Pauli spin matrices

The Pauli spin matrices introduced in Eq. (4.147) fulfill some important relations. First of all, the squared matrices yield the (2x2) unit matrix I2, 

This may be expressed in more compact form for all cyclic permutations of 1,2,3 as 

These relations may be generalized to the four-component case if we consider the even matrix E and the Dirac matrices ec and /5 cf. chapter 5, for which we have 

Relativistic Quantum Chemistry. Markus Reiher and Alexander Wolf 

The tensor product denoted by (g) is to be evaluated according to the general prescription 

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The strategy for representing this differential equation geometrically is to expand both H and p in tenns of the tln-ee Pauli spin matrices, 02 and and then view the coefficients of these matrices as time-dependent vectors in three-dimensional space. We begin by writing die the two-level system Hamiltonian in the following general fomi. 

Dirac s theory therefore leads to a Hamiltonian linear in the space and time variables, but with coefficients that do not commute. It turns out that these coefficients can be represented as 4 x 4 matrices, related in turn to the well-known Pauli spin matrices. I have focused on electrons in the discussion it can be shown... 

The spin operators may be taken to be the Pauli spin matrices.7... 

Particles spin Vz, 517 Dirac equation, 517 spin 1, mass 0,547 spin zero, 498 Partition function, 471 grand, 476 Parzen, E., 119,168 Pauli spin matrices, 730 PavM, W., 520,539,562,664 Payoff, 308 function, 309 discontinuous, 310 matrix, 309... 

Spin operators, taken as Pauli spin matrices, 730... 

Pauli spin matrices, geometric phase theory, eigenvector evolution, 14-17... 

We now consider how to eliminate the spin-orbit interaction, but not scalar relativistic effects, from the Dirac equation (25). The straightforward elimination of spin-dependent terms, taken to be terms involving the Pauli spin matrices, certainly does not work as it eliminates all kinetic energy as well. A minimum requirement for a correct procedure for the elimination of spin-orbit interaction is that the remaining operator should go to the correct non-relativistic limit. However, this check does not guarantee that some scalar relativistic effects are eliminated as well, as pointed out by Visscher and van Lenthe [44]. Dyall [12] suggested the elimination of the spin-orbit interaction by the non-unitary transformation... 

The second term on the right-hand side of the equation gives for point nuclei directly the one-electron spin-orhit operator (2) of the Breit-Pauli Hamiltonian and can he eliminated to give a spin-free equation that becomes equivalent to the Schrddinger equation in the non-relativistic limit. In a quaternion formulation of the Dirac equation the elimination becomes particularly simple. The algebra of the quaternion units is that of the Pauli spin matrices... 

It is also common in the literature to write the time-independent Dirac equation in terms of Pauli-spin matrices... 

Of course, the Spin Hamiltonian as given in Eq.(73) could also be directly derived from Eq. (77) for Dirac (81) pointed out that any permutation operator can be written in terms of vectors of Pauli spin matrices Oj and q,- as... 

These (without the h factor) are the Pauli spin matrices. 

Pauli principle, 45-47,178-182, 284-287 Pauli spin matrices, 96 P branch, 171-173,218,303 Peanuts, 320 Perpendicular band, 259, 265 Perturbation, spectroscopic, 283 Perturbation theory, 35-38,102 degenerate, 36-38 for nuclear motion, 149-159 time-dependent, 110-114 Phase, 13 Phenol, 225 Phosphorescence, 128 Phosphorous trichloride, structure of, 222, 223... 

Using the commutation properties of the Pauli spin matrices, eqs. (22), determine U as sy, apart from a phase factor exp(iy) which has no effect on eq. (22). 

Pauli introduced slightly different spin operators known as the Pauli spin matrices. They are defined by... 

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

Well-known realizations of the generators of this Lie algebra are given by the three components of the orbital angular momentum vector L = r x p, the three components of the spin S = a realized in terms of the Pauli spin matrices (Schiff, 1968), or the total one-electron angular momentum J = L + S. The components of each of these vector operators satisfy the defining commutation relations Eq. (4) if we use atomic units. We should also note that the vector cross-product example mentioned earlier also satisfies Eq. (4) if we define E = iey, j = 1, 2, 3. 

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