Broglie Matter Waves

We now come to a mysterious concept first put forth by Prince Louis De Broglie in 1923. Roughly, here is the reasoning due to De Broglie. 

The problem with interpretation of this idea of matter waves is that waves should exhibit diffraction which is not commonly observed macroscopically. Consider a typical rifle buUet of 4.2 g traveling at a speed of 965 m/sec. The De Broglie matter wavelength for that buUet would be 

485 X 10—8 cm = 5.485 A, which is comparable in size to a molecule. Thus, wave mechanics is useless for macroscopic calculations but essential for molecular calculations  

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The group velocity of de Broglie matter waves are seen to be identical with particle velocity. In this instance it is the wave model that seems not to need the particle concept. However, this result has been considered of academic interest only because of the dispersion of wave packets. Still, it cannot be accidental that wave packets have so many properties in common with quantum-mechanical particles and maybe the concept was abandoned prematurely. What it lacks is a mechanism to account for the appearance of mass, charge and spin, but this may not be an insurmountable problem. It is tempting to associate the rapidly oscillating component with the Compton wavelength and relativistic motion within the electronic wave packet. 

In this chapter we explored the question of how to use the De Broglie matter waves to describe... 

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. 

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... 

In order to describe microscopic systems, then, a different mechanics was required. One promising candidate was wave mechanics, since standing waves are also a quantized phenomenon. Interestingly, as first proposed by de Broglie, matter can indeed be shown to have wavelike properties. However, it also has particle-Uke properties, and to properly account for this dichotomy a new mechanics, quanmm mechanics, was developed. This chapter provides an overview of the fundamental features of quantum mechanics, and describes in a formal way the fundamental equations that are used in the construction of computational models. In some sense, this chapter is historical. However, in order to appreciate the differences between modem computational models, and the range over which they may be expected to be applicable, it is important to understand the foundation on which all of them are built. Following this exposition. Chapter 5 overviews the approximations inherent... 

In physics the concept is known as the property of all pairs of conjugate variables, such as position and momentum, mathematically related by Fourier transformation. The de Broglie formula that relates the momentum of matter waves to wavelength... 

De Broglie developed an equation that allowed him to calculate the wavelength of the matter wave associated with any object, from a bowling ball to an electron. Objects that we can see and can interact with have calculated wavelengths that are smaller than electrons. Their wavelengths are so tiny compared to their size that they do not have any measurable effect on the motion of the objects. 

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. 

We shall therefore briefly review the experimental efforts in this field throughout the last century. Soon after Louis de Broglie proposed his wave hypothesis for material particles matter wave phenomena were experimentally verified for electrons [Davisson 1927], atoms and dimers [Estermann 1930], and neutrons [Halban 1936 Gahler 1991]. A replica of Young s double-slit experiment with matter waves was demonstrated by Jonsson for electrons [Jonsson 1974], by Zeilinger et al. for neutrons [Zeilinger 1988], by Carnal and Mlynek for atoms [Carnal 1991] and by Schollkopf and Toennies for small molecules and noble gas clusters [Schollkopf 1994 Schollkopf 1996 Bruch 2002],... 

The answers to the above questions, not all of which need he presented here, were formulated between 1925 and 1926, in the revolution of modern quantum theory, which shook the foundations of physics and philosophy. Remarkably, the central theme of quantum theory was the nature of light, and what came to be called the wave-particle duality. But other broader implications of the new theory existed, and the first inkling of this was given in 1924 by Louis de Broglie (Fig. 3.26) in his doctoral dissertation. He postulated that particles may also possess wavelike properties and that these wavelike properties would manifest themselves only in phenomena occurring on an atomic scale, as dictated by Planck s constant. He also postulated that the wavelength of these matter waves, for a given particle such as an electron or proton, would be inversely proportional to the particle s momentum p, which is a product of its mass m and speed... 

It is reasonable to assume, as de Broglie did in 1924, that since photons can behave as moving particles moving particles may show wave properties. From the previous equation, we can devise that the wavelength of such matter waves is... 

Louis de Broglie proposes wave nature of matter. 

Figure 1. A binary optical multilayer (a) together with its quantum mechanical counterpart a particle with energy Ep = hco (co being the de Broglie frequency in terms of matter waves [6]) in ID periodic (b) and non-periodic (c) binary stepwise potential. The potential u replaces the refractive index n. Elementary wells are defined as fragments (m, di), i = 1,2.

Because de Broglie s hypothesis is applicable to all matter, any object of mass m and velocity v would give rise to a characteristic matter wave. However, Equation 6.8 indicates that the wavelength associated with an object of ordinary size, such as a golf ball, is so tiny as to be completely unobservable. This is not so for an electron because its mass is so small, as we see in Sample Exercise 6.5. 

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