168 Wavefunction of electrons

The wavefunctions of electrons in atoms are called atomic orbitals. The name was chosen to suggest something less definite than an orbit of an electron... 

The VB and MO theories are both procedures for constructing approximations to the wavefunctions of electrons, but they construct these approximations in different ways. The language of valence-bond theory, in which the focus is on bonds between pairs of atoms, pervades the whole of organic chemistry, where chemists speak of o- and TT-bonds between particular pairs of atoms, hybridization, and resonance. However, molecular orbital theory, in which the focus is on electrons that spread throughout the nuclear framework and bind the entire collection of atoms together, has been developed far more extensively than valence-bond... 

Wavefunctions of electrons in atoms are called atomic orbitals. The name was chosen to suggest something less definite than an orbit of an electron around a nucleus and to take into account the wave nature of the electron. The mathematical expressions for atomic orbitals—which are obtained as solutions of the Schrodinger equation—are more complicated than the sine functions for the particle in a box, but their essential features are quite simple. Moreover, we must never lose sight of their interpretation, that the square of a wavefunction tells us the probability density of an electron at each point. To visualize this probability density, we can think of a cloud centered on the nucleus. The density of the cloud at each point represents the probability of finding an electron there. Denser regions of the cloud therefore represent locations where the electron is more likely to be found. 

Ballistic transport, 191 Band structures, nanowire calculated subband energies as function of in-plane mass anisotropy, 188 carrier densities, 190-191 dispersion relation of electrons, 185 envelope wavefunction of electrons, 186 grid points transforming differential... 

Figure 18a depicts the harmonic potentials for a model system with one normal coordinate Q (vibrational coordinate) in the electronic ground state 0 and the excited state II. It is assumed that the potentials are (nearly) equal, but are shifted by AQ with respect to each other, i.e. the vibrational frequencies of both states are also (nearly) equal. Generally, the wavefunctions of electronic-vibrational states depend on both, the coordinates of the electrons and nuclei, and thus the wavefunctions are very difficult to handle. However, when it is taken into account that the electronic motion is much faster than the vibrational motion, one can factorize the vibrational and the electronic part of the wavefunction. This leads to the Born-Oppenheimer approximation with the... 

The wavefunctions of particles at different potentials or in differently shaped boxes differ. The orbitals of a particle in a planar circular box are Bessel functions, while the orbitals in a spherically symmetrical box have spherical harmonics that are characterized by three quantum numbers. The well-known hydrogen orbitals, which have an angular part and a radial part with quantum numbers n, /, and m, are examples of wavefunctions of electrons in the spherically symmetric Coulomb potential of the proton (considered immobile). 

The rigorous statement of the Pauli principle is that wavefunctions of electrons must be antisymmetric with respect to exchange of electrons. There is a simpler statement of the Pauli principle. It comes from the recognition that equation 12.8, the only acceptable wavefunction for helium, can be written in the form of a matrix determinant. 

The thermoluminescence intensity of nanoparticles can be expressed by the formula, I=-dm/dt=mnA, where m and n are the density of holes and electrons for recombination, respectively and A is the carrier recombination probability. In fact, with increase in the content of the surface states, holes and electrons of the particle become more accessible for the TL recombination, i. e., the m and n increases proportionally with the surface states. As the surface states increases with decrease in the size of the particle, nanoparticles with smaller size causes the increase in the TL efficiency. Furthermore, the wavefunctions of electrons and holes are effectively overlapped in nanoparticles, and this may also cause increase in their recombination probability, A. Because of these two factors, the TL of small nanoparticles is expected to be more than that of the bulk. Fig. 29 (d) shows the schematic diagram for the size dependence of the surface states. Note that the separation between the electron-hole states (similar to the donor-acceptor pairs) increases with the decreasing size of the nanoparticles because the trap-depth does not change much upon decreasing size,while the bandgap increases [161]. 

We shall be concerned mainly with the wavefunctions of electrons in atoms and in this case the predominant contribution to the potential comes from the Coulomb attraction of the nucleus. This potential is. spherically symmetric and therefore V(r) is a function of tlie radial coordinate r alone, This enables the Schrodinger equation to be separated into three differential equations which involve r, 0, and (j> separately. If we consider the motion of a single electron of mass m about a nucleus of mass M we can separate off the centre-of-mass motion and consider only the relative motion of the electron. In spherical polar coordinates equation C3.8) becomes (Problem 3.3)... 

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