Matter, wave nature

S. Inouye et al., Phase-Coherent Amphfication of Atomic Matter Waves, Nature 402, 641-644 (1999). 

L. Deng, E.W. Hagley, J. Wen, M. Trippenbach, Y. Band, P.S. Julienne, J.E. Simsarian, K. Helmersson, S.L. Rolston, W.D. PhiUips Four-wave mixing with matter waves. Nature 398, 218 (1999)... 

There was no experimental evidence for the wave nature of matter until 1927, when evidence was provided by two independent experiments. Davisson found that a diffraction pattern was obtained if electrons were scattered from a nickel surface, and Thomson found that when a beam of electrons is passed through a thin gold foil, the diffraction pattern obtained is very similar to that produced by a beam of X-rays when it passes through a metal foil. 

In classical mechanics both the position of a particle and its velocity at any given instant can be determined with as much accuracy as the experimental procedure allows. However, in 1927 Heisenberg introduced the idea that the wave nature of matter sets limits to the accuracy with which these properties can be measured simultaneously for a very small particle such as an electron. He showed that Ax, the product of the uncertainty in the measurement of the position x, and Ap, the uncertainty in the measurement of the momentum p, can never be smaller than M2tt ... 

As a characteristic feature, both the gap functions have nodes at poles (9 = 0,7r) and take the maximal values at the vicinity of equator (9 = 7t/2), keeping the relation, A > A+. This feature is very similar to 3P pairing in liquid 3He or nuclear matter [17, 18] actually we can see our pairing function Eq. (39) to exhibit an effective P wave nature by a genuine relativistic effect by the Dirac spinors. Accordingly the quasi-particle distribution is diffused (see Fig. 3)... 

Two main ideas are related to the development of this technique. The first one is the wave nature of the matter. As postulated by Louis de Broglie in 1924, a free electron with mass m, moving with speed v, has a wavelength % related to its momentum (p = mv) in exactly the same way as for a photon, that is,... 

The second idea related to the LEED technique is, as its name indicates, the diffraction phenomenon. With the wave nature of the electrons established, information on the interaction of a beam of light with matter can be extrapolated to understand the interaction of a beam of electrons with a crystal. In this sense, the corresponding relationship is that of a monochromatic beam of light—that with a single frequency and single wavelength—with that of a beam of electrons of fixed energy. 

Figure 6.17 shows a schematic of the LEED system. The sample is bombarded through the left by a beam of electrons. Only radiation or electrons (remember the wave nature of matter ) with the same energy as the incident beam are detected. These electrons are called elastic backscattered electrons. The detection system is a fluorescent screen placed in front of the sample. Holding the screen at a large positive potential accelerates the electrons. Once they reach it, they excite the phosphorus in the screen, marking it with bright spots characteristic of the diffraction pattern. Finally, a camera in front of the screen records the diffraction pattern. 

A note of caution The Bohr theory, even when improved and amplified, applies only to hydrogen and hydrogen-like species, such as He+ and Li+. The theory explains neither the spectra of atoms containing even as few as two electrons, nor the existence and stability of chemical compounds. The next advance in the understanding of atoms requires an understanding of the wave nature of matter. 

Combining the wave nature of matter and the probability concept of the Uncertainty Principle, M. Born proposed that the electronic wavefunction is no longer an amplitude function. Rather, it is a measure of the probability of an event when the function has a large (absolute) value, the probability for the event is large. An example of such an event is given below. 

The first convincing description of stationary quantum states was provided by the assumed wave nature of matter proposed by Louis de Broglie. The proposal had its roots in Einstein s explanation of the photoelectric effect and Compton s analysis of X-ray scattering. 

As integers always appear in Nature associated with periodic systems, with waves as the most familiar example, it is almost axiomatic that atomic matter should be described by the mechanics of wave motion. Each of the mechanical variables, energy, momentum and angular momentum, is linked to a wave variable by Planck s constant E = hu = h/r, p = h/X = hi), L = h/27r. A wave-mechanical formulation of any mechanical problem which can be modelled classically, can therefore be derived by substituting wave equivalents for dynamic variables. The resulting general equation for matter waves was first obtained by Erwin Schrodinger. 

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. 

Schrodinger in 1926 first proposed an equation for de Broglie s matter waves. This equation cannot be derived from some other principle since it constitutes a fundamental law of nature. Its correctness can be judged only by its subsequent agreement with observed phenomena (a posteriori proof). Nonetheless, we will attempt a heuristic argument to make the result at least plausible. 

