Constant quantum potential

In a region of constant quantum potential energy, the expression... 

Z-matriccs arc commonly used as input to quantum mechanical ab initio and serai-empirical) calculations as they properly describe the spatial arrangement of the atoms of a molecule. Note that there is no explicit information on the connectivity present in the Z-matrix, as there is, c.g., in a connection table, but quantum mechanics derives the bonding and non-bonding intramolecular interactions from the molecular electronic wavefunction, starting from atomic wavefiinctions and a crude 3D structure. In contrast to that, most of the molecular mechanics packages require the initial molecular geometry as 3D Cartesian coordinates plus the connection table, as they have to assign appropriate force constants and potentials to each atom and each bond in order to relax and optimi-/e the molecular structure. Furthermore, Cartesian coordinates are preferable to internal coordinates if the spatial situations of ensembles of different molecules have to be compared. Of course, both representations are interconvertible. 

Since TD-DFT is applied to scattering problems in its QFD version, two important consequences of the nonlocal nature of the quantum potential are worth stressing in this regard. First, relevant quantum effects can be observed in regions where the classical interaction potential V becomes negligible, and more important, where p(r, t) 0. This happens because quantum particles respond to the shape of K, but not to its intensity, p(r, t). Notice that Q is scale-invariant under the multiplication of p(r, t) by a real constant. Second, quantum-mechanically the concept of asymptotic or free motion only holds locally. Following the classical definition for this motional regime,... 

Quantum Free-Electron Theory Constant-Potential Model, The simple quantum free-electron theory (1) is based on the electron-in-a-box model, where the box is the size of the crystal. This model assumes that (1) the positively charged ions and all other electrons (nonvalence electrons) are smeared out to give a constant background potential (a potential box having a constant interior potential), and (2) the electron cannot escape from the box boundary conditions are such that the wavefunction if/ is... 

Bohm s failure to give an adequate explanation to support the pilot-wave proposal does not diminish the importance of the quantum-potential concept. In all forms of quantum theory it is the appearance of Planck s constant that signals non-classical behaviour, hence the common, but physically meaningless, proposition that the classical/quantum limit appears as h —> 0. The actual limiting condition is Vq —> 0, which turns the quantum-mechanical... 

A system of this type is not holistic, but partially holistic, which means that pairwise interaction occurs between the holistic units. The distinction drawn here between holistic and partially holistic systems is not in line with the terminology used in general philosophic discourse and in order to avoid any confusion it is preferable to distinguish between systems that interact either continuously, or discontinuously, with the quantum potential field. Quantum potential, like the gravitational potential, occurs in the vacuum, presumably with constant intensity. The quantum potential energy of a quantum object therefore only depends on the wave function of the object. 

The only difference between the classical and quantum formulations resides in the additional potential-energy term h2V2A/2mA, known as the quantum potential, Vq. In the classical case Vq 0. A quantum-mechanically stationary state occurs when Vq = k, a constant independent of x, i.e. 

It is noted that a system of particles reaches equilibrium when the resulting forces on them are zero, and hence the quantum force on a free particle must be perceived to vanish. This requires the quantum potential to be either zero or a constant, independent of position. The first condition relates to a classical particle, whereas the second condition implies... 

The intensity of a wave is proportional to the square of the amplitude, i.e. I = R2. Multiplication of the amplitude by a real constant therefore scales the intensity, but the quantum potential stays the same,... 

In terms of the causal model the kinetic energy in every stationary state with me = 0 is zero hence the total calculated energy is pure quantum potential energy. To confirm this, recall that Vq must be constant for the confined particle, i.e. 

In the following the use of the term quantum yield implies that it is either a true differential or a partial one however, another special definition is given. Derivations are carried out for thermal reactions and dependencies demonstrated and explained. This is just a matter of simplicity, since in the application of the formalism the photochemical quantum yields in contrast to the rate constants contain potential dependencies on the mechanism. This is demonstrated in Chapter 3 for complex reactions. 

As for the classical potential, the gradient of quantum potential energy defines a quantum force. A quantum object therefore has an equation of motion, m x= —VH — VV. For an object in uniform motion (constant potential) the quantum force must vanish, which requires = 0 or a constant, —k say. 

Concepts in theoretical science, which may appear difficult to comprehend on first encounter, often become accepted through familiarity, rather than insight. Such concepts end up as dogmatic belief, which casts a shadow on the understanding of related theories. The notion of a quantum/classical limit is discussed as an example. Analysed as a measure of quantum potential it is shown to clear up the related concepts of Compton wavelength, fine-structure constant, wave structures and the nature of the vacuum. In a Riemannian... 

Modeling electrochemical systems from first principles presents a considerable challenge. Quantum mechanical simulations are typically carried out within the canonical ensemble formalism where the number of electrons remains constant. The free energy is calculated with a constant temperature, volume and number of electrons F(T, F, Ne). Electrochemical systems, on the other-hand, are typically performed at a constant chemical potential in the grand canonical ensemble where p(T, F, Ne) is a constant. Throughout this book we have presented examples where the number of electrons is preserved upon chemical reaction. In order to model an electrochemical system, we would have to model... 

Montalti M, Credl A, Prod L, Gandolfi MT (2006), Handbook of photochemistry, 3rd edn. CRC Press, Boca Raton. An essential reference book containing data tables for a wide range of compounds, and a variety of reference materials including quantum yields, lifetimes, quenching rate constants, electrochemical potentials and solvent properties as well as information on standard procedures used in chemical actinometry, determination of emission and excitation spectra correction factors, and quantum yield measurements and also information on equipment such as lamps and filters. 

Parr et al. [1] for the first time put the qualitative concept, electronegativity, on a sound quantum mechanical basis when they defined, within the paradigm of density functional theory of Hohenberg and Kohn [28], that the electronegativity is the negative of a partial derivative of energy (E) of an atomic or molecular system with respect to the number of electrons N, at a constant external potential v(r). 

Here h(x) is the Heaviside step function with h(x > 0) = 1 and h(x > 0) = 0 (not to be confused with Planck s constant). The limit a(J.. . ) indicates that the sunnnation is restricted to channel potentials witir a given set of good quantum numbers (J.. . ). 

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