Quantum conditions, 4, 14

This adiabatic principle was one of the corner-stones of the old quantum theory. It allowed one to find the quantum conditions when an adiabatic change was imposed on a system. It was used successfully to account for the Stark and Zeeman effects in the spectrum of atomic hydrogen, resulting from the application of an electric and magnetic field respectively (Schwartzchild [1916] Epstein [1916]). 

Although Weyl s conjecture could not be substantiated in its original form it was pointed out soon afterwards by Schrodinger [43] and London [44] that the classical quantum conditions could be deduced from Weyl s world geometry by choosing complex components for the gauge factor, i. e. 

The quantum condition hu = w reduces to this expression by setting h = 1. 

Only even functions of the linear case remain acceptable in the cyclic system and hence the quantum condition becomes L(= 27rr) = nX. The energy levels for a cyclic 7r-electron system follows immediately as... 

Quantitative XPS, 24 92-94 Quantum and molecular mechanical simulations, combined, 76 750-751 Quantum condition equation, 23 803 Quantum cellular automata (QCA),... 

Quantization puts a serious constraint on five-dimensional motion. If there is an independence on x5, then the particle is freely moving in x5. However, being that dimension compactihed leads to an uncertainty in the position with the size of 2irRc, where Rc is the compactification radius. Thus a Bohr-type quantum condition appears formally ... 

Momentum and Energy Balance in an Axisymmetric Case Angular Momentum Quantum Conditions... 

In the case of the EMS mode of Eq. (45), the limit of zero rest mass corresponds to cos a = 0. In this limit where cos a and mo are exactly equal to zero, the result is like that of a conventional axisymmetric EM mode that either diverges at the axis or at infinity, and must be discarded as pointed out in Section VII.A.l on branch 1 of solutions. Therefore the present results hold only for a nonzero rest mass, but this mass can be allowed to become very small. This implies that the quantum conditions me2 = hv for the total energy and 5 = h/2n for the angular momentum are satisfied for a whole class of small values of cos a and the corresponding rest mass. 

The present theory has been developed in terms of an extended Lorentz invariant form of the electromagnetic field equations, in combination with an addendum of necessary basic quantum conditions. From the results of such a simplified approach, theoretical models have been obtained for a number of physical systems. These models could thus provide some hints and first... 

To specify the equilibrium state more in detail, the quantum conditions of Section III. now have to be imposed. 

With the adopted form (B.28)-(B.29) of the generating function, the independent variables consist of the exponential factor in the radial part (B.28) and 2n amplitudes ayl in the polar part (B.29) with p = 2v — 1 or 2v. There are two resulting quantum conditions (1) the combined spin and magnetic moment... 

The truly remarkable feature of these selection rules is that they only depend on the time-reversal and scalar character of the one-electron operator. This results in very strict quantum conditions which are of paramount importance for spectroscopy and magnetism of the (t2g )3 systems - as we intend to show in the next sections. 

Hence it appears that r cancels out and the quantum condition imposes... 

A solution to the Schrodinger equation for an electron must satisfy three quantum conditions corresponding to the three dimensions of space. Each quantum condition introduces an integer, called a quantum number, into... 

Bonino brought forward a further contribution to the theory of infrared spectra of organic liquids by incorporating the Bohr-Sommerfeld quantum conditions, including the correspondence principle of Bohr as well. This paved the way toward establishing a correlation between the physical and chemical image of molecules in the study of infrared spectra. From this series of papers on infrared spectroscopy, one can already observe the interdisciplinary character of Bonino s thought. In a lecture delivered some years later, Bonino offered these reflections on his chosen field of research ... 

All conclusions, drawn before the importance of Planck s quantum of action was appreciated, were strictly qualitative. Introduction of the quantum condition was Bohr s innovation, and it could have been more effectively combined with Nagaoka s stable orbits, rather than with electrodynamically unstable orbits. Whereas Bohr s was a one-electron theory, Nagaoka proposed a model for all atoms, with electrons spread across a set of concentric rings. Developing this into a quantum model remains an intriguing possibility. 

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 ... 

Note that the condition r>,u — rK(f) — (f M/c2) follows from the definition of p. Quantum conditions further yield r> 2/z. Note the incompatibility between the force defined in Eq. (50) and the energy law defined by the theory of special theory of relativity. One should note here that Einstein, in his studies of the general theory of relativity, started from the force law... 

---