Dirac problem

There are a couple of attempts to use the WK.B approximation to the two component radial Dirac problem for screened Coulomb fields. Kosaka and Yonei(17) review these and apply the form offered by Goldberg and Pratt(18) for calculations of bound state orbitals in the potentials suggested by Tomishima(16). Their procedure is rather more involved than the present one. 

In the case of an impulse input (Dirac problem, tp is infinitely small, the concentration Co is infinitely large and Ap is the area of the injection profile), Wade et al. [46] derived an analytical solution which can be written ... 

The result can easily be extended to all one-body Dirac problems encountered in atomic and molecular structure, providing a firm basis for variational calculations just as it does in nonrelativistic quantum mechanics. 

The initial conditions of system (20) coincide with those for the original equations X/,(0) = X" and V/i(0) = V . Appropriate treatments, as discussed in [72], are essential for the random force at large timesteps to maintain thermal equilibrium since the discretization S(t — t ) => 6nml t is poor for large At. This problem is alleviated by the numerical approach below because the relevant discretization of the Dirac function is the inner timestep At rather than a large At. 

The application in [24] is to celestial mechanics, in which the reduced problem for consists of the Keplerian motion of planets around the sun and in which the impulses account for interplanetary interactions. Application to MD is explored in [14]. It is not easy to find a reduced problem that can be integrated analytically however. The choice /f = 0 is always possible and this yields the simple but effective leapfrog/Stormer/Verlet method, whose use according to [22] dates back to at least 1793 [5]. This connection should allay fears concerning the quality of an approximation using Dirac delta functions. 

Most problems in chemistry [all, according to Dirac (1929)] could be solved if we had a general method of obtaining exact solutions of the Schroedinger equation... 

The year 1926 was an exciting one. Schrddinger, Heisenberg and Dirac, all working independently, solved the hydrogen atom problem. Schrddinger s treatment, which we refer to as wave mechanics, is the version that you will be fanuliar with. The only cloud on the horizon was summarized by Dirac, in his famous statement ... 

If Dirac was warning us that solution of the equations of quantum mechanics was going to be horrendous for everyday chemical problems, then history has proved him right. Fifty years on from there, Enrico dementi (1973) saw things differently ... 

Actually Schrddinger s original paper on quantum mechanics already contained a relativistic wave equation, which, however, gave the wrong answer for the spectrum of the hydrogen atom. Due to this fact, and because of problems connected with the physical interpretation of this equation, which is of second order in the spaoe and time variables, it was temporarily discarded. Dirac took seriously the notion of first... 

Before embarking on the problem of the interaction of the negaton-positon field with the quantized electromagnetic field, we shall first consider the case of the negaton-positon field interacting with an external, classical (prescribed) electromagnetic field. We shall also outline in the present chapter those aspects of the theory of the S-matrix that will be required for the treatment of quantum electrodynamics. Section 10.4 presents a treatment of the Dirac equation in an external field. 

Antilinear operator, antiunitary, 688 Antiunitary operators, 727 A-operation, 524 upon Dirac equation, 524 Approximation, 87 methods, successive minimax (Chebyshev), 96 problem of, 52 Arc, 258... 

This simplified treatment does not account for the fine-structure of the hydrogen spectrum. It has been shown by Dirac (22) that the assumption that the system conform to the principles of the quantum mechanics and of the theory of relativity leads to results which are to a first approximation equivalent to attributing to each electron a spin that is, a mechanical moment and a magnetic moment, and to assuming that the spin vector can take either one of two possible orientations in space. The existence of this spin of the electron had been previously deduced by Uhlenbeck and Goudsmit (23) from the empirical study of line spectra. This result is of particular importance for the problems of chemistry. 

Adopting those ideas to problem (1), (2), (14) concerning a point heat source, an excellent start in this direction is to replace the function f x) involved in formula (40) by f x) + 6 x — where 6 x - ) is Dirac s... 

Abstract. The Dirac equation is discussed in a semiclassical context, with an emphasis on the separation of particles and anti-particles. Classical spin-orbit dynamics are obtained as the leading contribution to a semiclassical approximation of the quantum dynamics. In a second part the propagation of coherent states in general spin-orbit coupling problems is studied in two different semiclassical scenarios. 

Spin-orbit coupling problems are of a genuine quantum nature since a priori spin is a quantity that only occurs in quantum mechanics. However, already Thomas (Thomas, 1927) had introduced a classical model for spin precession. Later, Rubinow and Keller (Rubinow and Keller, 1963) derived the Thomas precession from a WKB-like approach to the Dirac equation. They found that although the spin motion only occurs in the first semiclassical correction to the relativistic classical electron motion, it can be expressed in merely classical terms. 

Semiclassical studies of the propagation of coherent states have proven useful in many circumstances see, e.g., (Klauder and Skagerstam, 1982 Perelomov, 1986). Here we consider spin-orbit coupling problems that result from the Dirac equation either in a semiclassical or in a non-relativistic approximation (see, e.g., the Hamiltonians (30) and (31)). The Hamiltonians H that arise in such a context can be viewed as Weyl quantisations of symbols... 

Abstract. The relativistic periodically driven classical and quantum rotor problems are studied. Kinetical properties of the relativistic standard map is discussed. Quantum rotor is treated by solving the Dirac equation in the presence of the periodic -function potential. The relativistic quantum mapping which describes the evolution of the wave function is derived. The time-dependence of the energy are calculated. 

Nonrelativistic quantum chemistry has been discussed so far. But transition metal (starting already from the first row) and actinide compounds cannot be studied theoretically without a detailed account of relativity. Thus, the multiconfigurational method needs to be extended to the relativistic regime. Can this be done with enough accuracy for chemical applications without using the four-component Dirac theory Much work has also been done in recent years to develop a reliable and computationally efficient four-component quantum chemistry.25,26 Nowadays it can be combined, for example, with the CC approach for electron correlation. The problem is that an extension to multiconfigurational... 

With the help of previous two examples, we propose a new way of simulating dense QCD, which evades the sign problem. Integrating out quarks far from the Fermi surface, which are suppressed by 1 //j at high density, we can expand the determinant of Dirac operator at finite density,... 

Later reconstructing the history of this period, Coulson claimed that the development of wave mechanics from Schrodinger to Dirac came to a "full stop" about 1929 insofar as chemistry was concerned. It was one thing to deal with the simplest cases of the H2+ ion and the H2 molecule. These problems hardly were comparable to methane or benzene. "Despondency set in."46 Joseph Hirschfelder later recalled that there were many new techniques to be learned in the 1920s, but then the realization set in that although "nature might be simple and elegant, molecular problems were definitely more complicated.. .. At this point, the theoretical physicists left the chemist to wallow around with their messy molecules while they resumed their search for new fundamental laws of nature."47... 

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