Properties of Waves

Waves have properties that are often quite different from what we would associate with billiard balls or other particles. These differences will become very important when we discuss quantum mechanics. 

Particles (baseball) versus Waves (water wave) 

FIGURE 3.6 Comparison of the total distribution produced by two particle sources or two wave sources. Notice that waves exhibit interference, so the amplitude at some positions is far lower than the amplitude which would be produced by either source alone. 

Equation 3.30 is called the diffraction equation. The solution with n = 0 (9 = 0) is simple transmission of the light. The solutions for n f 0 give intensity in other directions, and the positions of these additional spots can be used to determine A, if d is known. Thus, optical scientists can use a manufactured diffraction grating with known line separations to measure the wavelength of an unknown light source. 

This technique, called X-ray crystallography, can be extended to measure the detailed structures of even complicated materials such as proteins with literally thousands of atoms. However, it is much harder to infer molecular structure if the material cannot be grown as a single crystal, and crystal growth can be exceedingly difficult. 

The speed of a wave depends on the type of wave and the nature of the medium through which the wave is traveling (e.g., air, water, or a vacuum). The speed of light through a vacuum, c, is 2.99792458 X 10 m/s. The speed, wavelength, and frequency of a wave are related by the equation 

Sunlamps Heat lamps Microwave ovens, UHFTV, FM radio, VHF TV AM radio 

The fundamental properties of waves are illustrated in Figme 6.2. Waves are characterized by their wavelength, frequency, and amphtude. Wavelength A (lambda) is the distance between identical points on successive waves (e.g., successive peaks or successive troughs). The frequency v (nu) is the number of waves that pass through a particular point in 1 second. Amplitude is the vertical distance from the midhne of a wave to the top of the peak or the bottom of the trough. 

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The Symmetry Properties of Wave Furictioris of Li3 Electronically Ground State in S3 Permutation Group... 

Valence bond and molecular orbital theory both incorporate the wave description of an atom s electrons into this picture of H2 but m somewhat different ways Both assume that electron waves behave like more familiar waves such as sound and light waves One important property of waves is called interference m physics Constructive interference occurs when two waves combine so as to reinforce each other (m phase) destructive interference occurs when they oppose each other (out of phase) (Figure 2 2) Recall from Section 1 1 that electron waves m atoms are characterized by their wave function which is the same as an orbital For an electron m the most stable state of a hydrogen atom for example this state is defined by the Is wave function and is often called the Is orbital The valence bond model bases the connection between two atoms on the overlap between half filled orbifals of fhe fwo afoms The molecular orbital model assembles a sef of molecular orbifals by combining fhe afomic orbifals of all of fhe atoms m fhe molecule... 

The bond orbitals of o, and relate to the other property of waves apart from the phase, that is, the amplitude. The bonding orbitals have large amplitudes on the low-lying atomic orbitals, i.e., on C of o, and on O of (Scheme 8). The antibonding orbitals have large amplitudes on the high-lying atomic orbitals. 

It is important to distinguish between mmetiy properties of wave functions on one hand and those of density matrices and densities on the other. The symmetry properties of wave functions are derived from those of the Hamiltonian. The "normal" situation is that the Hamiltonian commutes with a set of symmetry operations which form a group. The eigenfunctions of that Hamiltonian must then transform according to the irreducible representations of the group. Approximate wave functions with the same symmetry properties can be constructed, and they make it possible to simplify the calculations. 

The major difference between classical and quantum mechanical ensembles arises from the symmetry properties of wave functions which is not an issue in classical systems. 

It is important to note that the velocity of the wave in the direction of propagation is not the same as the speed of movement of the medium through which the wave is traveling, as is shown by the motion of a cork on water. Whilst the wave travels across the surface of the water, the cork merely moves up and down in the same place the movement of the medium is in the vertical plane, but the wave itself travels in the horizontal plane. Another important property of wave motion is that when two or more waves traverse the same space, the resulting wave motion can be completely described by the sum of the two wave equations - the principle of superposition. Thus, if we have two waves of the same frequency v, but with amplitudes A and A2 and phase angles

resulting wave can be written as ... 

Besides the modules of maximum values of ig-function radial part were compared with Po-parameter values, and the line dependence between these values was found. Using some properties of wave function for P-parameter, the wave equation of P-parameter was obtained. 

All terms on the left vanish except that for n = due to orthogonality property of wave functions and from the conditions of normalization,... 

It will be obvious from the content of Chapter 5 why such combinations are desired. First, only such functions can, in themselves, constitute acceptable solutions to the wave equation or be directly combined to form acceptable solutions, as shown in Section 5.1. Second, only when the symmetry properties of wave functions are defined explicitly, in the sense of their being bases for irreducible representations, can we employ the theorems of Section 5.2 in order to determine without numerical computations which integrals or matrix elements in the problem are identically zero. 

Second-quantization formalism was introduced into the theory of many-electron atoms by Judd [12]. This formalism enables one to give a simple and elegant description of both the rotation symmetry of a system and its permutational symmetry the tensorial properties of wave functions are translated to electron creation and annihilation operators, and the Pauli exclusion principle stems automatically from the anticommutation relations between these operators. 

For the basis (18.27) to be used effectively in practical computations an adequate mathematical tool is required that would permit full account to be taken of the tensorial properties of wave functions and operators in their spaces. In particular, matrix elements can now be defined using the Wigner-Eckart theorem (5.15) in all three spaces, so that the submatrix element will be given by... 

To sum up the potentialities of the isospin method are not exhausted by the results stated above. There is a deep connection between orthogonal transformations of radial orbitals and rotations in isospin space (see (18.40) and (18.41)). This shows that the tensorial properties of wave functions and operators in isospin space must be dominant in the Hartree-Fock method. This issue is in need of further consideration. 

The relationships describing the tensorial properties of wave functions, second-quantization operators and matrix elements in the space of total angular momentum J can readily be obtained by the use of the results of Chapters 14 and 15 with the more or less trivial replacement of the ranks of the tensors l and s by j and the corresponding replacement of various factors and 3nj-coefficients. Therefore, we shall only give a sketch of the uses of the quasispin method for jj coupling, following mainly the works [30, 167, 168]. For a subshell of equivalent electrons, the creation and annihilation operators a and a(jf are the components of the same tensor of rank q = 1/2... 

As in the case of LS coupling, the tensorial properties of wave functions and second-quantization operators in quasispin space enable us to separate, using the Wigner-Eckart theorem, the dependence of the submatrix elements on the number of electrons in the subshell into the Clebsch-Gordan coefficient. If then we use the relation of the submatrix element of the creation operator to the CFP... 

Spectroscopists usually use notations based upon the symmetry properties of wave functions to describe excited states. Since many molecules have no special symmetry properties, such devices are not really strictly applicable in general. However, symmetry notation applicable to related molecules of high symmetry is often extended to unsymmetrical systems. The procedure has been treated formally by Platt (7) who introduced the concept of local symmetry. ... 

This section reviews some basic ideas on the properties of waves, and is an introduction to the wave properties of radiation discussed later. 

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