Quantized eneigy

The Hydrogen Atom The solution to the Schrodinger equation for the hydrogen atom shows quantized eneigies for the electron and a set of wave functions called atomic orbitals. The atomic orbitals are labeled with specific quantum numbers the orbitals tell us the regions in which an electron can be located. The results obtained for hydrogen, with minor modifications, can be applied to more complex atoms. 

For example, the measured pressure exerted by an enclosed gas can be thought of as a time-averaged manifestation of the individual molecules random motions. When one considers an individual molecule, however, statistical thermodynamics would propose its random motion or pressure could be quite different from that measured by even the most sensitive gauge which acts to average a distribution of individual molecule pressures. The particulate nature of matter is fundamental to statistical thermodynamics as opposed to classical thermodynamics, which assumes matter is continuous. Further, these elementary particles and their complex substmctures exhibit wave properties even though intra- and interparticle eneigy transfers are quantized, ie, not continuous. Statistical thermodynamics holds that the impression of continuity of properties, and even the soUdity of matter is an effect of scale. 

Finally, by considering the turning point - eigen-eneigy equation from above the semiclassical quantization rewrites as... 

Number of microstates. The enagy of the other molecules is similarly quantized. Each quantized state of the systan of molecules is called a microstate, and at any instant, the total eneigy of the systan is dispersed throughout one microstate. In the next instant, it is dispersed throughout a different microstate. The number of microstates possible for a system of 1 mol of molecules is staggering, on the order of 10 ° ... 

Under a strong magnetic field the orbital motion of conduction elections is quantized and forms Landau levels. Therefore various pltysical quantities show a periodic variation with H since increasing field strength causes a sharp change in the free eneigy of the electron system when a Landau level crosses the Fermi level. In the three-dimensional system this sharp structure is observed at the extremal (maximum or minimum) cross-sectional area of the Fermi surface perpendicular to the field direction because the density of states also becomes extremal. 

According to quantum theoiy, energy is always emitted in whole-number multiples of hv. At the time Planck presented his theory, he could not explain why eneigies should be fixed or quantized in this manner. Startbg with this hypothesis, however, he had no difficulty correlating the experimental data for the emission by solids over the entire range of wavelengths the experimental data supported his new quantum theory. 

Here the Z/ are the nuclear charges, r/ the electron-nuclear separations, ri2 the electron-electron separation and Ru the intemuclear separations. The summations are over all nuclei. The scalar term (1.4.42) represents the nuclear-repulsion eneigy - it is simply added to the Hamiltonian and makes the same contribution to matrix elements as in first quantization since the inner product of two ON vectors is identical to the overlap of the determinants. The molecular one- and two-electron integrals (1.4.40) and (1.4.41) may be calculated using the techniques described in Chapter 9. 

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