Quantized orbit

To picture the spatial distribution of an electron around a nucleus, we must try to visualize a three-dimensional wave. Scientists have coined a name for these three-dimensional waves that characterize electrons they are called orbitals. The word comes from orbit, which describes the path that a planet follows when it moves about the sun. An orbit, however, consists of a specific path, typically a circle or an ellipse. In contrast, an orbital is a three-dimensional volume for example, a sphere or an hourglass. The shape of a particular orbital shows how an atomic or a molecular electron fills three-dimensional space. Just as energy is quantized, orbitals have specific shapes and orientations. We describe the details of orbitals in Section 7-1. 

The importance of N-representability for pair-density functional theory was not fully appreciated probably because most research on pair-density theories has been performed by people from the density functional theory community, and there is no W-representability problem in conventional density functional theory. Perhaps this also explains why most work on the pair density has been performed in the first-quantized spatial representation (p2(xi,X2) = r2(xi,X2 xi,X2)) instead of the second-quantized orbital representation... 

An electron remaining with a high degree of probability in the immediate neighborhood of a nucleus, where it occupies a quantized orbital. 

Atomic Number. The number of positive charges or the number of negative electrons around the nucleus was designated as the atomic number of the atom, These showed a close parallelism with the arrangement of atoms in the periodic system. Through the formulation of a number of rules based upon Bohr s picture of quantized orbits, the periodic system of tlie elements could be understood, Hydrogen was given one electron, and helium two, The two electrons in helinin constituted a closed shell which exhibited almost perfect spherical symmetry and chemical inactivity. 

The Kepler model was ceased upon by Sommerfeld to account for the quantized orbits and energies of the Bohr atomic model. By replacing the continuous range of classical action variables, restricting them to discrete values of... 

Result (3.1.17) or (3.1.18) is the Bohr energy for the hydrogen atom, except that Bohr had written the equation using a quantized orbital angular momentum 1 (it was discovered later by Schrodinger that for the H atom the lowest value for 1 is 0, while the principal quantum n = 1 is the correct one to use). 

Figure 6. Average adiabatic surfaces for the IHI system. The crosses denote the energies and locations of the quantized orbits. The solid lines show U-(r) determined from quantal computations. The coordinate on the right hand side denotes the asymptotic vibrational energies of HI.

These integrals, which are called action integrals, can be calculated only for conditionally periodic systems that is, for systems for which coordinates can be found each of which goes through a cycle as a function of the time, independently of the others. The definite integral indicated by the symbol is taken over one cycle of the motion. Sometimes the coordinates can be chosen in several different ways, in which case the shapes of the quantized orbits depend on the choice of coordinate systems, but the energy values do not. 

With these equations and Equation 7-11, the energy values of the quantized orbits and the values of the major and minor semiaxes can be expressed in terms of the quantum numbers and the physical constants involved. The energy is seen to have the value... 

The main planets between t and Sol follow the wave pattern generated by the perturbation of the solar pattern by the presence of Jupiter. The central inner planet, Earth is the most massive of these. Each planetary system is surrounded by a wave pattern similar to (a), with moons on well-defined quantized orbits and with rings and shepherding satellites in the inner sector. 

A spiral galaxy, such as M51, shown in Plate 5.1 has a structure, which is self-similar to the generating spiral of the solar system. Major stars in the spiral arms are expected to occupy quantized orbits like the planets in the solar system. This conjecture is confirmed by the observation of two distinct redshifts in the light from the different arms of M51 and other spiral galaxies. More importantly the jump in redshifts tends to have a constant value for all of these galaxies (Tifft Cocke, 1987). 

Even without knowing the cause of galactic redshifts their quantum nature rules out the Doppler interpretation and conhrms that galaxies have structures self-similar to that of the solar system. The major source of galactic light is stars on characteristically quantized orbits around the galactic... 

The prominent role of the golden ratio that conditions the observed periodic table of the elements hints at a general self-similarity between atomic and celestial structures. By exploiting this similarity the Bode -Titius law is shown to be based on the same number theory as nuclide periodicity. All planets, moons and rings in the solar system obey the same rules of com-mensurability and move on quantized orbits like those assumed in planetary... 

The quantum restriction postulate Only certain quantized orbits will be allowed. These orbits are restricted to the condition where the angular momentum (I) is an integral multiple of hlljr ... 

Bohr postulated that there can be only certain discrete orbits for the electron around a nucleus—called stationary states—and that to go from one state to another, an atom must absorb or emit a packet of just the right amount of energy—a quantum. He then proceeded to predict the position of the lines in the hydrogen spectrum based on Balmer s formula, Planck s energy packets, the mass and charge on an electron, and his quantized orbits. 

Electrons exist in quantized orbits at specific, fixed energies and specific, fixed distances from the nucleus. 

De BrogUe pointed out that in many ways the behavior of electrons in Bohr s quantized orbits was simUar to the known behavior of waves. For example, scientists at the time knew that any wave confined to a space can have only certain frequencies. De BrogUe suggested that electrons be considered waves confined to the space around an atomic nucleus. It foUowed that the electron waves could exist only at specific frequencies. And according to the relationship E = hv, these frequencies corresponded to specific energies—the quantized energies of Bohr s orbits. 

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