Estimation of Energy Levels for Arbitrary Potential Wells We want to draw attention to a method far the estimation of energy levelsrelated to a potential well of an arbitrary shape. The method is closely related to the principle of correspondence which has already found wide application for other estimations. We shall start from the postulate that the frequencies f,! and f,,of theemitted light according to the classical and the quantum mechanical theory, respectively, are equal to one another. This pastulate will be referred to as frequency correspondence. For the value of fclwe make use of fd = lit*
c
(1)
where t , ~is the classical oscillation time. For any given potential well and for any value of the energy of the particle, one can easily estimate the value oft,^. Even if high accursry is aimed for, the calculations will not involve other than standard computational procedures. According to Bohr's postulate the frequency fqmsatisfies hfqm = AE
(2)
m = hit,'
(3)
With frequency correspondence we ohtain Equation (3) relates the difference between the energy levels to the classical oscillation time. In case of a harmonic mcillator it straightforwardly shows that the energy levels are equidistant as t , ~is independent of the particle energy. Similarly eqn. (3) can he applied to the other standard examdes commonlv used in ouantum mechanics. such as the oarticle in the hou and the rigid rotator. I t is particularly advantageous to appl; eqn. (3) io problems where the shape of tke potential well is less simple. In Figure 1an arbitrary potential well is shown; a particle in the A region will behave as a harmonic oscillator (parabolic shape of the well): a particle in the B region will loose much time when moving near the flat part of the potential cuwe and therefore its t , ~value will be very large. Consequently (see eqn. (3)),the density of energy levels will be great. In the C region this effect will loose its relevance, which leads to greater distances between the energy levels. Apart from didactical applications, frequency correspondence may well he used in scientific discussions, as it rapidly provides a global picture of the energy levels when a potential curve ia given. In cases where the potential curve is not precisely known' involving a certain degree of guessing or estimating (as in the case of gas adsorption) the results of the exact quantum mechanical treatment may often be expected to be not more relevant than those obtained from frequency correspondence. 'Poulis, J. A., Thomas, J. M., and Massen, C. H., Thermochim. Aeto, 5, (1912) I. Physics Department University of Teohnology Eindhoven. the Netherlands
490 I Journal of ChemlcalEducation
J. A. Poulis C. H.Massen
