ХИМИЧЕСКАЯ ФИЗИКА, 2004, том 23, № 2, с. 110-115
ЭЛЕМЕНТАРНЫЕ ^^^^^^^^^^
ФИЗИКО-ХИМИЧЕСКИЕ ПРОЦЕССЫ
УДК 535.11
QUANTUM MECHANICAL RELATIONS FOR THE ENERGY, MOMENTUM AND VELOCITY OF SINGLE PHOTONS IN DISPERSIVE MEDIA
© 2004 r. R. J. Buenker*, P. L. Muin o**
*Bergische Universität-Gesamthochschule Wuppertal, Fachbereich 9-Theoretische Chemie, Gaussstr. 20, D-42097 Wuppertal, Germany **Department of Chemistry, Mathematics, and Physical Science, St. Francis College. Loretto, Pennsylvania 15940 USA
Received 16.11.2002
Attempts to explain the refraction of light in dispersive media in terms of a photon or "corpuscular" model have heretofore been unable to account for the observed decrease in the speed of light as it passes from air into a region of higher refractive index n such as water or glass. In the present work it is argued on the basis of the quantum mechanical relations p = hk and E = ha that the energy of photons satisfies the equation E=pc/n. It is possible to obtain an exact prediction of the observed speed of the photons in a given medium by application of Hamilton's equations of motion to the above formula, but at the same time to conclude, in agreement with the arguments of Newton and other classical physicists, that the photon momentum increases in direct proportion to n, thereby producing the well-known bending of light rays toward the normal when entering water from air. The corresponding relativistic particle theory of light indicates that the potential V encountered by the photons in a given medium is attractive for n > 1 and is momentum-dependent, which suggests the microscopic interactions responsible for the refraction of light are non-Coulombic in nature and are instead akin to the spinorbit and orbit-orbit terms in the Breit-Pauli Hamiltonian for electrons moving in an external field. The present theory concludes that the reason photons are slowed down upon entering water from air is that their relativistic massp/vincreases faster with n than does their momentum, which in turn requires that Einstein's famous E = mc2 formula does not hold for light dispersion because the energy of the photons is expected to be the same in both media. In summary, all the known experimental data regarding light dispersion can be successfully explained in terms of a particle theory of light once it is realized that photons possess exceptional properties by virtue of their zero proper mass and their capacity to undergo electromagnetic interactions with surrounding media.
1. INTRODUCTION
The refraction of light as it passes through transparent media is a phenomenon of everyday experience which has occupied the attention of scientists dating back at least to the time of the ancient Greek philosophers. Since the experiments of Foucault in 1850, it has been known that light moves more slowly in water and other materials than it does in free space. Newton [1] and his followers had come to the opposite conclusion based on his corpuscular theory of light, and this fact had a decisive effect in promoting the competing wave model of electromagnetic radiation [2, 3]. It was simply argued that if light were really composed of particles, classical mechanics should have been able to predict that it would be slowed down in dispersive materials, and having failed in this, the above premise should be discarded entirely. The pioneering developments of the early 20th century [4, 5] caused a rethinking of this conclusion, however, so that there is now a consensus among physicists that some experiments are best explained in terms of light waves, while others seemingly require a particle formulation to remain consistent with the observed results. Moreover, the concept of wave-particle duality has been generalized by de Broglie [6] to apply to all types of matter.
The question that will be explored in the present study is whether there is really no way of understanding light dispersion in terms of a particle model. Recent experiments [7] have shown, for example, that single photons travel through glass at the group velocity of light vg = c/ng. The technique employed makes use of a two-photon interferometer and thus takes advantage of the wave properties of light, but one can reasonably conclude that what is being measured is simply the speed of individual photons in the apparatus. To explore this point further, new experiments have been carried out in our laboratory [8] in which time-correlated single-photon counting (TCSPC) detection has been employed to measure the ratio of the speeds of light in air and water. The most interesting aspect of this study was that the shapes of the photon counting profiles (instrument response) are not affected by the medium through which the light passes (over a distance of nearly 1.0 m). It is possible to explain this result in a very simple manner, namely that, as in the earlier experiments of Steinberg et al. [7], each photon is decelerated by an amount which is inversely proportional to the group index of refraction ng as it passes from air into a dispersive medium. These findings certainly add support to the particle theory of light advocated by Newton [1], but at the same time underscore the need to better understand why the classical theory leads to an errone-
ous prediction of the dependence of the speed of light on the nature of the medium through which it passes.
