John G. Foss
Department of Biochemistry and Biophysics Iowa State University Ames, Iowa 50010
A
Photonic Angular Momentum and Selection Rules for Rotational Transitions *
simple rationale for the ill = 1 selection rules for rotational transitions was presented by Moynihan in a recent note.' The object of this note is to present an alternate rationale which emphasizes the quantization of the light rather than the molecular rotator. No claim is made that this alternative is superior to Moynihan's approach, hut it does present another simple physical picture which will appeal to many students, and it emphasizes an interesting property of light which is usually ignored in elementary discussions of spectroscopy. The argument to be presented is based on the fact that all photons carry either left or right angular momentum depending on whether they are right or left circularly polarized. (Linearly polarized light can be thought of as containing an equal number of right and left circularly polarized photons.) All photons carry either * h / 2 r units of angular momentum independent of their frequency, where h is Planck's constant. This bare statement of fact will not be very convincing for most students, but there are two simple ways to show that photons do carry angular momentum. The first is an argument by analogy and the second is a description of an experimental demonstration of the existence of the angular momentum. In classical mechanics the moment of inertia I, and angular velocity o,are exact angular counterparts of the more familiar linear quantities mass, M and velocity, v. The angular momentum of a rotating body is given by Iw and the angular kinetic energy by ' / J w l . (Compare Mu and 1/2Mv2 for the linear momentum and kinetic energy.) For our purposes it will he useful to express the angular momentum in terms of the angular kinetic energv, thus
Therefore, by analogy, if a photon has an angular momentum it might be calculated by dividing its circular frequency, o ( = Z s v ) , into twice its energy, hu. This gives h/rr which is only twice the correct value of h / 2 s = fi.% Is this quantity fi "real" or is it just a convenient mathematical fiction? A convincing demonstration of its reality was provided by the very difficult experiments of Beth3 in 1936. A simplified version of his experiment is shown in the figure. A torsion pendulum of resonant frequency f is made by suspending a thin mica disc from a fine quartz thread. The mica is of an appropriate thickness to he a half wave plate for light of the wavelength used for the experiment. A circularly polarized beam of light, chopped at a frequency f (to maximize the effect), is sent in a normal direction through the mica disc. Now a properly oriented half wave plate has the property of changing the handedness of circular light passing through it. Thus if the incom-
1 MOYNIHAN, C. T., J. CHEM.EDUC., 46, 431 (1969). % A ninteresting mnemonic or pseudo proof that even elimirttes this factor of two follows from an assumption that an equipartition principle is operative which divides the total energy hu equally between linear and angular kinetic energies. This then leads to the correct result both for the photonic angular and linear momenta. (The linear momentum is
where i is the reciprocal wavelength.) 3 BETA, R. A,, Physical Rmim, 50, 115 (1936). For a brief description of this experiment, see page 561 of R. W. Dithburn's "Light," (1st Ed.), Blackie and Son, London, 1952.
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A rimpliRed version of Beth's experiment. A, Torsion pendulum; B, emerging beam left circvlarly polarized; C, half ware plate Ifor example, sheet of mica);
D, 6ght circvlorly polarized light beam.
ing photons carry +6 units of angular momentum, those leaving will have -6 and, to conserve angular momentum, the plate must acquire (or lose) 26 for each photon passing through it. If the number of photons per second passing through the pendulum is known, and the angular momentum of the oscillating pendulum is measured, the angular momentum per photon can then be calculated. When this was done by Beth an angular momentum was measured which was reasonable agreement with 6 per photon. Once it is agreed that photons carry 6 units of angular momentum, there are a number of interesting consequences. Consider, for example rotational spectra. Classically the kinetic energy of a rotating molecule is given by ('/,)Iw2 and it can assume all values of w. If E, is the classical energy corresponding to a particular value of w (obtained for example by collisions), then the energy of a molecule which has absorbed J photons is
Now spectroscopic methods normally only permit
measurements of the difference in energy between two states and this difference is given by
which only differs from the usual quantum mechanical result in having (2J 1) instead of (2J 2). (It has been implicitly assumed that the time needed for the rotational transition is short compared to the time for E, to change.) In summary then, the quantization of the angular momentum of photons leads quite directly to the A J = + 1 selection rule and it in turn to the spacing of the rotational levels in eqns. (2) and (3). There are several other obvious consequences of the constant photonic angular momentum. For example, it can be used to rationalize the selection rules for atomic transitions A1 = + 1 and in general to rationalize selection rules involving changes in angular momentum. I t can also be used to justify the Bohr postulate that changes in the angular momentum of the hydrogen atom occur in multiples of6.
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