Quantization of energies

According to quantum theoiy, energy is always emitted in whole-number multiples of hv. At the time Planck presented his theory, he could not explain why eneigies should be fixed or quantized in this manner. Startbg with this hypothesis, however, he had no difficulty correlating the experimental data for the emission by solids over the entire range of wavelengths the experimental data supported his new quantum theory. 

The idea that energy is quantized rather than continuous may seem strange, but the concept of quantization has many everyday analogies. For example, vending machines dispense cans or bottles of soft drinks only in whole numbers (you can t buy part of a can or bottle from a machine). Each can or bottle is a quantum of its soft drink. Even processes in living systems involve quantized phenomena. The eggs laid by hens are quanta (hens lay only whole eggs). Similarly, when a 

The National Institute of Standards and Technology (NISO gives a value of 6.6260693 x 10 J - s for PlanckS constant. Typically, three 

Max Karl Emsi Ludwig Planck (1858-1947). German physicist. Planck received the Nobel Prize in Physics in 1918 for his quantum theory. He also made significant contributions in thermodynamics and other areas of physics. 

CHAPTER 6 Quantum Theory and the Electronic Structure of Atoms 

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The energy E is said to be quantized in discrete packets, or quanta, each of energy hv. It is because of the extremely small value of h that quantization of energy in macroscopic systems had escaped notice, but, of course, it applies to all systems. 

An electron in an atom is like a particle in a box, in the sense that it is confined within the atom by the pull of the nucleus. We can therefore expect the electron s wavefunctions to obey certain boundary conditions, like the constraints we encountered when fitting a wave between the walls of a container. As we saw for a particle in a box, these constraints result in the quantization of energy and the existence of discrete energy levels. Even at this early stage, we can expect the electron to be confined to certain energies, just as spectroscopy requires. 

We cannot extrapolate our knowledge of everyday macroscopic world to the world of subatomic dimensions. The Heisenberg uncertainty principle, the wave character of particle motion and quantization of energy become important when the masses of the particles become comparable to Planck s constant h. 

X3 process. The sample loses its alignment due to rotation but due to quantization of energy the molecules will be aligned after a characteristic time over and over again. The time dependence of the third order polarisability has been described by Mukamel and is given below ... 

In his contribution Nemst dealt with the application of the Theory of Quanta to a few Physico-Chemical Problems, in particular the connection between Nemst theorem8 and the quantization of energy. 

There are several reasons for starting this account with a discussion of electromagnetic radiation. Historically, it was in this area that the quantum theory first developed. It is easier here to understand the evidence for the theory, and to appreciate some of its paradoxical consequences, than it is in the quantum theory of matter. The applications of the light-quantum hypothesis, as it was first called, also provide key pieces of evidence for the quantization of energy in atoms and molecules. Studies of the absorption and emission of radiation—the field of spectroscopy—and of the effect of light on chemical reactions—photochemistry—are very important areas of modem chemistry, in which the quantum nature of radiation is crucial. 

Most other forms of spectroscopy do not involve emission of extra particles such as electrons, but the straightforward absorption or emission of photons. These processes increase or decrease the energy of an atom ex molecule, by an amount equal to the photon energy. The results all reinforce the conclusion of photoelectron spectroscopy that only discrete energy levels occur (see Fig. 1.12). For example, the line spectra of atoms, known since the early nineteenth century, only contain lines at certain well-defined wavelengths. The quantization of energy, not only in electromagnetic radiation but in material systems, is an inescapable conclusion rtf spectroscopy. 

This chapter deals with some solutions of Schrddinger s equation in situations that are relatively simple, but which nevertheless have important applications in chemistry. Most of these deal with the quantization of energy. There is, however, another remarkable prediction of quantum mechanics, known as tunnelling, which is treated first. 

Two papers by Albert Einstein ultimately led to acceptance of the idea of quantization of energy for radiation, and were central to the development of the quantum theory (ironically, in later years Einstein became the most implacable critic of this same theory). The first of these papers, in 1905, concerned the photoelectric effect. Light ejected electrons from a metallic surface if the light had a greater frequency than some threshold frequency v0 which depended on the particular metal. The kinetic energy K of the emitted electrons was proportional to the excess frequency, v — v0 (Figure 5.4). Only the number of emitted electrons, not the kinetic energy, increased as the intensity increased. 

Another paper of Einstein s showed that quantization of energy also predicted that Dulong and Petit s heat capacity rule would only be valid at high temperatures. Assume that the only allowed energies are E = 0, hv, 2hv,... nhv, where n can be arbitrarily large. The average energy is... 

Section 5.5 Atomic Structure and Spectra-quantization of Energy... 

Motion along the reaction coordinate was limited to classical mechanics, whereas the sum and density (or, to be precise, the degeneracy) of states should be evaluated according to quantum mechanics. The integral in Eq. (7.49) should really be replaced by a sum N (E) is not a continuous function of the energy, but due to the quantization of energy, it is only defined at the allowed quantum levels of the activated complex. That is, the sum of states G (E ) should be calculated exactly by a direct count of the number of states ... 

By now, it was becoming clear that there was a connection between electrons in bodies, the radiant energy emitted by those bodies, and the distribution of that energy in the spectrum. But a more detailed theory with more information was needed. Rutherford had proposed an atom modeled on the solar system, with electrons orbiting around a positive nucleus and a lot of empty space between the electrons and the nucleus. In 1913 the Danish physicist Niels Bohr (1885-1962), who worked with Rutherford for four years and on his return to Copenhagen made Denmark a world center of theoretical physics, published one of the twentieth century s most important papers. He applied Planck s equation and the notion of quantization of energy to Rutherford s... 

This chapter introduces the quantum mechanics required for the analyses in this text. The state of an electron is represented by a wave funetion ji. Kach observable is represented by an operator O. Quantum theory asserts that the average of many measurements of an observable on electrons in a certain state is given in terms of these by ji 0 d r. The quantization of energy follows, as does the determination of states from a Hamiltonian matrix and the perturbative solution. The Pauli principle and the time-dependence of the state are given as separate assertions. 

What does this equation mean We have simply specified that A and k are constants. What values can these constants have Note that if they could assume any values, this equation would lead to an infinite number of possible energies—that is, a continuous distribution of energy levels. However this is not correct. For reasons we will discuss presently, we find that only certain energies are allowed. That is, this system is quantized. In fact, the ability of wave mechanics to account for the observed (but initially unexpected) quantization of energy in nature is one of the most important factors in convincing us that it may be a correct description of the properties of matter. 

The key new ideas of qnantnm mechanics include the quantization of energy, a probabilistic description of particle motion, wave-particle duality, and indeterminacy. These ideas appear foreign to ns because they are inconsistent with our experience of the macroscopic world. We have accepted them because they have provided the most comprehensive account of the behavior of matter and radiation and because the agreement between theory and the results of all experiments conducted to date has been astonishingly accurate. 

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