Wave mechanics conditions

Schrodinger s equation is widely known as a wave equation and the quantum formalism developed on the basis thereof is called wave mechanics. This terminology reflects historical developments in the theory of matter following various conjectures and experimental demonstration that matter and radiation alike, both exhibit wave-like and particle-like behaviour under appropriate conditions. The synthesis of quantum theory and a wave model was first achieved by De Broglie. By analogy with the dual character of light as revealed by the photoelectric effect and the incoherent Compton scattering... 

To represent observables in n-dimensional space it was concluded before that Hermitian matrices were required to ensure real eigenvalues, and orthogonal eigenvectors associated with distinct eigenvalues. The first condition is essential since only real quantities are physically measurable and the second to provide the convenience of working in a cartesian space. The same arguments dictate the use of Hermitian operators in the wave-mechanical space of infinite dimensions, which constitutes a Sturm-Liouville problem in the interval [a, 6], with differential operator C(x) and eigenvalues A,... 

The only assumption, in addition to Bohr s conjecture, is that the electron appears as a continuous fluid that carries an indivisible charge. As already shown, Bohr s conjecture, in this case, amounts to the representation of angular momentum by an operator L —> ihd/dp, shown to be equivalent to the fundamental quantum operator of wave mechanics, p —> —ihd/dq, or the difference equation (pq — qp) = —ih(I), the assumption by which the quantum condition enters into matrix mechanics. In view of this parallel, Heisenberg s claim [13] (page 262), quoted below, appears rather extravagent ... 

The first objective of quantum theory is indeed aimed at the electron. The wave-mechanical version of quantum theory, which is the most amenable for chemical applications, starts with solution of Schrodinger s wave equation for an electron in orbit about a stationary proton. There is no rigorous derivation of Schrodinger s equation from first principles, but it can be obtained by combining the quantum conditions of Planck and de Broglie with the general equation6 for a plane wave, in one dimension ... 

With the quantum conditions E = hv, p = h/X, the wave-mechanical analogue becomes ... 

Having examined the leading interpretations of the quantum formalism, a more general theory of atomic structure, consistent with all points of view, could conceivably now be recognized. The first aspect, never emphasized in chemical theory, but fundamental to matrix mechanics, is that the observed frequencies that determine the stationary energy states of an atom, always depend on two states and not on individual electronic orbits. The same conclusion is reached in wave mechanics, without assumption. It means that an electronic transition within atoms requires the interaction between emitter and receptor states and the frequency condition AE(An) = hu, for all pairs in n. This condition by itself offers no rationale for the occurrence of the... 

The way in which the spin factor modifies the wave-mechanical description of the hydrogen electron is by the introduction of an extra quantum number, ms = Electron spin is intimately linked to the exclusion principle, which can now be interpreted to require that two electrons on the same atom cannot have identical sets of quantum numbers n, l, mi and rns. This condition allows calculation of the maximum number of electrons on the energy levels defined by the principal quantum number n, as shown in Table 8.2. It is reasonable to expect that the electrons on atoms of high atomic number should have ground-state energies that increase in the same order, with increasing n. Atoms with atomic numbers 2, 10, 28 and 60 are... 

Extrapolation of the hem lines to Z/N = 1 defines another recognizable periodic classification of the elements, inverse to the observed arrangement at Z/N = t. The inversion is interpreted in the sense that the wave-mechanical ground-state electronic configuration of the atoms, with sublevels / < d < p < s, is the opposite of the familiar s < p < d < f. This type of inversion is known to be effected under conditions of extremely high pressure [52]. It is inferred that such pressures occur in regions of high space-time curvature, such as the interior of massive stellar objects, a plausible site for nuclear synthesis. 

To satisfy the conditions of wave mechanics the relationships are written in the form... 

Atomic and sub-atomic particles behave fundamentally different from macroscopic objects because of quantum effects. The more closely an atom is confined the more classical its behaviour. (Compare 5.2.1). Mathematically, the boundary condition on the particle wave function ip —> 0 as r —> oo, is replaced by limr >ro xp — 0, where r0 oo. It means that the influence of the free particle has a much longer reach through its wave function than a particle confined to a bulk phase. Wave-mechanically, the wavelength of the particle increases and approaches infinity for a completely localized, or classical particle. Electrons and atoms in condensed phases, where their motion is... 

The first sentence of Schrodinger s classic paper reads as follows In this paper I wish to consider, first, the simplest case of the hydrogen atom, and show that the customary quantum conditions can be replaced by another posmlate, in which the notion of whole numbers, merely as such, is not introduced. Two things about this sentence are noteworthy. First, an explanation of the hydrogen atom is clearly the objective of Schrodinger s wave mechanics. Second, in the development that follows this introductory sentence, quantum numbers ( whole numbers ), which appeared in Bohr s model of the hydrogen atom in a somewhat ad hoc fashion, appear as a namral consequence of Schrodinger s physical and mathematical approach. 

Quantization enters the wave mechanical description of the particle in a box via the boundary conditions. Boundary conditions arise from the physical requirements of natural systems. That is, we must insist that our descriptions of natural systems make physical sense. For example, assume that in describing an aqueous solution containing an acid, we arrive at the expression H + ]2 = 4.0 X 10 s M2. The solutions to this expression are... 

The steps leading to equation 1.34 must not be regarded as a rigorous derivation of the Schrodiugcr equation, since we have cmploycid an equation of classical mechanics to obtain a wave mechanical expression from wliich the quantum condition of discrete energy levels rmiy be devised. 

Equation (58) is the normalization condition that was postulated by Max Bom, in his interpretation of Schrodinger s nonrelativistic wave mechanics as a probability calculus. As we see here, the derived normalization is not a general relation in the full, generally covariant expression of the field theory. 

Here we conclude our account of Bohr s theory. Although it has led to an enormous advance in our knowledge of the atom, and in particular of the laws of line spectra, it involves many difficulties of principle. At the very outset, the fundamental assumption of the validity of Bohr s frequency condition amounts to a. direct and unexplained contradiction of the laws of the classical theory. Again, the purely formal quantisation rule, which stands at the head of the theory, is a foreign element which in the first instance is absolutely unintelligible from the physical point of view. We shall see later how both of these difficulties are removed in a perfectly natural way in wave mechanics. 

These commutation laws (Born and Jordan, 1925) take here the place of the quantum conditions in Bohr s theory. The considerations by which their adoption is justified, as also the further development of matrix mechanics as a formal calculus, are for brevity omitted here. In the next section, however ( 4, p. 121), it will be found that the analogous commutation laws in wave mechanics are mere matters of course. In Appendix XV (p. 291), taking the harmonic oscillator as an example, we show how and why they lead to the right result. 

Here A depends on the mechanical conditions (tension, thickness of the string) and represents in effect the square of the frequency of vibration. (It may be remarked that in classical vibrational processes the proper value parameter always contains the square of the frequency of vibration, while in wave-mechanical problems the proper value parameter is given in general by the energy E hv, and therefore contains the frequency in the first power.) The solutions of this differential equation are... 

The intensity of the spectral line is the product of two factors, the irumber of excited atoms and the radiating strength J of an individual atom, which we have just calculated. Thus, with regard to the conditions of excitation of lines, those ideas in Bohr s theory which are brilliantly verified by experiment are just the ideas which are retained in their entirety in the wave mechanics. The latter theory adds a more exact calculation of the intensity J of the individual elementary act, depending on evaluation of the integrals occurring in the matrix elements, while on this question Bohr s theory could only with difficulty make a few statements, with the help of very considerable use of the correspondence principle. 

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