Diracs Relation

A very useful equation employing the Pauli spin matrices is the so-called Dirac relation. For any pair of vector operators u and v... 

Since the B operator can by means of the Dirac relation (55) be separated into scalar relativistic and spin-orbit terms, even better approximations to the lORA (MIORA-2) and ERA (MERA-2) equations that do not have any spin-dependent terms in the metric can be obtained by only omitting the spin-orbit contribution to the B operator in the metric. 

When calculating this in Bose-Einstein (relation [4.65]) and Fermi-Dirac (relation [4.70]) statistics, it is impossible to determine this coefficient. We must therefore settle for a limiting development iwN/z in the form ... 

We now make use of the Dirac relation (4.14) to eliminate the spin dependence where possible. The kinetic energy term is... 

The only term remaining that has any spin dependence is the term involving the potential in the second equation, and this can also be separated out using the Dirac relation. 

Spin-dependent parts using the Dirac relation (4.14) follows a similar pattern for each of these operators ... 

Dirac equation. This method of eliminating the small component is not a procedure that leads to a simplification. It does, however, have some motivation, both physical and practical. First, it projects out the negative-energy states, and leaves a Hamiltonian that may have a variational lower bound, avoiding the potential problem of variational collapse. Second, it removes from explicit consideration the small component, and with the use of the Dirac relation (4.14) it yields a one-component operator that can be used in nonrelativistic computer programs. 

This is the equivalent of the second-order Douglas-KroU operator, but it only involves operators that have been defined in the free-particle Foldy-Wouthuysen transformation. As for the Douglas-Kroll transformed Hamiltonian, spin separation may be achieved with the use of the Dirac relation to define a spin-fi ee relativistic Hamiltonian, and an approximation in which the transformation of the two-electron integrals is neglected, as in the Douglas-Kroll-Hess method, may also be defined. Implementation of this approximation can be carried out in the same way as for the Douglas-Kroll approximation both approximations involve the evaluation of kinematic factors, which may be done by matrix methods. 

Considering first the term involving the square of O, and making use of the Dirac relation, we have... 

From the Dirac relation (4.14), (or p)(or p) = p, and the first term in the series gives p /2w = f, the nonrelativistic kinetic energy operator. Making use of the Dirac relation in the second term gives... 

This is not exactly the same as the Pauli operator, but we must remember that the relativistic correction to the property is a second-order property in the Pauli approximation and includes a relativistic correction to the wave function. However, by applying the Dirac relation we can see that it contains the spin-orbit correction to the property and a spin-free correction as well. 

Convince yourself that the last factor, (g, — n, + 1) in Eq. (10.1.10) correctly accounts for the Fermi-Dirac relation when all n, particles are distributed among g, states associated with energy e,. Show that the arbitrary constant of integration in Eq. (10.3.2) vanishes. Incidentally, a and jS as used here are standard notation and are not to be confused with the thermal parameters introduced in the first three chapters. 

The Fermi level in electrochemical systems describes the average energy of charge carriers at equilibrium. The occupancy of electrons at a particular energy in a semiconductor is given by the Fermi-Dirac relation - ... 

Irrespective of such complications as the thermal expansion, the above approximation fails at high concentration and the more general result for the conduction band (see Ref. [128]) follows, by taking account of the exact Fermi-Dirac relation in Elq. (5.45) (see Eqs. (5.40), (5.41)) as... 

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