Broglie waves

Consider first a very small but macroscopic body, with a mass of 1 pg 10-9 kg), moving at a minute velocity of 0.1 nm s-1 (less than 1 mm per ay). The momentum is 10-19 kg m s l, and using the value of Planck s 2-11 anstant in eqn 2.21, a wavelength of 6 x 10-15 m is predicted. This is very tuch smaller than an atom (about 100 pm), and in fact the same order of t 2 i tagnitude as the size of the atomic nucleus. It is hard to think of any 

Consider now an electron, with mass 9.1 x 10-31 kg, accelerated in a potential of IV. The kinetic energy is then 1 eV, or 1.6 x 10 19 J. The momentum is most easily found from 

These experiments show conclusively that beams of electrons do have 

I wave-like properties, and allow de Broglie s formula (eqn 2.21) to be verified quantitatively. No-one who has used an LEED apparatus can be left in any doubt that this theory is correct. Simply turning a knob on the control panel changes the accelerating voltage for electrons, and hence their kinetic energy and momentum. The diffraction spots then move in exactly the way predicted by eqn 2.21, in conjunction with simple diffraction theory (see Problem 2 below). 

Electrons are peculiarly light, so that their wave properties are easy to detect. Heavier particles must have correspondingly lower velocities (and kinetic energies according to eqn 2.22) to give momenta corresponding to suitable wavelengths. Neutrons are commonly used for diffraction. They me 

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In wave mechanics the wave function 4>a(x) of a system in a state is specified by the label a. For example, in the case of a free particle a could be the momentum p, and ipa(x) would be the de Broglie wave function,... 

Atomic-beam diffraction was first demonstrated in 1930, as a verification of the concept of the de Broglie wave (Estermann and Stem, 1930). In the 1970s, it was developed into an extremely informative method for determining topography and atomic structure of solid surfaces (Steele, 1974 Goodman and Wachman, 1976). 

Figure 15. Laboratory-scale experiment for studying the behavior of de Broglie waves.

Matter is classically particulate in nature, but it also manifests wave character. The wave property of matter is related to its particle nature by de Broglie s relation A = hip, where A is known as the de Broglie wave length. 

A nonrelativistic particle is moving five times as fast as a proton. The ratio of their de Broglie wave lengths is 10. Calculate the mass of the particle. 

This is an example of the de Broglie wave-particle duality. The resulting wave equation is... 

It is assumed that the eigenfunction (x) operated on by these infinitesimal field generators is such that the same relation (803) holds between eigenvalues of the field. In order for this to be true, the eigenfunction must be the de Broglie wave function, specifically, the phase of the classical electromagnetic field. On the 0(3) level, this is a line integral, as we have seen. 

J. P. Vigier, Explicit mathematical construction of relativistic non-linear de Broglie waves described by three-dimensional (wave and electromagnetic) solitons piloted (controlled) by corresponding solutions of associated linear Klein-Gordon and Schrodinger equations, Found. Phys. 21(2) (1991). 

K. P. Sinha, E. C. G. Sudarshan, and J. P. Vigier, Superfluid vacuum carrying real Einstein-de Broglie waves, Phys. Lett. A 114A(6), 298-300 (1986). 

C. Dewdney, A. Kyprianidis, J. P. Vigier, A. Garuccio, and P. Gueret, Time dependent neutron interferometry Evidence in favor of de Broglie waves, Lett. Nuovo Cimento 40(16) (Ser. 2), 481 187 (1984). 

A. Garuccio and J. P. Vigier, An experiment to interpret E.P.R. action-at-a-distance The possible detection of real de Broglie waves, Epistemol. Lett. (1981) (includes reply by O. Costa de Beauregard). 

A. Garrucio and J. P. Vigier, Possible experimental test of the causal stochastic interpretation of quantum mechanics Physical reality of de Broglie waves, Found. Phys. 10(9—10), 797-801 (1980). 

However, the results obtained for de Broglie waves of the particles in the deuteron nucleus are in favor of presence of muons rather than pions, in the deuteron. The reason for this conclusion is that there is resonance between a nucleon s and a muon s de Broglie waves [5], which would not be the case if pions were taken as participating objects in the deuteron.2... 

The possibility for resonance between photons with de Broglie waves of the particles is elaborated on in Refs. 4 and 14. 

It is however possible to discuss several special cases analytically. The zero temperature correlation length can still be observed as long as this is smaller than the thermal de Broglie wave length At which can be rewritten for K not too close to Ku as t < f/y Kt(, KtK" 1 with tu Lpl, where we defined tk via = jj-, analogously to the definition of At, and used (30). We call this domain the quantum disordered region. 

Fig. 5. The low temperature crossover diagram of a one-dimensional CDW. t and K are proportional to the temperature and the strength of quantum fluctuations, respectively. The amount of disorder corresponds to a reduced temperature tu 0.1. In the classical and quantum disordered region, respectively, essentially the t = 0 behavior is seen. The straight dashed line separating them corresponds to At 1, i.e., K 1, where At is the de Broglie wave length. In the quantum critical region, the correlation length is given by At- Pinning (localization) occurs only for t = 0, K

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