Wave function Wavelength

Schrodinger equation valence electrons wave function wavelength, X wave mechanical model wave-particle duality of nature... 

Another related issue is the computation of the intensities of the peaks in the spectrum. Peak intensities depend on the probability that a particular wavelength photon will be absorbed or Raman-scattered. These probabilities can be computed from the wave function by computing the transition dipole moments. This gives relative peak intensities since the calculation does not include the density of the substance. Some types of transitions turn out to have a zero probability due to the molecules symmetry or the spin of the electrons. This is where spectroscopic selection rules come from. Ah initio methods are the preferred way of computing intensities. Although intensities can be computed using semiempirical methods, they tend to give rather poor accuracy results for many chemical systems. 

Equation (1.8) represents a plane wave exp[i(A x — mt)] with wave number k, angular frequency m, and phase velocity m/A, but with its amplitude modulated by the function 2 cos[(AA x — Amt)/2]. The real part of the wave (1.8) at some fixed time to is shown in Figure 1.2(a). The solid curve is the plane wave with wavelength X = In jk and the dashed curve shows the profile of the amplitude of the plane wave. The profile is also a harmonic wave with wavelength... 

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... 

If the wavelengths of the reacting nuclei become comparable to barrier widths, that is, the distance nuclei must move to go from reactant well to product well, then there is some probability that the nuclear wave functions extend to the other side of the barrier. Thus, the quantum nature of the nuclei allows the possibility that molecules tunnel through, rather than pass over, a barrier. 

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... 

The approximation in which only the leading term in the expansion is retained is referred to as the dipolar approximation. The dipolar approximation will be more closely obeyed for small values of k = 2n/k, that is, for longer wavelengths and, in particular, for visible light. The higher-order terms are also smaller when the scattering object is compact relative to the wavelength used, that is, if either the initial state or the final state has a compact core-type wave function. 

The resolution of this paradox is easily obtained once it is remembered that the NFE bands in aluminium are formed from the valence 3s and 3p electrons. These states must be orthogonal to the s and p core functions, so that they contain nodes in the core region as illustrated for the 2s wave function in Fig. 2.12. In order to reproduce these very short wavelength oscillations, plane waves of very high momentum must be included in the plane wave expansion of . Retaining only the two lowest energy plane waves in eqn (5.35) provides an extremely bad approximation. 

It is, of course, not correct to treat the wave function of a polyatomic molecule as localized in the chromophoric group considered responsible for the optical absorption. The carbonyl group in aldehydes and ketones gives rise to absorption which extends from about 3430 A to about 2200 A (as well as to absorption at shorter wavelengths). Nevertheless the carbon-oxygen bond is never broken by absorption at these wavelengths. Frequently an adjacent bond is broken but often more complex processes occur. It is sometimes possible to describe these processes in terms of quantum mechanics but some of them should not be treated as direct dissociations. 

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