page_title: Rotational transition
section_title: Nuclear wave function
hierarchical_section_title: Rotational transition / Diatomic molecules / Nuclear wave function
caption_attribution_description: English: Vector addition triangle for orbital angular momentum of a diatomic molecule with components of orbital angular momentum of nuclei and orbital angular momentum of electrons, neglecting coupling between electron and nuclear orbital motion and spin-dependent coupling.
context_page_description: A rotational transition is an abrupt change in angular momentum in quantum physics. Like all other properties of a quantum particle, angular momentum is quantized, meaning it can only equal certain discrete values, which correspond to different rotational energy states. When a particle loses angular momentum, it is said to have transitioned to a lower rotational energy state. Likewise, when a particle gains angular momentum, a positive rotational transition is said to have occurred.
  Rotational transitions are important in physics due to the unique spectral lines that result. Because there is a net gain or loss of energy during a transition, electromagnetic radiation of a particular frequency must be absorbed or emitted. This forms spectral lines at that frequency which can be detected with a spectrometer, as in rotational spectroscopy or Raman spectroscopy.
context_section_description: Quantum theoretical analysis of a molecule is simplified by use of Born–Oppenheimer approximation. Typically, rotational energies of molecules are smaller than electronic transition energies by a factor of m/M ≈ 10⁻³ – 10⁻⁵, where m is electronic mass and M is typical nuclear mass. From uncertainty principle, period of motion is of the order of Planck's constant h divided by its energy. Hence nuclear rotational periods are much longer than the electronic periods. So electronic and nuclear motions can be treated separately. In the simple case of a diatomic molecule, the radial part of the Schrödinger Equation for a nuclear wave function Fₛ(R), in an electronic state s, is written as (neglecting spin interactions)
  where μ is reduced mass of two nuclei, R is vector joining the two nuclei, Eₛ(R) is energy eigenvalue of electronic wave function Φₛ representing electronic state s and N is orbital momentum operator for the relative motion of the two nuclei given by
  The total wave function for the molecule is
  where rᵢ are position vectors from center of mass of molecule to iᵗʰ electron. As a consequence of the Born-Oppenheimer approximation, the electronic wave functions Φₛ is considered to vary very slowly with R. Thus the Schrödinger equation for an electronic wave function is first solved to obtain Eₛ(R) for different values of R. Eₛ then plays role of a potential well in analysis of nuclear wave functions Fₛ(R).
caption: Vector addition triangle for orbital angular momentum of a diatomic molecule with components of orbital angular momentum of nuclei and orbital angular momentum of electrons, neglecting coupling between electron and nuclear orbital motion and spin-dependent coupling.Since angular momentum N of nuclei is perpendicular to internuclear vector R, components of electronic angular momentum L and total angular momentum J along R are equal.
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