Einstein-Planck equation

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... 

At a quantum-mechanical level, there is a simple relationship that ties together the twin modes by which we visualize photons we say that the energy of a photon particle is E and the frequency of a light wave is v. The Planck-Einstein equation, Equation (9.3), says... 

In practice, most chemists prefer to talk in terms of wavelength rather than frequency. For this reason, we often combine Equations (9.2) and (9.3), and so obtain a modified form of the Planck-Einstein equation ... 

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. 

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 ... 

Of course, there is ample evidence that light consists of waves (diffraction and interference experiments), and this must also have been clear to Einstein. How is it possible, then, that light appears to be at the same time a point-like particle and a wave extended over space In the photoelectric effect and the Planck s equation, EM radiation appears as particles with energy, hv, but, in reality, they are dealing with absorption and emission events. We do not really know whether EM radiation after emission and before reabsorption consists of particles. 

Helfferich [90] had given a mathematical description of the diffusion of ions in ion-exchange beads based on the Nernst-Planck and the Nernst-Einstein equations, Hornby et al [82] combined a case of this treatment with the Michaelis-Menten kinetic equation (Table IX) in order to calculate diffusion reactions with linear profiles in an idealized quasi-Nernst layer of thickness x limited on one side by a bulk solution of concentration Sq and on the other one by a charged enzyme-layer the rate of reaction in the steady state equals the sum of and Jj the fluxes of the charged substrate due respectively to the electrical and concentration gradients ... 

The miderstanding of the quantum mechanics of atoms was pioneered by Bohr, in his theory of the hydrogen atom. This combined the classical ideas on planetary motion—applicable to the atom because of the fomial similarity of tlie gravitational potential to tlie Coulomb potential between an electron and nucleus—with the quantum ideas that had recently been introduced by Planck and Einstein. This led eventually to the fomial theory of quaiitum mechanics, first discovered by Heisenberg, and most conveniently expressed by Schrodinger in the wave equation that bears his name. 

When Max Planck wrote his remarkable paper of 1901, and introduced what Stehle (1994) calls his time bomb of an equation, e = / v , it took a number of years before anyone seriously paid attention to the revolutionary concept of the quantisation of energy the response was as sluggish as that, a few years later, whieh greeted X-ray diffraction from crystals. It was not until Einstein, in 1905, used Planck s concepts to interpret the photoelectric effect (the work for which Einstein was actually awarded his Nobel Prize) that physicists began to sit up and take notice. Niels Bohr s thesis of 1911 which introduced the concept of the quantisation of electronic energy levels in the free atom, though in a purely empirical manner, did not consider the behaviour of atoms assembled in solids. 

In the course of his research on electromagnetic waves Hertz discovered the photoelectric effect. He showed that for the metals he used as targets, incident radiation in the ultraviolet was required to release negative charges from the metal. Research by Philipp Lenard, Wilhelm Hallwachs, J. J. Thomson, and other physicists finally led Albert Einstein to his famous 1905 equation for the photoelectric effect, which includes the idea that electromagnetic energy is quantized in units of hv, where h is Planck s con-... 

A hundred years ago it was generally supposed that all the properties of light could be explained in terms of its wave nature. A series of investigations carried out between 1900 and 1910 by Max Planck (1858-1947) (blackbody radiation) and Albert Einstein (1879-1955) (photoelectric effect) discredited that notion. Today we consider light to be generated as a stream of particles called photons, whose energy E is given by the equation... 

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