Matrix spin

We summarize here the principal spin and rotation matrices for I = /2, first the matrices for a single spin, then those for a two-spin system I—S in which the eigenfunctions are products of the basis vectors a and /3. 

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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). 

Exercise 13.3-1 Verify explicitly, by using the spin matrices from eq. (11.6.8), that the matrix representative (MR) of U=sy satisfies the matrix representation of eq. (22). 

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