but opposite angular momentum and opposite probability current. Equal density does not imply identical states.

On a sphere, the eigenfunctions are spherical harmonics, and

$$E_\ell=\frac{\hbar^2\ell(\ell+1)}{2I},\qquad \ell=0,1,\ldots,\qquad m=-\ell,\ldots,\ell.$$

Here the normalization uses $d\Omega=\sin\theta\,d\theta\,d\phi$. This distinction is visible even in the isotropic ground state $Y_{00}=1/\sqrt{4\pi}$. Its probability density per unit solid angle is constant, while its polar-angle marginal is

$$p(\theta)=\int_0^{2\pi}|Y_{00}|^2\sin\theta\,d\phi =\frac12\sin\theta.$$

Equal polar-angle intervals near the equator cover more area than equal intervals near a pole. Isotropy therefore does not mean a uniform histogram in $\theta$.

For an ideal linear rigid molecule, the same spectrum is written $E_J=B_{\mathrm{rot}}J(J+1)$, with the energy constant $B_{\mathrm{rot}}=\hbar^2/(2I)$. Spectroscopy also uses rotational constants in frequency or wavenumber units; check the units before applying a formula. Molecules with identical nuclei require the exchange symmetry of the complete molecular state, including nuclear spin, when deciding which rotational states occur and with what statistical weights.

Reading sequence

Page	What you will learn
Particle on a Ring	Periodic domain, integer angular momentum, spectrum, current, and flux preview
Particle on a Sphere	Angular Laplacian, spherical-harmonic eigenstates, and $2\ell+1$ degeneracy
Rigid Rotor	Molecular moment of inertia, rotational Hamiltonian, and ideal spectrum
Spherical Harmonics as Wavefunctions	Normalization, phases, nodes, real bases, and plots
Rotational Spectra	Rotational constants, dipole selection rules, line positions, and model limits
Angular Probability Distributions	Circle and sphere measures, marginals, and visualization pitfalls
Rotor in External Fields: First Encounter	Residual symmetry, field-induced mixing, and degeneracy breaking

Choosing a starting point

Start with the ring to see how periodicity quantizes angular momentum. Continue to the sphere and angular probability before interpreting orbital lobes. For molecular applications, study the rigid rotor and rotational spectra; external fields then show how orientation-dependent interactions mix angular states.

Exercises

For an isotropic state on a sphere, what is the probability of lying within polar angle $\theta_0$ of the north pole? Evaluate it for a hemisphere.

Solution

Integrating the polar marginal gives $P(0\leq\theta\leq\theta_0)=(1-\cos\theta_0)/2$. At $\theta_0=\pi/2$, this is $1/2$, as rotational symmetry requires. Sampling $\cos\theta$ uniformly, rather than $\theta$ uniformly, produces isotropic directions.

A polar linear molecule has a permanent electric dipole and follows the ideal rigid-rotor spectrum. Find the photon energy for an allowed absorption $J\to J+1$. Are all such lines equally intense?

Solution

The energy difference is $B_{\mathrm{rot}}[(J+1)(J+2)-J(J+1)]=2B_{\mathrm{rot}}(J+1)$. The line positions are equally spaced in energy in this ideal model. Their intensities also depend on populations and transition matrix elements, so they are not generally equal. A molecule without a permanent dipole lacks these pure rotational electric-dipole lines, although other spectroscopic processes can probe its rotation.

References

• P. W. Atkins and R. S. Friedman, Molecular Quantum Mechanics, 5th ed., Oxford University Press, 2011.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
• G. Herzberg, Molecular Spectra and Molecular Structure I: Spectra of Diatomic Molecules, 2nd ed., Van Nostrand, 1950.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• C. H. Townes and A. L. Schawlow, Microwave Spectroscopy, Dover, 1975.
• R. N. Zare, Angular Momentum: Understanding Spatial Aspects in Chemistry and Physics, Wiley, 1988.