DeBroglie waves

X(b, g) Angle of deflection b Impact parameter pgb Angular momentum g Initial relative speed pg Relative momentum 9t(a) Phase shift l Angular momentum quantum number h ft l +1) Angular momentum a — pgffi = 2jr/k Wavenumber of the deBroglie wave ah Relative momentum... 

The time evolution of the wavefunetion E is determined by solving the time-dependent Sehrodinger equation (see pp 23-25 of EWK for a rationalization of how the Sehrodinger equation arises from the elassieal equation governing waves, Einstein s E=hv, and deBroglie s postulate that )i=h/p)... 

DeBroglie, in the mid 1920s, proposed the idea that particles could be treated as waves by the relationship, X = h/mv. This equation related the mass (m) and velocity (v) of a particle to its wavelength (X) by using Planck s constant (h). 

Thermal neutrons have energies comparable to the excitation energies of molecular solids and because of their mass they carry momentum. This momentum or wave-vector, k, is conventionally represented through the characteristic deBroglie wavelength, X and hence, k = 2n/X. Typical neutron wavelengths match the interatomic distances in solids, ca. 2 A and, unlike the... 

But we know more than the fact that a particle has attributes of a wave. From the work of Einstein and deBroglie, we know that the energy of the particle is related to the frequency of the corresponding wave by... 

Max Planck noted that in certain situations, energy possessed particlelike properties. A French physicist, Louis deBroglie, hypothesized that the reverse could be true as well Electrons could, at times, behave as waves rather than particles. This is known today as deBroglie s wave-particle duality. 

Werner Heisenberg, a German physicist, building on deBroglie s hypothesis, argued that it would be impossible to exactly specify the location of a particle (such as the electron) because of its wavelike character (a wave travels indefinitely in space in contrast to a particle that has fixed dimensions). This hypothesis in turn led to the Heisenberg uncertainty principle (1927), which states that it is impossible to specify both the location and the momentum (momentum is the product of mass and velocity) of an electron in an atom at the same time. 

As we noted at the end of Chapter 2, the success of Bohr s theory was short-lived. Emission spectra of multi-electron atoms (recall that the hydrogen atom has only one electron) could not be explained by Bohr s theory. DeBroglie s statement that electrons have wave properties served to intensify the problem. Bohr stated that electrons in atoms had very specific locations. The very nature of waves, spread out in space, defies such an exact model of electrons in atoms. Furthermore, the exact model is contradictory to Heisenberg s Uncertainty Principle. 

DeBroglie considered electrons to have both wave and particle properties. 

The word radiation was used until about 1900 to describe electromagnetic waves. Around the turn of the century, electrons. X-rays, and natural radioactivity were discovered and were also included under the umbrella of the term radiation. The newly discovered radiation showed characteristics of particles, in contrast to the electromagnetic radiation, which was treated as a wave. In the 1920s, DeBroglie developed his theory of the duality of matter, which was soon afterward proved correct by electron diffraction experiments, and the distinction between particles and waves ceased to be important. Today, radiation refers to the whole electromagnetic spectrum as well as to all the atomic and subatomic particles that have been discovered. 

And what does that mean In the 1920s Louis DeBroglie described electrons as both particles and waves because they have precise mass, go splat-splat-splat (or click-click-click ) into Geiger counters yet show interference like radio and light waves. It is one thing to say particle-waves and quite another to really picture them. Try it. Our problem is that electrons are outside of both our direct senses and experiences. As Bronowski notes, twentieth-century physics introduced abstraction and uncertainty and the need for what he describes as tolerance in modeling nature." The nineteenth-century satire Flatland by Shakespearean scholar Edwin A. Abbott illustrates our limitations. ... 

The electrons in a three-dimensional metal spread as waves of various wavelengths usually called the DeBroglie wavelength. 

The union of deBroglie s, Planck, Heisenberg, Schrodinger and Bohr s ideas lead to the development of the wave-mechanical view of the atom. The solution to Schrodinger s equation provides for 3 quantum numbers added to Pauli s idea that no two electrons could have the same solution set for Schrodinger s equation (known as Pauli exclusion principle), we have four quantum numbers that describe the most probable orbital for an electron of a given energy or the quantum mechanical model of the atom. 

Neutron scattering involves interactions of a neutron beam with atomic nuclei (Figure 4.3). The wave like nature of a beam of neutron particles each with mass M is incorporated in the deBroglie equation which states that the wave length of a neutron beam, A, is inversely proportional to the particle velocity, v, or... 

E. Schrodinger (1926), following the earher work of L. deBroglie (1924), advanced a fundamental equation of wave mechanics. The Schrodinger equation of wave mechanics was developed for waves oscillating in three dimensions with co-ordinates x, y and z ... 

The cause of this kind of decoherence is evidently thermal because of the alteration of the deBroglie wavelength of the wave packet associated with the quantum particle. Dissipation, on the other hand, arises from the exchange of energy between the system, which, in this case, comprise the tunneling particle and the environment of the electron cloud, or, for that matter, phonons that get excited because of the elastic distortion created by the particle (which are called interstitial sites). In either case, the coupling with the environment can be modeled as... 

In order to more completely understand reciprocal space relative to real space defined by Cartesian coordinates, we need to first recall the deBroglie relationship that describes the wave-particle duality of matter ... 

In wave mechanics the momentum p is connected with the deBroglie wavelength, X-B, of the electron... 

Referring to Fig. 2.2A, we see that wavefunction is equivalent to a standing wave with a wavelength X of 2Z/ . We therefore also could write the momentum (nhl2l) as /i//l or hv, where v is the wavenumber (1/A). This is consistent with deBroglie s expression linking the momentum of a free particle to the wavenumber of an associated wave (Eq. B2.3.1). But the... 

---