Quantum length

F.F. Abraham et al Spanning the continuum to quantum length scales in a dynamic simulation of brittle fracture. Europhys. Lett. 44, 783-787 (1998)... 

Zero-point energy effects can be traced to the light masses of the constituents in quantum clusters and to the extremely low temperatures in the optical molasses and condensates. The zero-point energy effects can be described in terms of the ratio A of the quantum lengths... 

The de Boer parameter, Eq. (3), corresponds to the quantum length ratio, Eq. (2), at temperature T =G /3fes, where ks is the Boltzmann factor. For weakly interacting particles in ultracold clouds (categories 4 and 5) the thermal de Broglie wavelength... 

The quantum lengths ratio A involving the thermal de Broglie wavelength determines the critical temperatures for noninteracting bosons (Section I.C). 

Spanning the Continuum to Quantum Length Scales in a Dynamic Simulation of Brittle Fracture. 

It is now necessary to examine the partition function in more detail. The energy states for translation are assumed to be given by the quantum-mechanical picture of a particle in a box. For a one-dimensional box of length a. 

Diffraction is based on wave interference, whether the wave is an electromagnetic wave (optical, x-ray, etc), or a quantum mechanical wave associated with a particle (electron, neutron, atom, etc), or any other kind of wave. To obtain infonnation about atomic positions, one exploits the interference between different scattering trajectories among atoms in a solid or at a surface, since this interference is very sensitive to differences in patii lengths and hence to relative atomic positions (see chapter B1.9). 

As a scientific tool, ab initio quantum chemistry is not yet as accurate as modem laser spectroscopic measurements, for example. Moreover, it is difficult to estimate the accuracies with which various methods predict bond energies and lengths, excitation energies and the like. In the opinion of tlie author, chemists who... 

It is eonnnon praetiee in elassieal eomputer simulations not to attempt to represent intramoleeular bonds by tenns in the potential energy fiinetion, beeause these bonds have very high vibration frequeneies and should really be treated in a quantum meehanieal way rather than in tire elassieal approximation. Instead, the bonds are treated as being eonstrained to have fixed length, and some straightforward ways have been devised to ineorporate these eonstraints into the dynamies (see later). 

Quantum mechanical calculation of molecular dynamics trajectories can sim ulate bon d breakin g and frtrm ation.. Although you dt) n ot see th e appearance or disappearan ce ofhonds, you can plot the distan ce between two bonded atom s.. A distan ce excccdi n g a theoretical bond length suggests bond breaking. 

Quantum mechanics is primarily concerned with atomic particles electrons, protons and neutrons. When the properties of such particles (e.g. mass, charge, etc.) are expressed in macroscopic units then the value must usually be multiplied or divided by several powers of 10. It is preferable to use a set of units that enables the results of a calculation to he reported as easily manageable values. One way to achieve this would be to multiply eacli number by an appropriate power of 10. However, further simplification can be achieved by recognising that it is often necessary to carry quantities such as the mass of the electron or electronic charge all the way through a calculation. These quantities are thus also incorporated into the atomic units. The atomic units of length, mass and energy are as follows ... 

We shall examine the simplest possible molecular orbital problem, calculation of the bond energy and bond length of the hydrogen molecule ion Hj. Although of no practical significance, is of theoretical importance because the complete quantum mechanical calculation of its bond energy can be canied out by both exact and approximate methods. This pemiits comparison of the exact quantum mechanical solution with the solution obtained by various approximate techniques so that a judgment can be made as to the efficacy of the approximate methods. Exact quantum mechanical calculations cannot be carried out on more complicated molecular systems, hence the importance of the one exact molecular solution we do have. We wish to have a three-way comparison i) exact theoretical, ii) experimental, and iii) approximate theoretical. 

Figure 9.40 shows the IgSg band of HCN involving one quantum of Vi, the CN stretching vibration, and six quanta of V3, the CH stretching vibration. This extremely weak band was observed using a cavity length of 1.3 m. 

The two-dimensional carrier confinement in the wells formed by the conduction and valence band discontinuities changes many basic semiconductor parameters. The parameter important in the laser is the density of states in the conduction and valence bands. The density of states is gready reduced in quantum well lasers (11,12). This makes it easier to achieve population inversion and thus results in a corresponding reduction in the threshold carrier density. Indeed, quantum well lasers are characterized by threshold current densities as low as 100-150 A/cm, dramatically lower than for conventional lasers. In the quantum well lasers, carriers are confined to the wells which occupy only a small fraction of the active layer volume. The internal loss owing to absorption induced by the high carrier density is very low, as Httie as 2 cm . The output efficiency of such lasers shows almost no dependence on the cavity length, a feature usehil in the preparation of high power lasers. 

Section 4.04.1.2.1). The spectroscopic and the diffraction results refer to molecules in different vibrational quantum states. In neither case are the- distances those of the hypothetical minimum of the potential function (the optimized geometry). Nevertheless, the experimental evidence appears to be strong enough to lead to the conclusion that the electron redistribution, which takes place upon transfer of a molecule from the gas phase to the crystalline phase, results in experimentally observable changes in bond lengths. 

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