Particle on a Ring Spectroscopic Selection Rules Determined by Group Theory

摘要

Early introduction of symmetry concepts in the Physical Chemistry curriculum has been shown to help students understand the relative simplicity of spectroscopic selection rules. Most Quantum Mechanics and Spectroscopy textbooks begin with various one-dimensional problems. Application of symmetry arguments to the particle-in-a-box problem is presented in some books, but no sources have been found where symmetry arguments are used to determine the selection rules for particle-on-a-ring spectroscopic transitions. This hinders the early introduction of symmetry concepts. This article removes this hindrance by deriving the particle-on-a-ring rotational selection rules using group theory symmetry arguments. The D-infinity h character table is used to define rotational wave function symmetry, and the Doh direct product table is used to determine the nonzero behavior of the transition dipole moment integral. A survey of the symmetry of allowed transitions leads to the well-known rotational selection rules of Delta m = 0 for Rayleigh scattering, Delta m = +/- 1 for direct absorption and emission, and Delta m = +/- 2 for Raman scattering. This approach will allow the use of symmetry arguments for spectroscopic selection rule determination for the one-dimensional problems that establish the early foundation in Quantum Mechanics and Spectroscopy courses.