Quantum assumption

As photon momentum p = E/c, the quantum assumption E = hu implies that p = hu/c = h/X. This relationship between mechanical momentum and wavelength is an example of electromagnetic wave-particle duality. It reduces the Compton equation into ... 

The explanation for these regular series lies in the existence of discrete, quantized energy levels. In 1913 Niels Bohr was able to derive the formula for these series in terms of the ad hoc quantum assumptions of the BOHR THEORY. In the mid-1920s the formula was derived in a deductive way from quantum mechanics. 

Carrying this condition through as in the hydrogen atom case and again making the quantum assumption... 

Most basic quantum mechanics was developed by 1930. However, the development of quantum mechanics as applied to electrons also led to new theories of the nucleus, all of which today inherently contain quantum assumptions. Today, quantum mechanics encompasses the entire behavior of the atom. Because chemistry starts with atoms, quantum mechanics provides the very basis of modem chemical science. 

Applications of quantum mechanics to chemistry invariably deal with systems (atoms and molecules) that contain more than one particle. Apart from the hydrogen atom, the stationary-state energies caimot be calculated exactly, and compromises must be made in order to estimate them. Perhaps the most useful and widely used approximation in chemistry is the independent-particle approximation, which can take several fomis. Conuiion to all of these is the assumption that the Hamiltonian operator for a system consisting of n particles is approximated by tlie sum... 

In general, the phonon density of states g(cn), doi is a complicated fimction which can be directly measured from experiments, or can be computed from the results from computer simulations of a crystal. The explicit analytic expression of g(oi) for the Debye model is a consequence of the two assumptions that were made above for the frequency and velocity of the elastic waves. An even simpler assumption about g(oi) leads to the Einstein model, which first showed how quantum effects lead to deviations from the classical equipartition result as seen experimentally. In the Einstein model, one assumes that only one level at frequency oig is appreciably populated by phonons so that g(oi) = 5(oi-cog) and, for each of the Einstein modes. is... 

In either case, the structure of the solvation shell has to be calculated by otiier methods supplied or introduced ad hoc by some fiirther model assumptions, while charge distributions of the solute and within solvent molecules are obtained from quantum chemistry. 

Figure A3,12.2(a) illnstrates the lifetime distribution of RRKM theory and shows random transitions among all states at some energy high enongh for eventual reaction (toward the right). In reality, transitions between quantum states (though coupled) are not equally probable some are more likely than others. Therefore, transitions between states mnst be snfficiently rapid and disorderly for the RRKM assumption to be mimicked, as qualitatively depicted in figure A3.12.2(b). The situation depicted in these figures, where a microcanonical ensemble exists at t = 0 and rapid IVR maintains its existence during the decomposition, is called intrinsic RRKM behaviour [9].

With the assumption of hannonic oscillators, the molecule s quantum energy levels are... 

Apparently, the most natural choice for the electronic basis functions consist of the adiabatic functions / and tli defined in the molecule-bound frame. By making use of the assumption that A" is a good quantum number, we can write the complete vibronic basis in the form... 

This paper is meant as a contribution to systematize the quantum-classical modeling of molecular dynamics. Hence, we are interested in an extended theoretical understanding of the models rather than to further contribute to the bunch of numerical experiments which have been performed on certain models by applying them to particular molecular systems. Thus, we will carefully review the assumptions under which our models are known to approximate the full quantum dynamical (QD) evolution of the system. This knowledge... 

We have assumed that the order of the subscripts on the atomic orbitals p is immaterial in writing a, p, and S. In the general case, these assumptions are not self-evident, especially for p. The interested reader should consult a good quantum mechanics text (e.g., Hanna, 1981 McQuarrie, 1983 Atkins and Eriedman, 1997) for their justification or critique. 

Hamiltonian quantum mechanical operator for energy, hard sphere assumption that atoms are like hard billiard balls, which is implemented by having an infinite potential inside the sphere radius and zero potential outside the radius Hartree atomic unit of energy... 

But probably the most serious barrier has been the paralysis that overtakes the inexperienced mind when it is faced with an explosion. This prevents many from recognizing an explosion as the orderly process it is. Like any orderly process, an explosive shock can be investigated, its effects recorded, understood, and used. The rapidity and violence of an explosion do not vitiate Newton s laws, nor those of thermodynamics, chemistry, or quantum mechanics. They do, however, force matter into new states quite different from those we customarily deal with. These provide stringent tests for some of our favorite assumptions about matter s bulk properties. 

A usual, but not always valid, assumption about fj is fj(Q) = CjQ. A great deal of the literature is devoted to the analysis of this Hamiltonian, both classical and quantum mechanical. 

As argued in section 2.3, when the asymmetry e far exceeds A, phonons should easily destroy coherence, and relaxation should persist even in the tunneling regime. Such an incoherent tunneling, characterized by a rate constant, requires a change in the quantum numbers of the vibrations coupled to the reaction coordinate. In section 2.3 we derived the expression for the intradoublet relaxation rate with the assumption that only the one-phonon processes are relevant. 

The theory of crystal growth accordingly starts usually with the assumption that the atoms in the gaseous, diluted, or hquid mother phase will have a tendency to arrange themselves in a regular lattice structure. We ignore here for the moment the formation of poly crystalhne solids. In principle we should start with the quantum-mechanical basis of the formation of such lattice structures. Unfortunately, however, even with the computational effort of present computers with a performance of about 100 megaflops... 

We have assumed so far, implicitly, that the interactions are strictly local between neighboring atoms and that long-ranged forces are unimportant. Of course the atom-atom interaction is based on quantum mechanics and is mediated by the electron as a Fermi particle. Therefore the assumption of short-range interaction is in principle a simplification. For many relevant questions on crystal growth it turns out to be a good and reasonable approximation but nevertheless it is not always permissible. For example, the surface of a crystal shows a superstructure which cannot be explained with our simple lattice models. 

Since quantum mechanics allows us to predict, with certainty, the component of the second spin by measuring the same spin component of the first (and remotely positioned) particle - and to do so without in any way disturbing that second particle - BPR s first two assumptions attribute an element of physical reality to the value of any spin component of either particle i.e. the spin components must be determinate. On the other hand, assuming that the particles cannot communicate information any faster than at the speed of light, the only way to stay consistent with BPR s third postulate is to assume the existence of hidden variables. 

Bell s result, made all the more remarkable for the very few assumptions he makes to derive it, rather dramatically asserts that cither EPR s three premises are wrong or quantum mechanics is incorrect. However, recent experiments by A.spect, et.al. ([aspect82a], [aspect82b]). On and Mandel [01188], and others have shown, virtually conclusively, that nature satisfies the quantum mechanical prediction (equation 12.54) and not Bell s inequality (equation 12.55), thus strongly denying the possibility of local hidden variables. We are thus left with what is arguably one of the deepest mysteries in the foundations of physics the existence of a profoundly nonclassical correlation between spatially-far separated systems, or nonscparability. 

However, the taxonomic effectiveness of electronic configurations is not a basis for thinking that quantum mechanics can successfully account even for the restricted field of atomic chemistry. Clearly, molecular quantum chemistry is even less secure due to the additional assumptions which must be made apart from the validity of atomic orbitals. 

Before proceeding further it should be noted that there is a great difference in severity of the restrictions imposed by the earlier assumptions (a)-(d) and the present ones, (e) and (f). The earlier ones mainly restrict the size of the enclosed molecules to such a range that they neither distort the lattice, nor give rise to multiple occupancies or quantum effects. 

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