The diffraction of H2 and He beams by the surfaces of alkali halide crystals was observed originally in 1930 and provided direct evidence for the wave-nature of matter. Interest in the technique has revived in recent years and considerable experimental advances have been made, particularly in the production of well-collimated and approximately mono-energetic beams for reviews, including experimental details, see refs. 231, 232. 

The interference of microscopic particles leads to a diffraction pattern with deviations with respect to the mere sum of the individual probabilities. The two events are no longer independent. If we wish to state in advance where the next particle will appear, we are unable to do so. The best we can do is to say that the next particle is more likely to strike in one area than another. A limit to our knowledge, associated with the wave-matter duality, becomes apparent. In the double-slit experiment, we may know the momentum of each particle but we do not know an5 hing about the way the particles traverse the slits. Alternatively, we could think of an experiment that would enable us to decide through which slit the particle has passed, but then the experiment would be substantially different and the particles would arrive at the screen with different distributions. In particular, the two slits would become distinguishable and independent events would occur. No interference would be detected, that is, the wave nature of the particle would be absent. In such an experiment, in order to obtain information about the particle position just beyond the slits, we would change its momentum in an unknown way. Indeed, recent experiments have shown that interference can be made to disappear and reappear in a quantum eraser (ref. 6 and references therein). 

Nobel laureate in Physics) in 1924. In quantum mechanics, two ensembles which show the same distributions for all the observables are said to be in the same state. Although this notion is being introduced for statistical ensembles, it can also be applied to each individual microsystem (see, for example, ref. 8), because all the members of the ensemble are identical, non-interacting and identically prepared (Fig. 1.4). Each state is described by a state function, ip (see, for example, ref. 3). This state function should contain the information about the probability of each outcome of the measurement of any observable of the ensemble. The wave nature of matter, for example the interference phenomena observed with small particles, requires that such state fimctions can be superposed just like ordinary waves. Thus, they are also called wavefunctions and act as probability amplitude functions. 

One of the remarkable properties of quantum mechanics is that the wave nature of matter completely escapes perception in our everyday life, although this feature is a cornerstone of the theory. The smallness of Planck s constant and therefore of the de Broglie wavelength of a macroscopic object is certainly largely responsible for the non-observability of quantum effects in the classical world. However, it is important to ask whether there are fundamental limits to quantum physics and how far we can push the experimental techniques to visualize quantum effects in the mesoscopic world for objects of increasing size, mass and complexity. Where are the fundamental limits on the way towards larger objects ... 

Radiography was thus initiated without any precise understanding of the radiation used, because it was not until 1912 that the exact nature of x-rays was established. In that year the phenomenon of x-ray diffraction by crystals was discovered, and this discovery simultaneously proved the wave nature of x-rays and provided a new method for investigating the fine structure of matter. Although radiography is a very important tool in itself and has a wide field of applicability, it is ordinarily limited in the internal detail it can resolve, or disclose, to sizes of the order of 10 cm. Diffraction, on the other hand, can indirectly reveal details of internal structure of the order of 10 cm in size, and it is with this phenomenon, and its applications to metallurgical problems, that this book is concerned. The properties of x-rays and the internal structure of crystals are here described in the first two chapters as necessary preliminaries to the discussion of the diffraction of x-rays by crystals which follows. 

The conventional conceptual content of quantum mechanics was initiated by the Copenhagen School when it was recognized that one could express the linear Schrodinger wave mechanics [28] in terms of a probability calculus, whose solutions are represented with a Hilbert function space. Max Bom then interpreted the wave nature of matter in terms of a spatially distributed probability amplitude—a wave represented by a complex function—to accompany the material particle as it moves from one place to another. The Copenhagen view was then to define the basic nature of matter in terms of the measurement process, with an underlying probability calculus, wherein the probability densities (for locating the particles of matter/volume) are the real-number-valued moduli of the matter wave amplitudes. 

But this was not Schrodinger s intention in his formulation of wave mechanics [29] Rather, it was to complete the Maxwell field formulation of electromagnetic theory by incorporating the empirically verified wave nature of matter in the source terms on the right-hand sides of Maxwell s equations. 

Bohr s explanation of spectra, which we expounded in 3, p. 72, points out the road we must follow in setting up a new atomic mechanics. In fact, long even before the discovery of the wave nature of matter, an at least provisional atomic mechanics was successfully founded by Bohr and developed by himself and his collaborators, the most prominent of whom is Kramers. 

Louis de Broglie introduces the theory of the wave nature of matter. 

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