De Broglie

The miderstanding of molecular motions is necessarily based on quaiitum mechanics, the theory of microscopic physical behaviour worked out in the first quarter of the 20th century. This is because molecules are microscopic systems in which it is impossible—or at least very dangerous —to ignore the dual wave-particle nature of matter first recognized in quaiitum theory by Einstein (in the case of classical waves) and de Broglie (in the case of classical particles). 

The first tenn is what one would expect to obtain classically for a particle of momentum hk, and it is much bigger than the second tenn provided k a 1. Since the de Broglie wavelength X is In/kQ, this condition is equivalent to the statement that the size of the wavepacket be much larger than tire de Broglie wavelength. 

Atom-surface interactions are intrinsically many-body problems which are known to have no analytical solutions. Due to the shorter de Broglie wavelengdi of an energetic ion than solid interatomic spacings, the energetic atom-surface interaction problem can be treated by classical mechanics. In the classical mechanical... 

Figure C 1.4.1. The situation of various physical phenomena along a scale of temperature plotted against de Broglie wavelengdi.

This is often written in terms of the de Broglie thermal wavelength, A ... 

De Broglie s and Schrodinger s contributions to our present understand ing of electrons were de scribed in Section 1 1... 

The picture of the electron in an orbit as a standing wave does, however, pose the important question of where the electron, regarded as a particle, is. We shall consider the answer to this for the case of an electron travelling with constant velocity in a direction x. The de Broglie picture of this is of a wave with a specific wavelength travelling in the x direction as in Figure 1.4(a), and it is clear that we cannot specify where the electron is. 

According to (2.29), dissipation reduces the spread of the harmonic oscillator making it smaller than the quantum uncertainty of the position of the undamped oscillator (de Broglie wavelength). Within exponential accuracy (2.27) agrees with the Caldeira-Leggett formula (2.26), and similar expressions may be obtained for more realistic potentials. 

In conclusion of this section, we write out the expressions for the density matrix of a free particle and a harmonic oscillator. In the former case p(x, x P) is a Gaussian with the half-width equal to the thermal de Broglie wavelength... 

The underlying principle of RHEED is that particles of matter have a wave character. This idea was postulated by de Broglie in (1924). He argued that since photons behave as particles, then particles should exhibit wavelike behavior as well. He predicted that a particle s wavelength is Planck s constant h divided by its momentum. The postulate was confirmed by Davisson and Germer s experiments in 1928, which demonstrated the diffraction of low-energy electrons from Ni. ... 

When Davisson and Germer reported in 1927 that the elastic scattering of low-energy electrons from well ordered surfaces leads to diffraction spots similar to those observed in X-ray diffraction [2.238-2.240], this was the first experimental proof of the wave nature of electrons. A few years before, in 1923, De Broglie had postulated that electrons have a wavelength, given in A, of ... 

Band gap engineetring confined hetetrostruciutres. When the thickness of a crystalline film is comparable with the de Broglie wavelength, the conduction and valence bands will break into subbands and as the thickness increases, the Fermi energy of the electrons oscillates. This leads to the so-called quantum size effects, which had been precociously predicted in Russia by Lifshitz and Kosevich (1953). A piece of semiconductor which is very small in one, two or three dimensions - a confined structure - is called a quantum well, quantum wire or quantum dot, respectively, and much fundamental physics research has been devoted to these in the last two decades. However, the world of MSE only became involved when several quantum wells were combined into what is now termed a heterostructure. 

The existence of carbon nanotubes with diameters small compared to the de Broglie wavelength has been described by Iijima[l,2,3] and others[4,5]. The energy band structures for carbon nanotubes have been calculated by a number of authors and the results are summarized in this issue by M.S. Dresselhaus, G. Dres-selhaus, and R. Saito. In short, the tubules can be either metallic or semiconducting, depending on the tubule diameter and chirality[6,7,8]. The calculated density of states[8] shows singularities... 

So, despite the very small diameter of the MWCNT with respeet to the de Broglie wavelengths of the charge carriers, the cylindrical structure of the honeycomb lattice gives rise to a 2D electron gas for both weak localisation and UCF effects. Indeed, both the amplitude and the temperature dependence of the conductance fluctuations were found to be consistent with the universal conductance fluctuations models for mesoscopic 2D systems applied to the particular cylindrical structure of MWCNTs [10]. 

---