The Wave Nature of Matter

Evidently light has wave and particle aspects, and we can describe it in terms of photons, which are associated with waves of frequency v = E/h. Now photons are rather peculiar particles in that they have zero rest mass. In fact, they can exist only when traveling at the speed of light. The more normal particles in our experience have nonzero rest masses and can exist at any velocity up to the speed-of-light limit. Are there also waves associated with such normal particles  

De Broglie s relation can be reached as follows. Einstein s relation for photons is 

But a photon carrying energy E has a relativistic mass given by 

A normal particle, with nonzero rest mass, travels at a velocity v. If we regard Eq. (1 -40) as merely the high-velocity limit of a more general expression, we arrive at an equation relating particle momentum p and associated wavelength A  

m refers to the rest mass of the particle plus the relativistic correction, but the latter is usually negligible in comparison to the former. 

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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 ... 

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 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. 

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 ... 

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. 

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. 

Uncertainty Arising from the Wave Nature of Matter... 

Instead, even when a system hypothetically experiences zero quantum fluctuations, the wave nature of the system will be still slightly dominant over its partiele side at both observed and free evolutions see the upper branehes of Eqs. (4.556) and (4.562). These extremes show that the wave nature of matter will never be fully transferred to particle contents and the mesosystems will never be fully charaeterized by pure particle (or meehanieal) features. 

Because of the wave nature of matter, a quantum particle can sometimes overcome energy barriers that the particle, if behaving classically, would have insufficient energy to cross. This phenomenon, called quantum tunneling, is the basis for a number of scientific applications, such as the scanning tunneling microscope (STM). 

Experimental evidence for the wave nature of matter comes from interference and diffraction experiments involving sub-atomic particles, such as electrons, neutrons and photons (light particles). When a beam of electrons is directed at a thin metal foil with two narrow slits as shown in Figure 12.8, the interference patterns can be recorded on photographic film. Similar interference patterns can be observed with protons, neutrons and with some molecules. 

The variation of reflection from thin films (e.g., soap bubbles) of light of different wavelengths results in the perception of colors and is a fanuliar example of scattering interference. Less familiar is the variation in reflection of a particle beam, outlined above. However, once we recognize the wave nature of matter, we must expect particles to manifest the same sort of wave properties we associate with light. 

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