Waves matter

Louis de Broglie proposed that electrons, known as particles or corpuscles with mass and negative charge, can be considered as waves. He derived the wavelength X for a particle by applying Einstein s theory of relativity on a wave packet. His conclusion was that X should be related to the momentum p as 

This is the sinus form for wave expansion in one dimension. According to Equation 1.8, at a certain point (x = constant), waves with time period 2%l i will pass. The frequency will be V = a 2%. On the other hand, if the time is kept constant, waves with wavelength X = 2 i/k will be distributed on the x-axis. 

In the mind of de Broglie, the wave accompanies the particle. He also talked about a pilot wave that guided the particle in its path. It is unclear in the theory of de Broglie what the physical meaning of the matter wave is. Is it a longitudinal wave like sound waves How then could waves appear in a vacuum The meaning of the waves was not cleared up until the interpretation of Max Born some years later. 

if the electron has achieved a kinetic energy of e V = 1 eV (in atomic units), the de Broglie wavelength is roughly 12 A (1.2 nm). 

Equation 1.8 may be generalized to a complex three-dimensional function  

Mathematical aside in this equation, / represents the square root of minus one. Since 

The first term in parentheses is the expansion of cosx and the second term in parentheses is the expansion of sinx. Thus e = cosx + /sinx. 

In three dimensions the constant k becomes a vector, which is often referred to as the wave vector. Here, k will be used to indicate the magnitude of the wave vector, and this can be related to the wavelength A by noting that the magnitude of the wavefunction y/ does not change 

The kinetic energy, T, is equal to p /2m, and when this is combined with equation (1.13) we obtain  

The use of 7k as a shorthand way of writing h/(2/c) wili be extensively used throughout this book, 

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The history of EM (for an overview see table Bl.17,1) can be interpreted as the development of two concepts the electron beam either illuminates a large area of tire sample ( flood-beam illumination , as in the typical transmission electron microscope (TEM) imaging using a spread-out beam) or just one point, i.e. focused to the smallest spot possible, which is then scaimed across the sample (scaiming transmission electron microscopy (STEM) or scaiming electron microscopy (SEM)). In both situations the electron beam is considered as a matter wave interacting with the sample and microscopy simply studies the interaction of the scattered electrons. 

Inertial sensors are useful devices in both science and industry. Higher precision sensors could find practical scientific applications in the areas of general relativity (Chow et ah, 1985), geodesy and geology. Important applications of such devices occur also in the field of navigation, surveying and analysis of earth structures. Matter-wave interferometry has recently shown its potential to be an extremely sensitive probe for inertial forces (Clauser, 1988). First, neutron interferometers have been used to measure the Earth rotation (Colella et ah, 1975) and the acceleration due to gravity (Werner et ah, 1979) in the end of the seventies. In 1991, atom interference techniques have been used in... 

Principle of a light pulse matter-wave interferometer... 

For matter waves hk is the particle momentum and the uncertainty relation AxAp > h/2, known as the Heisenberg uncertainty principle. 

For matter waves wavelength and frequency are assumed to relate to particle momentum and energy according to A = h/p and 1/ = E/h. The proportionality factor is Planck s constant h. Hence... 

De Broglie s hypothesis of matter waves received experimental support in 1927. Researchers observed that streams of moving electrons produced diffraction patterns similar to those that are produced hy waves of electromagnetic radiation. Since diffraction involves the transmission of waves through a material, the observation seemed to support the idea that electrons had wave-like properties. 

W. Kohn, Nobel Lecture Electronic Structure of Matter-Wave Functions and Density Functionals, Rev. Mod. Phys. 71 (1999), 1253. 

Ultrafast laser excitation gives excited systems prepared coherently, as a coherent superposition of states. The state wave function (aprobabihty wave) is a coherent sum of matter wave functions for each molecule excited. The exponential terms in the relevant time-dependent equation, the phase factors, define phase relationships between constituent wave functions in the summation. 

The matter wave function is formed as a coherent superposition of states or a state ensemble, a wave packet. As the phase relationships change the wave packet moves, and spreads, not necessarily in only one direction the localized launch configuration disperses or propagates with the wave packet. The initially localized wave packets evolve like single-molecule trajectories. 

When Schrodinger introduced matter wave groups or wave packets in 1926 they were strictly theoretical, quantum mechanical formalisms. There were no experimentally accessible ways to prepare matter wave packets from molecules. Now... 

Excitations of molecules with femtosecond laser pulses lead to excited-state matter wave packets coherently, launching them with such well-defined spatial resolution and coherence in nuclear motions that they evolve like single-molecule trajectories. Both electronically excited and vibrationally excited ground-state species may be studied. The structural change versus time profile of a reaction turns out to be compatible with classical modes of thinking. 

Here, is the angular frequency of a matter wave, such as that of an electron, is its wave number magnitude, and m ) is the rest mass of the particle corresponding to the matter wave. The rest mass could be the photon s rest mass, estimated to be less than 10 68 kg. 

This result is true for all matter waves and also in the Michelson-Gale experiment, where it has been measured to a precision of one part in 1023 [49]. Hasselbach et al. [51] have demonstrated it in electron waves. We have therefore shown that the electrodynamic and kinematic explanation of the Sagnac effect gives the same result in a structured vacuum described by 0(3) gauge group symmetry. 

The starting point of de Broglie s theory is the belief that the reality is observer-independent even if the observer interacts and therefore modifies in greater or lesser degree the external reality. Therefore in this model it is assumed that the matter waves are real physical waves different from the common statistical wave , Active and arbitrarily normalized. This real wave is composed of an extended, yet finite wave 0, plus a singularity , such that... 

A second interpretation of the Aharonov-Bohm effect was devised by Boyer [65,66], who used matter waves associated to moving electrons. Waves coming from each slit interfere with a phase shift = 2jidistance between two slits. If P is the impulse of an electron in the beam, the de Broglie relation gives us P 2nh/X. This results in the fact that the phase... 

Therefore, it has been shown convincingly that electrodynamics is an 0(3) invariant theory, and so the 0(3) gauge invariance must also be found in experiments with matter waves, such as matter waves from electrons, in which there is no electromagnetic potential. One such experiment is the Sagnac effect with electrons, which was reviewed in Ref. 44, and another is Young interferometry with electron waves. For both experiments, Eq. (584) becomes... 

The shaded area in this sketch is not arbitrary, as it is determined by the right-hand side of Eq. (587). The line integrals OA and AO change sign, and this accounts for reflection of matter waves and for the Sagnac and Young effects in matter waves, such as electron waves. Therefore, the electron is an 0(3) invariant entity, as shown by the Sagnac effect for electron waves [44]. It follows that the Dirac equation should be developed as an 0(3) invariant equation. 

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