Dirac solutions

The complete solutions to (5) will be referred to as Dirac solutions whereas the solutions to (8) with spin-orbit coupling omitted will be referred to as modified PauU solutions. 

Department of Chemistry, Washington State University Dirac Solutions... 

Dirac showed in 1928 dial a fourth quantum number associated with intrinsic angidar momentum appears in a relativistic treatment of the free electron, it is customary to treat spin heiiristically. In general, the wavefimction of an electron is written as the product of the usual spatial part (which corresponds to a solution of the non-relativistic Sclnodinger equation and involves oidy the Cartesian coordinates of the particle) and a spin part a, where a is either a or p. A connnon shorthand notation is often used, whereby... 

We thus obtain a Lagrangean density, whieh is equivalent to Eq. (149) for all solutions of the Dirac equation, and has the structure of the nonrelativistic Lagrangian density, Eq. (140). Its variational derivations with respect to v / and v / lead to the solutions shown in Eq. (152), as well as to other solutions. 

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... 

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 ... 

In this paper, for functions (pi r) we shall use the four-component spinors r) being solutions of the Dirac equation... 

The solutions (q, r) of the Dirac equation (1) for the SDW are represented as a linear combination of r) with variational parameters... 

In the relativistic KKR method the trial function inside the MT-sphere is chosen as a linear combination of solutions of the Dirac equation in the center-symmetrical field with variational coefficients C7 (k)... 

Morrison, J., and Moss, R., 1980, Approximate solution of the Dirac equation using the Foldy-Wouthuysen Hamiltonian , A/o/. Phys. 41 491. 

One can actually prove a stronger result all nondeterministic LG models that satisfy semi-detailed balance and possess no spurious conservation laws have universal equilibrium solutions whose mean populations are given by the Fermi-Dirac distribution (equation 9.93) [frishc87]. 

Next we investigate the physical content of the Dirac equation. To that end we inquire as to the solutions of the Dirac equation corresponding to free particles moving with definite energy and momentum. One easily checks that the Dirac equation admits of plane wave solutions of the form... 

For tiie case of a Coulomb field, special methods exist that reduce the equation to a form where the nonrelativistic solutions can be used.11 Thus, if we denote by tji the wave function of the electron moving in a Coulomb field, then t/i obeys the Dirac equation... 

The solutions of Eq. (10-405) include the solutions to two first-order equations, namely, the first-order Dirac equation and this first-order... 

Before doing so we note a certain peculiarity of negative energy spinors. Let (p,a) be a positive energy solution of the Dirac equation corresponding to helicity s so that... 

Following the hypothesis of electron spin by Uhlenbeck and Goudsmit, P. A. M. Dirac (1928) developed a quantum mechanics based on the theory of relativity rather than on Newtonian mechanics and applied it to the electron. He found that the spin angular momentum and the spin magnetic moment of the electron are obtained automatically from the solution of his relativistic wave equation without any further postulates. Thus, spin angular momentum is an intrinsic property of an electron (and of other elementary particles as well) just as are the charge and rest mass. 

No theoretical proof of the Pauli principle was given originally. It was injected into electronic structure theory as an empirical working tool. The theoretical foundation of spin was subsequently discovered by Dirac. Spin arises naturally in the solution of Dirac s equation, the relativistic version of Schrodinger s equation. 

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