The quantum mechanical picture

The quantum mechanical calculation of the internal energy in terms of phonons gives the following expression  

There is a neat mathematical trick that allows us to express this in a more convenient form. Consider the expression 

It is evident that when these products are expanded out we obtain exponentials with exponents (n + )p hco with all possible non-negative values of n. Therefore the expressions in these two equations are the same, and we can substitute the second expression for the first one in Eq. (6.49). Notice further that the geometric series summation gives 

This quantity represents the average occupation of the phonon state with frequency 

We examine first the behavior of this expression in the low-temperature limit. In this limit, becomes very large, and therefore (r) becomes negligibly small, except when happens to be very small. We saw in an earlier discussion that cu goes to zero linearly near the center of the BZ (k 0), for the acoustic branches. 

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It is now necessary to examine the partition function in more detail. The energy states for translation are assumed to be given by the quantum-mechanical picture of a particle in a box. For a one-dimensional box of length a. 

The classical model predicts that the largest probability of finding a particle is when it is at the endpoints of the vibration. The quantum-mechanical picture is quite different. In the lowest vibrational state, the maximum probability is at the midpoint of the vibration. As the quantum number v increases, then the maximum probability approaches the classical picture. This is called the correspondence principle. Classical and quantum results have to agree with each other as the quantum numbers get large. 

Figure 5-5. The quantum mechanical picture discrete population histograms take the place of continuous distributions. The overall paramagnetism increases with increasing field strength.

Lawton, R. T., and Child, M. S. (1980), Excited Stretching Vibrations of Water, the Quantum Mechanical Picture, Mol. Phys. 40, 773. 

A new quantum theory called wave mechanics (as formulated by Schrodinger) or quantum mechanics (as formulated by Heisenberg, Born and Dirac) was developed in 1926. This was immediately successful m accounting for a wide variety of experimental observations, and there is little doubt that, in principle, the theory is capable of describing any physical system. A strange feature of the new mechanics, however, is that nowhere does the path or velocity of the electron enter the description. In fact it is often impossible to visualize any classical motion that could be consistent with the quantum mechanical picture of the atom,... 

Both the classical and quantum approaches ultimately lead to a model in which the polarizability is related to the ease with which the electrons can be displaced within a potential well. The quantum mechanical picture presents a more quantitative description of the potential well surface, but because of the number of electrons involved in nonlinear optical materials, theoreticians often use semi-empirical calculations with approximations so that quantitative agreement with experiment is not easily achieved. 

According to the viewpoint of local realism, the recurring correlations in the Bohm experiment can be attributed to the existence of hidden variables which determine the spin state in every possible direction. It is as if each particle carried a little code book containing all this detailed information, a situation something like the left-hand drawing in Fig. 16.4. It must be concluded—so far—that both local realism and the quantum-mechanical picture of the world are separately capable of giving consistent accounts of the EPR and Bohm experiments. In what follows, we will refer to the two competing worldviews as local realism (LR) and quantum mechanics ("QM). By QM we will understand the conventional formulation of the theory, complete as it stands, without hidden variables or other auxilliary constructs. 

In addition to these direct measurements, we shall soon see that two important aspects of alkene chemistry are consistent with the quantum mechanical picture of the double bond, and are most readily understood in terms of that picture. These are (a) the concept of hindered rotation and the accompanying phenomenon of geometric isomerism (Sec. 5.6), and (b) the kind of reactivity characteristic of the carbon-carbon double bond (Sec. 6.2). 

The atoms in a molecule undergo vibrations around their equilibrium configuration within the quantum mechanical picture, even at zero temperature. The application of elementary djmamical principles to these small amplitude vibrations leads to normal mode analysis. Crystalline solids can naively be thought of as big molecules but solving the equations becomes impossible imless the periodicity of the unit cell is included whereupon major simplifications of the algebra are introduced. 

In the vector model free precession involves a rotation at frequency Q about the z-axis in the quantum mechanical picture the Hamiltonian involves the z-angular momentum operator, Iz - there is a direct correspondence. 

In this section we discuss how shock compression produces electronic and vibrational excitations that can cause chemical reactions. It is an outline for a theory that connects the fluid-mechanical picture of shock compression to the quantum mechanical picture of chemical reaction dynamics. 

Hybridization is purely a mathematical procedure, originally invented to reconcile the quantum mechanical picture of electron density in s, p, etc. orbitals with traditional views of directed valence. For example, it is sometimes said that in the absence of hybridization combining a carbon atom with four unpaired electrons with four hydrogen atoms would give a methane molecule with three equivalent, mutually perpendicular bonds and a fourth, different, bond (Fig. 4.6). Actually, this is incorrect the 2s and three 2p orbitals of an unhybridized carbon along with the four orbitals of four hydrogen atoms provide, without invoking hybridization, a tetrahedrally symmetrical... 

Describe the main features of the quantum mechanical picture of the atom... 

The Franck-Condon principle says that the intensities of the various vibrational bands of an electronic transition are proportional to these Franck-Condon factors. (Of course, the frequency factor must be included for accurate treatments.) The idea was first derived qualitatively by Franck through the picture that the rearrangement of the light electrons in the electronic transition would occur quickly relative to the period of motion of the heavy nuclei, so the position and momentum of the nuclei would not change much during the transition [9]. The quantum mechanical picture was given shortly afterwards by Condon, more or less as outlined above [10]. 

In the discussions of the kinetic theory of gases and of intermolecular forces, we obtained expressions for properties of matter in bulk in terms of the properties of the individual molecules. In this chapter we will describe the cohesive energy of ionic crystals in terms of the interactions of the ions in the crystals, and some of the properties of metals and covalent crystals in terms of the quantum mechanical picture obtained from the Schrodinger equation. In Chapter 29 we will describe the method for calculating the thermodynamic properties of bulk systems from a knowledge of structure. 

The quantum-mechanical picture of hyperfine structures presented by the spin-spin nuclear magnetic resonance (NMR) and electron-spin resonance (ESR) spectra involves a variety of spin Hamiltonian parameters of molecular origin whose magnitude determines that of the coupling constants. In such an analysis, the most characteristic term arises from the Fermi -or contact -operator ... 

Electronic enagy hypasuifaces represent the PES for the motion of the nuclei. In the quantum mechanical picture, only some eneigies will be allowed we will have the vibrational and rotational energy levels, as for diatomics. 

Fig. 8.20l The same reasoning is used as in Fig. 8.19, but this time, the key charge distributions are drawn in a realistic way instead of a sehematie diagram. The sections r. = 0 of the two crucial electronic charge distributions are drawn, (a) In the quantum mechanical picture, what decides about chemical bonding is an electron cloud — l n l fc = that contains the charge...

Because the quantum mechanical picture of the atom is both complex and non-intuitive, we begin by pointing out some key comparisons between it and the more famihar Bohr model. In the Bohr model, electrons are viewed as particles traveling along circular orbits of fixed radius. In the quantum model, electrons are viewed as waves rather than particles, and these waves are considered to be spread... 

The quantum mechanical picture of the atom, and hence our pictures of various orbitals, arises from ideas of probability and average position. To understand the notion of an instantaneous dipole, though, we ll need to step away ftom that viewpoint. Instead, we ll need to consider electron positions or distributions at a single instant. For an electron in an s orbital, for example, the ideas presented in... 

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