Rigid rotor quantum treatment

A major success of statistical mechanics is the ability to predict the thermodynamic properties of gases and simple solids from quantum mechanical energy levels. Monatomic gases have translational freedom, which we have treated by using the particle-in-a-box model. Diatomic gases also have vibrational freedom, which we have treated by using the harmonic oscillator model, and rotational freedom, for which we used the rigid-rotor model. The atoms in simple solids can be treated by the Einstein model. More complex systems can require more sophisticated treatments of coupled vibrations or internal rotations or electronic excitations. But these simple models provide a microscopic interpretation of temperature and heat capacity in Chapter 12, and they predict chemical reaction equilibria in Chapter 13, and kinetics in Chapter 19. 

A proper quantum-mechanical treatment ofH makes no presumptions about the distance of an electron from a nucleus. Thus, the description of the hydrogen atom is the same as for the 3-D rigid rotor except it includes variation of r, which ranges from 0 to 00. This is illustrated in Figure 11.16. The hydrogen atom is defined like a 3-D rigid rotor, only now the radius can vary also. Wavefunctions describing the... 

At present the derivation of this equation from the fully quantum-mechanical theory relies upon a heuristic extension of the semiclassical treatment for rigid rotors in which the rotational heat capacity and rotational collision number are replaced by the total internal heat capacity and the internal collision number respectively. This (approximate) extension seems to be justified by a result of van den Oord Korving (1988). 

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