Wavelengths, de Broglie

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. 

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... 

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]. 

The time-dependent eikonal approximation for the density operator can be introduced by analogy to the treatment for stationary states. Here the steps leading to the working equations are first outlined for a general system. For nuclear motions with short de Broglie wavelengths A = ft/P, it is assumed that the xi vary little over a wavelength so that... 

The surface sensitivity of most electron probe techniques is due to the fact that the penetration depth of electrons into metals falls to a minimum of 4 to 20 A when their kinetic energy is between 10 and 500 eV. It is also convenient that electrons at these energies have de Broglie wavelengths on the order of angstroms. With a monochromatic beam, it is possible to do LEED. 

In an experiment of thoughts the 3D piece of metal shall be reduced in the z-direction to only a few nanometres, comparable with the electronic de Broglie wavelength (t4 = k), whereas in x- and y-direction it is kept infinite a 2D quantum well is formed. Compared with the former 3D situation, the electrons in x-and y-direction can still freely move in these directions, but not in the z-direction. Electrons in this direction are confined like in a box. The states are quantized, whereas in x- and y-direction the situation does not differ from that in the 3D case. Figure 5a shows the quantized situation in z-direction with well-defined Ak values n— I, 2, etc., and Figure 5b indicates the Ak values close to 0. 

Figure 5. Electrons in a two-dimensional system, (a) The z-direction is reduced to an extension corresponding to the de Broglie wavelength of free electrons resulting in discrete AA values, (b) Instead of a quasi-continuous situation, a so-called staircase results, (c) In X- and y-direction the situation corresponds to that in Figure 4a. (Reprinted from Ref [5], 2004, with permission from Wiley-VCH.)...

We have found that only distinct values for the de Broglie wavelength satisfy the requirement that the wave function represents the motion of the particle. These distinct values are denoted as and are given by... 

Tunneling may be conceptualized in several ways. For example, the wave-particle duality of quantum mechanics indicates that particles of mass m and velocity v have an associated de Broglie wavelength given in Eq. 10.6, where h is Planck s constant."... 

Damping effects, transition state trajectory deterministic driving, 209-213 stochastically moving manifolds, 215-222 De Broglie wavelength, two-pathway excitation, coherence spectroscopy 165-166 Decoherence, two-pathway excitation, coherence spectroscopy ... 

The only size-related quantity for the incident electron is its de Broglie wavelength,... 

In the second line, we have carried the integral over the ideal gas part, which results in the temperature-dependent de Broglie wavelength, A. The final expression is similar to the familiar casting of the canonical partition function,... 

Indicated by a thermal de Broglie wavelength which is a substantial fraction of the average intermolecular separation. 

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