2. PHOTON MOMENTUM IN DISPERSIVE MEDIA
Newton's argument was based on observations of the refraction of light in dispersive media (see Figure). According to Snell's Law of Refraction, the angles of incidence 01 and refraction 02 at an interface are inversely proportional to the respective indices of refraction which are characteristic for each medium:
«1sin 01 = n2sin 02. (1)
The bending of light rays was regarded as evidence for the existence of a force acting at the interface between two media. Since light always travels in a straight line within a given medium, it follows that the corresponding potentials are constant throughout and the resultant force acting on the light must be in the direction normal to the interface. Because of Newton's Second Law this means that the component of the momentum of the particles of light which is parallel to the interface must also be constant; hence, according to Figure,
p1sin 01 = p2sin 02, (2)
which implies by comparison with Eq. (1) that the total momentum of the particles p is always proportional to the refractive index n of a given medium. Since light is bent more toward the normal in water than in air, one is led unequivocally to the conclusion that the momentum of the photons is greater in water.
From there it was only a short step for Newton to conclude that the velocity of light v must also be greater than in air, since by definition,
p = m v, (3)
and there was no evidence at that time to indicate that the inertial mass m of mechanical particles could be anything but a constant. This conclusion becomes far less obvious, however, once the results of Einstein's Special Theory of Relativity [9] are taken into consideration. It is known that the mass of a particle varies with the relative velocity of the observer, for example, and also that mass is not conserved in reactive processes.
On closer examination, it is clear that Newton's arguments are only directly applicable to the momentum of photons in dispersive media. Since there have never been any quantitative measurements of photon momenta in condensed media, such as by the observation of X-ray scattering with electrons therein or of nuclear recoil following high-energy emission processes, it seems fair to say that there is still a good possibility that Eq. (2) is correct. Indeed, there is independent evidence obtained from application of the quantum mechanical relation,
p = h k, (4)
Normal
Schematic diagram showing the refraction of light at an interface between air and water. The fact that the light is always bent more toward the normal in water (Snell's Law) led Newton to believe that there is an attractive potential in the denser medium which causes the particles of light to be accelerated.
to the wave theory of light that this is so (k = 2tcA). The fundamental equation for the phase velocity of light in dispersive media is:
ffl/k = c/n. (5)
In this equation it is known that the frequency o> is independent of n and therefore that the wave vector k is proportional to n. Comparison with Eqs. (2) and (3) shows that according to Newton's theory, the momentum of the photons must be proportional to k, consistent with Eq. (4).
The latter equation is acknowledged to be valid for photons in free space [10] and, following de Broglie's hypothesis [6], to hold for free particles in general. Furthermore, in the Davisson and Germer experiments [11], it was necessary to take into account the fact that the momentum of the electrons increases upon entering the Ni crystal interior in order to correctly predict the wavelength of the maximum in the observed electron diffraction pattern. This shows that Eq. (4) is also valid for particles in the presence of a potential. Dicke and Wittke [12] have also pointed out that a consistent definition of a refractive index can be made for electrons passing between different regions of constant potential if one assumes the de Broglie relation, in which case one again finds that p is proportional to n. It is interesting to note that it is not necessary to know the value of h in Eq. (4) to arrive at such relationships, only that
there is a proportionality. Thus, the pioneering experiments of the late 19th century that led to the formulation of the quantum theory of matter are not actually needed to infer that the momentum of photons in dispersive media is directly proportional to the wave vector k, and is thus larger in water and glass than in air. Since both the photon energy E and the light frequency ffl are independent of n, it is also possible to infer their proportionality in ligh
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