Quantum rules

Quantization of the Electromagnetic Field.—Instead of proceeding as in the previous discussion of spin 0 and spin particles, we shall here adopt essentially the opposite point of view. Namely, instead of formulating the quantum theory of a system of many photons in terms of operators and showing the equivalence of this formalism to the imposition of quantum rules on classical electrodynamics, we shall take as our point of departure certain commutation rules which we assume the field operators to satisfy. We shall then show that a... 

Proceeding in the spirit above it seems reasonable to inquire why s is equal to the number of equivalent rotations, rather than to the total number of symmetry operations for the molecule of interest. Rotational partition functions of the diatomic molecule were discussed immediately above. It was pointed out that symmetry requirements mandate that homonuclear diatomics occupy rotational states with either even or odd values of the rotational quantum number J depending on the nuclear spin quantum number I. Heteronuclear diatomics populate both even and odd J states. Similar behaviors are expected for polyatomic molecules but the analysis of polyatomic rotational wave functions is far more complex than it is for diatomics. Moreover the spacing between polyatomic rotational energy levels is small compared to kT and classical analysis is appropriate. These factors appreciated there is little motivation to study the quantum rules applying to individual rotational states of polyatomic molecules. 

Notice the similarities between the two parts of Table 1.1 and the form of the Periodic Table given in Figure 1.3. The quantum rules, the Pauli exclusion principle and the aufbau principle combine to explain the general structure of the Periodic Table. 

Suppose now that two gaseous atoms are made to approach each other. As long as the electron clouds of the two atoms do not overlap, the electron-energy states continue to follow the quantum rules for gaseous atoms. When, however, the electron clouds begin to overlap and the electrons interact with both atoms, the rules for electron-energy states are upset and they start changing. 

Electron Motion Around the Nucleus. The first approach to a treatment of these problems was made by Niels Bohr in 1913 when he formulated and applied rules for quantization of electron motion around the nucleus. Bohr postulated states of motion of the electron, satisfying these quantum rules, as peculiarly stable. In fact, one of them would be really permanently stable and would represent the ground state of the atom, The others would be only approximately stable. Occasionally an atom would leave one such state for another and, in the process, would radiate light of a frequency proportional to the difference in energy between the two states. By this means, Bohr was able to account for the spectrum of atomic hydrogen in a spectacular way. Bohr s paper in 1913 may well be said to have set the course of atomic physics on its latest path. 

Quantum rules restrict the vibrational energy in each normal mode to the discrete values given by the equation,... 

Nucleons have a quantum ordering, analogous to the quantum energy levels that electrons occupy. However, the quantum rules for protons and neutrons are much more complicated and heyond our purposes here. 

Theoretical quantum mechanics had its origin in two seminal papers, starting from apparently different points of view, and published independently and almost simultaneously by Heisenberg and Schrodinger, respectively. The major unsolved problem of physics, addressed by both, was to find a fundamental basis for the ad hoc quantum rules, formulated by Som-... 

Light absorbed by an atom or molecule excites it from the initial ground (or excited) state to a higher-energy excited state for low-intensity light, this occurs, provided that the various applicable quantum rules for the transition are satisfied (electric-dipole "allowed" transitions). If quantum rules "forbid" a transition, then the transition is either absent ("strongly forbidden transition") or very weak ("weakly allowed transition"). The "Jablonski"110 diagram (Fig. 3.16) depicts various forms of absorption and emission from... 

Bohr s application of Planck s ideas to Rutherford s atomic structure solved the impossible-atom problem. The energy of an electron in an atom was fixed. An atom could go from one energy state to another, but an electron could not emit a continuous stream of radiation and spiral into the nucleus. Quantum rules prohibit it. 

In Bohr s theory, the atom consisted of electrons circling the nucleus, but only at specific distances from the nucleus, orbits with diameters restricted by quantum rules. Add a quantum of energy to the atom and a Bohr electron would jump from an orbit closer to the nucleus to one farther away. Then, falling back to a more stable orbit, it would release a quantum of energy, sometimes in the form of visible light. 

Use of the Heisenberg spin Hamiltonian (equation 1) to represent the energy difference of the singlet and triplet spin states is easily demonstrated. Two spins, Sj and Sj, can be added to produce a maximum spin of 5 max = Sj + Sj, and lower values - 1, max - 2 down to a minimum of Si - Sj. When the two spins are both one-half, the two possible values for the total spin are 5 tot = 1 and 5 tot = 0, the spin triplet and singlet, respectively. To evaluate the energies of these states from equation f, it is necessary to know the value of Si Sj for the two states. This can be found by evaluating the vector sum of the spins and employing the basic quantum rule... 

It has been shown in the last section, and will further appear in succeeding sections, that the hypothesis of the spinning electron has made it possible to understand the splitting of terms (multiplets), a phenomenon which the orbital picture by itself was quite incapable of explaining. The phenomenon in fact depends upon the possession of angular momentum by the electron itself this internal angular momentum, by the quantum rules, can be directed in different ways with respect to the direction of the orbital moment, or with respect to a direction marked out by external means. 

Nevertheless, it should be emphasized that the epistemological and the theoretical connections between quantum laws and quantum chemical explanations of bonding are far removed from deductive reasoning they are not mere extensions of quantum rules to chemistry. They do not conform to a reductive model for relation between physics and chemistry. 

The rigid rotator in space can be described by polar coordinates of the figure axis,

total angular momentum is given by Equation 6-8, and the component of angular momentum along the z axis by... 

For this crystal it is seen that a cycle for the coordinate z is the identity distance d, so that (p, being constant in the absence of forces acting on the crystal) the quantum rule becomes... 

The Wilson-Sommerfeld quantum rules, in terms of the polar coordinates r, d, and [Pg.39]

In this communication I wish to show, first for the simplest case of the non-relativistic and unperturbed hydrogen atom, that the usual rules of quantization can be replaced by another postulate, in which there occurs no mention of whole numbers. Instead, the introduction of integers arises in the same natural way as, for example, in a vibrating string, for which the number of nodes is integral. The new conception can be generalized, and I belieVe that it penetrates deeply into the true nature of the quantum rules. 

A much more rigorous test of the quantum rules is made possible by applying Bohr s frequency condition to the frequencies of spectral lines. 

Astronomical observations of star movements support the existence of black holes. For example, from movements of stars close to our galactic center, it is believed that a black hole is located at SgrA in the center of the Milky Way, with a mass > 3 x 10 Mq. The radius of such a hole would be of the same size as that of our sun. The density of matter in the hole would be several million times the density of our sun (average value for the sun is about 1400 kg/m ). Obviously matter cannot be in the same atomic state (i.e. nuclei surrounded by electrons) as we know on earth. Instead we must assume that the electron shells are partly crushed we refer to this as degenerate matter, because the electron quantum rules. Tables 11.1 -11.2, cannot be upheld. For completely crushed atoms, matter will mainly consist of compact nuclei. For example, for calcium the nuclear density is -2.5x10 7 kg/m (cf. Fig. 3.4). 

It wiU remain to be decided whether subsequently the intervals in the various series of coordinates or energies are to be made vanishingly small. Later developments will show that this procedure is unnecessary and indeed incorrect, and that a discrete series of states, properly defined by what will be ealled quantum rules, is what corresponds to nature. For the present, however, the assumption of the series of numerous and fairly closely defined states may be regarded as a convenient simplification. 

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