Introduction. Matter waves

According to de Broglie s idea, wavelike properties are associated with any particle of matter, and the de Broglie wavelength is defined by the fundamental relation 

The next natural objects are neutral atoms or molecules. The wave properties of atoms and molecules, and various types of their interaction with matter and electromagnetic fields (from static to optical) make it possible to implement atom and molecular optics. It is precisely the great variety of methods for exerting an effect on an atom (or molecule) possessing a static electrical or magnetic moment, a quadrupole 

The known methods to implement atom optics (atomic-optical effects) can be classed into the following three categories  

The main effects of laser-controlled atomic-beam optics are as follows (1) colli-mation of atomic beams, (2) reflection, (3) focusing, (4) guiding in hollow fibers, and (5) interferometry. The collimation of an atomic beam by making use of 2D laser cooling has already been considered in Section 5.4.2, and the laser guiding of atoms in a hollow optical fiber in Section 6.1.3. Therefore, we shall briefly consider the atomic reflection (atomic mirrors), atomic-beam focusing, and interferometry effects. In the latter case, laser radiation is used to produce the atomic beams necessary for the observation of the interference of atoms. 

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Umklapp process In the interaction of a continuous wave (photon, electron, etc.) with the lattice, the quasi-momentum of the wave is conserved, modulo a vector in the reciprocal lattice. The introduction of these quanta of momentum leads to the Umklapp process. In many macroscopic treatments the matter is treated as a continuous medium and Umklapp processes are neglected. In our treatment, Umklapp processes are included in the coulombic interactions (calculation of the local field), but implicitly omitted in the retarded interactions, since we dropped the term (cua/c)2 in (1.64). 

As already pointed out, terms such as wave function, electron orbit, resonance, etc., with which we describe the formulations and results of wave mechanics, are borrowed from classical mechanics of matter in which concepts occur which, in certain respects at least, show a correspondence to the wave mechanical concepts in question. The same is the case with the electron spin. In Bohr s quantum theory, Uhlenbeck and Goudsmit s hypothesis meant the introduction of a fourth quantum number j, which can only take on the values +1/2 and —1/2- In wave mechanics it means that the total wave function, besides the orbital function, contains another factor, the spin function. This spin function can be represented by a or (3, whereby, for example, a describes the state j = +1/2 and P that with s = —1/2. The correspondence with the mechanical analogy, the top, from which the name spin has been borrowed, is appropriate in so far that the laevo and dextro rotatory character, or the pointing of the top in the + or — direction, can be connected with it. A magnetic moment and a... 

I ve been using marbles and atom-size insects as an analogy for electrons, but I don t want to leave you with the misconception that electrons can only be thought of as solid objects. In the introduction to this book and in the first chemistry book, I discussed how we can think of electrons (and all particles, for that matter) as collections of waves. It is this wave nature of electrons that is the basis for quantum mechanics, which is the math we use to come up with the uncertainty principle. So, while it is often convenient to consider electrons to be tiny, solid objects, you should always be aware of the model of electrons as waves. 

The development of wave mechanics has been made possible through the introduction by de Broglie of a new principle dealing with the wave character of matter. The basis of this principle is the recognition that different interpretations are appropriate to different kinds of measurements thus atoms and electrons which have hitherto been regarded as discrete particles arc considered to possess a dual character, in the sense that they may possess both corpuscular and wave properties. A duality of a similar kind had been revealed earlier in studies on the propagation of light. 

To conclude this introduction to quantum mechanics, it is interesting to note the omnipresence and the agglutinating role of Planck s constant. Indeed, if it was set equal to zero, all the construction which began with black-body radiation and the quantization of radiation energy, followed by the wave-matter duality and the Heisenberg principle. .. would fall down. In addition, the intrinsic angular momentum (spin) of some particles, including the electron, would be forced to be zero, with many consequences at the theoretical and practical levels. 

Surface waves on a finite FSS will radiate just like the Floquet currents will radiate. These matters—and, in particular, how they are being excited—will be the subject of detailed discussions in Chapter 4. It suffices in this introduction to present a typical example as shown in Fig. 1.4. We show here 25 columns with the same element dimension as earher (see insert). The angle of incidence is... 

Between the introduction of Bohr s theory and the development of quantum mechanics, there was very little in the way of new contributions to the understanding of matter—except for an important idea put forth by Louis de Broglie (Figure 9.20) in 1924. De Broglie, a scientist whose family was part of the French aristocracy, hypothesized that if a wave like light can have particle properties, why can t particles like electrons, protons, and so on have wave properties ... 

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