Planck-Einstein relation

The Einstein-Planck relation indicates that the energy of a photon of monochromatic (single-frequency) radiation depends only on its wavelength or frequency. A beam of radiation is more or less intense depending on the quantity of photons per unit time and per unit area, but the quantum energy (E) per photon is always the same for a given frequency of the radiation. 

For a long time, electrons were perceived to be particles—infinitesimal planets that orbit the nucleus of an atom. In 1924, however, a French physicist named Louis de Broglie showed that electrons also have wavelike properties. He did this by combining a formula developed by Albert Einstein that relates mass and energy with a formula developed by Max Planck that relates frequency and energy. The realization that electrons have wavelike properties spurred physicists to propose a mathematical concept known as quantum mechanics. 

We discuss colour in Chapter 9, so we restrict ourselves here to saying the colour of a substance depends on the way its electrons interact with light crucially, absorption of a photon causes an electron to promote between the two frontier orbitals. The separation in energy between these two orbitals is E, the magnitude of which relates to the wavelength of the light absorbed X according to the Planck-Einstein equation, E = hc/X, where h is the Planck constant and c is the... 

We will perform a simple calculation in two parts. First, we will divide by the Avogadro number to obtain the energy per bond. Second, we convert the energy to a wavelength X with the Planck-Einstein relation (Equation (9.4)), E = hc/X. 

Next, we determine the wavelength, and start by rearranging the Planck-Einstein relation to make wavelength X the subject ... 

This expression can be compared with Planck s formula - Equations (A3.2) or (2.2) - to obtain the following two relations among the Einstein coefficients ... 

At low pressure, the only interactions of the ion with its surroundings are through the exchange of photons with the surrounding walls. This is described by the three processes of absorption, induced emission, and spontaneous emission (whose rates are related by the Einstein coefficient equations). In the circumstances of interest here, the radiation illuminating the ions is the blackbody spectrum at the temperature of the surrounding walls, whose intensity and spectral distribution are given by the Planck blackbody formula. At ordinary temperatures, this is almost entirely infrared radiation, and near room temperature the most intense radiation is near 1000 cm". ... 

On the other hand, when introducing the phase and group velocities v of expressions (88), the energy relations due to Planck, Einstein, and de Broglie result in... 

Recalling the fundamental relations of quantum mechanics, Planck-Einstein, and de Broglie... 

However, right at the beginning of the twentieth century, Planck and Einstein again introduced the corpuscular view with the notion of the photon. The energy E of such a particle is given by de Broglie s relation... 

The energy of a photon of light is related to its frequency v by the relationship E = hv, where h is Planck s constant. Blue light has an energy of about 250 kJ (60 kcal) per einstein (an einstein is a mole of photons). 

It can also be shown that Schrodinger s wave equation is none other than a form of the classical differential equation for a wave phenomenon in which the new feature is to be found in the application of it to electrons by means of the experimentally verified De Broglie relation which, in turn follows, as was seen, from a combination of the fundamental relations of Planck and Einstein in the form Av = me2 (p. 107). 

Planck s resolution of the problem of blackbody radiation and Einstein s explanation of the photoelectric effect can be summarized by a relation between the energy of a photon to its frequency ... 

Einstein in 1917 showed the relation between the three radiative processes and Planck s blackbody radiation law. Suppose the radiation in a cavity is in equilibrium at a temperature T. This means that the rate of upward and downward transitions between every pair of energy levels E and in the walls of the enclosure must exactly balance. Thus,... 

Clearly, a general theory able to naturally include other solvent modes in order to simulate a dissipative solute dynamics is still lacking. Our aim is not so ambitious, and we believe that an effective working theory, based on a self-consistent set of hypotheses of microscopic nature is still far off. Nevertheless, a mesoscopic approach in which one is not limited to the one-body model, can be very fruitful in providing a fairly accurate description of the experimental data, provided that a clever choice of the reduced set of coordinates is made, and careful analytical and computational treatments of the improved model are attained. In this paper, it is our purpose to consider a description of rotational relaxation in the formal context of a many-body Fokker-Planck-Kramers equation (MFPKE). We shall devote Section I to the analysis of the formal properties of multivariate FPK operators, with particular emphasis on systematic procedures to eliminate the non-essential parts of the collective modes in order to obtain manageable models. Detailed computation of correlation functions is reserved for Section II. A preliminary account of our approach has recently been presented in two Letters which address the specific questions of (1) the Hubbard-Einstein relation in a mesoscopic context [39] and (2) bifurcations in the rotational relaxation of viscous liquids [40]. 

Both the SRLS and the FT inertial models were discussed in the context of the Hubbard-Einstein relation, that is, the relation between the momentum correlation time Tj and the rotational correlation time (second rank) for a stochastic Brownian rotator [39]. It was shown that both models can cause a substantial departure from the simple expression predicted by a one-body Fokker-Planck-Kramers equation ... 

Transition strengths can be given in terms of Einstein rate coefficients. For a pair of states j > and k > it is shown in elementary texts that these are related in a simple way. If one assumes that, for any pair of microstates i and j, the rate from i to j is equal to the rate from j to i one has the principle of detailed balance). Then, the relation between the coefficients is consistent with thermodynamics (Planck s black-body radiation law). 

The general form of this relationship agrees with the well-known Nernst-Planck-Einstein formula, according to which the diffusion coefficient is given by the relation... 

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