The de Broglie Wavelength

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. 

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]. 

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... 

The chemical potential pa on the left is the full chemical potential including ideal and excess parts. In this chapter we will scale the chemical potentials by (3 and often refer to this unitless quantity as the chemical potential. 3/ia yields the absolute activity. The first term on the right is the ideal-gas chemical potential, where pa is the number density, Aa is the de Broglie wavelength, and q 1 is the internal (neglecting translations) partition function for a single molecule without interactions with any other molecules. 

Intermolecular vibration involving nearest-neighbor molecules are easily excited in liquid hydrocarbons, since the de Broglie wavelength of the electron at a few tenths of electron-volt energy is comparable to the intermolecular separation. The quantum for this vibration lies in the far IR and can be observed indirectly by Raman spectra. Raman shifts in many hydrocarbon liquids have... 

Although the De Broglie wavelength of free electrons is 0.1 nm, the value of an electron in a small crystallite can be much larger because the effective mass of electrons in a small particle is considerably smaller. Energy levels evolve from HOMO and LUMO to those of clusters, Q-sized particles, and finally bulk semiconductor. Figure 7.7 shows the energy levels in bulk- and Q-sized particulate semiconductors. 

For typical lepton energies of a few MeV, the de Broglie wavelength is of order 100 times the nuclear radius and when orbital angular momentum is zero, one can use the allowed approximation for their wave functions... 

Since a sine function is continuous, single-valued and finite for all real values of its argument, the only restriction on Ex — p2x/2m is that it be positive. The energy spectrum is therefore continuous as for a classical particle and the constant k = p/h, or in terms of the de Broglie wavelength, k = 2tt/ is also known as the wave vector of the particle. 

At very short distances (of the order of the De Broglie wavelength), quantum effects become very large because the localization energy is getting important. 

The electronic structure and hence optical properties of nanomaterials depend on the core size. For example, nanoparticles of core size >3 nm show surface plasmon resonance, which is due to the excitation of surface plasmons of nanoparticles by light. When the size of gold nanoparticles comes down to around 1 nm, which is equal to the de Broglie wavelength of the conduction electrons, the electronic bands... 

The de Broglie wavelength of thermal He atoms is comparable with the interatomic distances of surfaces and adsorbed layers. Thus, from measurements of the angular positions of the diffraction peaks the size and orientation of the 2D unit cell, i.e. the structure of the outermost layer, can be straightforwardly determined. Analysis of the peak intensities yields the potential corrugation, which usually reflects the geometrical arrangement of the atoms within the 2D unit celP. 

We recovered here the usual WKB formula for tunneling probability, which exhibits an exponentially decaying behavior. On the other hand, from Eq. (2.10), we observed immediately that resonances occur when the thickness of the barrier equals integer multiples of one half of the de Broglie wavelength in the barrier region. 

The second view is based on the uncertainty relation between coordinate and momentum. In the region of a classically allowed barrier, the de Broglie wavelength of an electron is... 

Using a nozzle. He beams with narrow kinetic energy distribution can be obtained. The de Broglie wavelength of He with kinetic energy E is... 

The de Broglie wavelength associated with helium atoms is given by... 

The latter equation implies that a particle of momentum, p, has an associated wavelength, A = h/p, the de Broglie wavelength. This was verified a year... 

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