Dirac spin operator

PROBLEM 3.6.5. Define the 4x4 Dirac spin operator a from the three 2x2 Pauli spin operators [Pg.153]

The Dirac operators a, . ..04, work on the basis vectors ei,. ..04 just as the Pauli spin operators work on the spin eigenvectors a, / . The only other property to be noted is that the Dirac spin-orbitals have two large components and two small components , their ratio being of the order (2moc)" ... 

If we refrain from such a restriction and consider a spin-operator-dependent Hamiltonian, such as the 4-component KS Hamiltonian or the Dirac-Coulomb Hamiltonian, the Hamiltonian does not commute with the square of the spin operator. The square of the spin operator and the Hamiltonian then do not share the same set of eigenfunctions, and hence spin is no longer a good quantum number. In this noncollinear framework we must therefore find a different solution and may define a spin density equal to the magnetization vector (32). 

Dirac (81,84,85) showed that the potential energy of two interacting electron spins in a non-relativistic framework depends on their spin operators sx and 2,... 

The quantum mechanical mechanisms that underlie exchange coupling are complex, but can be modeled by a phenomenological Hamiltonian that involves the coupling of local spin operators Sa and SB, the so-called Heisenberg-Dirac-Van... 

Pauli spin vector Dirac spin vector electron spin magnetic moment nuclear spin magnetic moment rotational magnetic moment electric dipole moment Ioldy Wouthuysen operator gradient operator Laplacian... 

The vectors and denote the Dirac 4x4 matrices for electron i (in standard representation) and the nuclear spin operator for nucleus a. The constants and Kg are nuclear parameters, while is the nuclear spin quantum number. 

We introduce the change of the metric characteristic of DPT, and expand in powers of c. To 0(c ) we get the non-relativistic Hartree-Fock equations in Levy-Leblond form. The leading relativistic correction to the energy is then expressible in terms of nonrelativistic HF spin orbitals or rather the corresponding lower components xf - For the Dirac-Coulomb operator we get after some rearrangement [17, 18] ... 

In projective relativity the field equations contain, in addition to the gravitational and electromagnetic fields, also the relativistic wave equation of Schrodinger and, as shown by Hoffinann (1931), are consistent with Dirac s equation, although the correct projective form of the spin operator had clearly not been found. The problem of spin orientation presumably relates to the appearance of the extra term, beyond the four electromagnetic and ten gravitational potentials, in the field equations. It correlates with the time asymmetry of the magnetic field and spin. 

The spin moment operator has been introduced arbitrarily here, but appears naturally if the equation is derived by reducing the relativistic Dirac equation. With the spin operator included the wavefimction must be treated as a two component spin function and the operators as 2 X 2 matrices. 

Because the Pauli matrices enter the Dirac matrices we already note the connection to the spin operator through Eq. (4.146). For the physical interpretation of the oc parameters we may quote Dirac [100] The Oi s are new dynamical variables which it is necessary to introduce in order to satisfy the conditions of the problem. They may be regarded as describing some internal motions of the electron, which for most purposes may be taken to be the spin of the electron postulated in previous theories. However, they are also connected to the velocity operator to be shown in section 5.3.3, which was also known to Dirac as is clear from a footnote in a paper by Breit [101]. Since this matter is not that straightforward to interpret, Breit later devoted a whole new paper [102] to this issue. 

We stress that this interaction enters the quantum mechanical Hamiltonian because of the unretarded vector potential. Hence, we understand that the instantaneous magnetic interaction of moving electrons shows up in four-component first-quantized Dirac theory as the interaction of the electron s two spin momenta as expressed by the spin operators of the two electons on the right-hand... 

The relativistic Hamilton operator for an electron can be derived, using the correspondence principle, from its relativistic classical Hamiltonian and this leads to the one-electron Dirac equation, which does contain spin operators. From the one-electron Dirac equation it seems trivial to define a many-electron relativistic equation, but the generalization to more electrons is less straightforward than in the non-relativistic case, because the electron-electron interaction is not unambiguously defined. The non-relativistic Coulomb interaction is often used as a reasonable first approximation. The relativistic treatment of atoms and molecules based on the many-electron Dirac equation leads to so-called four-component methods. The name stems from the fact that the electronic wave functions consist of four instead of two components. When the couplings between spin and orbital angular moment are comparable to the electron-electron interactions this is the preferred way to explain the electronic structure of the lowest states. 

The extension to the case of the four-component Dirac Hamiltonian above follows readily by noting that the spin operator and the orbital angular momentum operator for this case are... 

The most obvious new feature of the Dirac equation as compared with the standard nonrelativistic Schrodinger equation is the explicit appearance of spin through the term a p. Any spin operator trivially commutes with a spin-free Hamiltonian, but the introduction of spin-dependent terms may change this property, as demonstrated in the case of ji/-coupling. A further scrutiny of spin symmetry is therefore a natural first step in discussing the symmetry of the Dirac Hamiltonian. This requires a basis of spin functions on which to carry out the various operations, and a convenient choice is the familiar eigenfunctions of the operator, i, i) and j, — ), also called the a and spin functions. 

---