Standing waves three dimensional

A three-dimensional (3D) piece of metal can be considered as a crystal of infinite extension in the directions x, y and z with standing waves with the wave numbers k, ky and k, each being occupied with two electrons as a maximum. In a piece of bulk metal the energy differences Sk y are so small that A->0, identical with quasi-free continuously distributed electrons. Since the energy of free electrons varies with the square of the wave numbers, its dependence on k describes a parabola. Figure 4a shows these relations. 

We have already seen (p. 2) that the individual electrons of an atom can be symbolised by wave functions, and some physical analogy can be drawn between the behaviour of such a wave-like electron and the standing waves that can be generated in a string fastened at both ends—the electron in a (one-dimensional) box analogy. The first three possible modes of vibration will thus be (Fig. 12.1) ... 

In eqn. 2.7 the number n is a quantum number5. It is in fact related to the number of nodes in the wavefunction and must in this case be a positive integer (n = 1, 2, etc.). This would apply to a wave which follows one direction only. Since real space is three-dimensional a standing wave must be defined by three quantum numbers. The motions of electrons around nuclei are essentially circular, so that the use of polar coordinates is preferable and the three quantum numbers are ... 

Another method recently developed for manipulating small particles uses the forces created by a two- or three-dimensional sound field that is excited by a vibrating plate, the surfaces of which move sinusoidally and emit an acoustic wave into a layer of fluid. Such a wave is reflected by a rigid surface and generates a standing sound field in the fluid, the forces of which act on particles by displacing them in one, two or three dimensions. In this way, particles of sizes between one and several hundred microns can be simultaneously manipulated in a contactless manner. Equations describing this behaviour have been reported [63]. 

We examine the effect of the standing waves across the plate on "the three dimensional stress fields at the crack-tip. We consider those sound waves which propagate with the crack, i.e., o> = /5 v. In this case the quantity, given by Eq. (25b), for example, can be written as. 

In order to visualize the effect of the standing waves across the plate, the three dimensional plot of the normalized hoop stress, dg°g + Ogg, is shown for the identical case of Fig. 2 at the fixed crack velocity, v/ Cj =0.8. 

In this paper we have calculated the three dimensional stress fields at the tip of the crack associated with the standing waves across the plate, assuming the waves modulate the stress intensity factor of the crack. Thus the present analysis takes account of these important experimental observations for the microcrack branching instability described above. We are unable, however, to clarify the direct cause of this instability but we have pointed out that the transient interference pattern of the standing waves could enhance the stress fields at the tip of the crack, as shown in Fig. 5, and possibly change the dynamics of the propagating crack. 

The variation in the intensity of the electron charge can be described in terms of a three-dimensional standing wave like the standing wave of the guitar string. 

An electron behaves like a standing wave, but— unlike the wave created by a vibrating guitar string—it is three dimensional. This means that the node of a 2s atomic orbital is actually a surface—a spherical surface within the 2s atomic orbital. Because the electron wave has zero amplitude at the node, there is zero probability of finding an electron at the node. 

Figure2.3 Electrons in athree-dimensional (3-D) bulk solid [16], (a) Such a solid can be modeled as an infinite costal along all three dimensions, x, y, and z (b) The assumption of periodic bounda7 conditions yields standing waves as solutions forthe Schrodingerequation for free electrons. The associated wavenumbers (kx, ky, kx) are periodically distributed in the reciprocal (t-space [17], Each of the dots shown in the figure represents a possible electronic state kx, ky, k ). Each state in fc-space can be only occupied by two electrons. In a large solid, the spacing A/tx,y,z between individual electron states is very small and therefore the fc-space is quasi-continuously filled with states. A sphere with radius ftp includes all states with k = k +l +kiy < kp. In the ground...

If the atoms are sufficiently cold, they can be trapped in the potential minima of a three-dimensional standing laser wave field (optical lattice). The idea has been already proposed by Lethokov [1184], but could not be realized at that time because the lowest obtainable temperatures were still too high. Nowadays many experiments have been performed with cold atoms or molecules in optical lattices (see Sect. 9.1.16). 

Sufficiently cold atoms can be trapped in optical lattices, formed by a three-dimensional overlap of standing laser waves in x-, y-, and z-directions. If the kinetic energy of the atoms is lower than the potential energy minima of such lattices they are trapped in these minima and cannot leave their fixed positions. Such a state is called a Mott insulator (in analogy to the situation in solid state physics explained by Sir Nevil Mott where a Mott insulator describes a situation where electrical conductivity should be possible according to the band structure but the repulsive interaction between the electrons prevents electron transport). The potential well depth can be tuned by changing the intensity of the three laser beams. 

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