Pauli equation matrices

In order to understand the physical meaning of the equation, the electron mass m and the speed of light c, which are both unity in atomic units, are explicitly written in this and the next sections. The a in Eq. (6.55) is called the Pauli spin matrix (Pauli 1925). This equation is invariant for the Lorentz transformation, because the momentum p = —/ V is the first derivative in terms of space. More importantly, this equation rsqu-irts four-component wavefunctions. 

Generalized first-order kinetics have been extensively reviewed in relation to teclmical chemical applications [59] and have been discussed in the context of copolymerization [53]. From a theoretical point of view, the general class of coupled kinetic equation (A3.4.138) and equation (A3.4.139) is important, because it allows for a general closed-fomi solution (in matrix fomi) [49]. Important applications include the Pauli master equation for statistical mechanical systems (in particular gas-phase statistical mechanical kinetics) [48] and the investigation of certain simple reaction systems [49, ]. It is the basis of the many-level treatment of... 

Note that, since the von Neumann equation for the evolution of the density matrix, 8 j8t = — ih H, / ], differs from the equation for a only by a sign, similar equations can be written out for p in the basis of the Pauli matrices, p = a Px + (tyPy -t- a p -t- il- In the incoherent regime this leads to the master equation [Zwanzig 1964 Blum 1981]. For this reason the following analysis can be easily reformulated in terms of the density matrix. 

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... 

Since the Pauli-Schrodinger equation is of the form ih = //vjz, we may write the wave functions as /,y = elh ltlh i(())ij-. We then have the transition matrix element written as,... 

The Pauli matrices a (r) operate in the Nambu (Keldysh) space. The counting current I x) is to be found from the quantum kinetic equations [15] for the 4x4 matrix Keldysh-Green function G in the mesoscopic normal region of the interferometer confined between the reservoirs,... 

The Hartree-Fock equations (5.47) (in matrix form Eqs. 5.44 and 5.46) are pseudoeigenvalue equations asserting that the Fock operator F acts on a wavefunction i//, to generate an energy value ,-, times i/q. Pseudoeigenvalue because, as stated above, in a true eigenvalue equation the operator is not dependent on the function on which it acts in the Hartree-Fock equations F depends on i// because (Eq. 5.36) the operator contains J and K, which in turn depend (Eqs. 5.29 and 5.30) on i//. Each of the equations in the set (5.47) is for a single electron ( electron 1 is indicated, but any ordinal number could be used), so the Hartree-Fock operator F is a one-electron operator, and each spatial molecular orbital i// is a one-electron function (of the coordinates of the electron). Two electrons can be placed in a spatial orbital because the, full description of each of these electrons requires a spin function 7 or jl (Section 5.2.3.1) and each electron moves in a different spin orbital. The result is that the two electrons in the spatial orbital i// do not have all four quantum numbers the same (for an atomic Is orbital, for example, one electron has quantum numbers n= 1, / = 0, m = 0 and s = 1/2, while the other has n= l,l = 0,m = 0 and s = —1/2), and so the Pauli exclusion principle is not violated. 

Expanding t and g in terms of Pauli matrices oc and the unit matrix we obtain four coupled one-dimensional integral equations for the components t (cp, ff) of the matrix t, which contain the energy integrated normal and anomalous retarded Green functions... 

We next use the Pauli matrix representations of the spin angular momentum operator components in the instantaneous molecule-fixed axis system from equation (2.92) to rewrite the above relationships ... 

Pauli s exclusion principle imposes that either (ri, r2) or x(si, S2) must be odd (the other one remaining even) with respect to the interchange of indices 1 and 2. If the Coulomb repulsion t/(ri, r2) = e /47reo ri - r2 between both electrons is small, this contribution to the Hamiltonian H may be considered as a perturbation. Let (Pair) (respectively, b(r)) be the eigenstate of the Hamiltonian Hi = Ti =p l2m (respectively, H2 = T2 =p /2m) characterized by the eigenvalue (respectively, E, ). In this framework, this approximation allows us to solve the well-known secular equation det(H - El) = 0 (where 1 is the identity matrix and H=Ti + T2 + U(ri, V2)). hi the subspace spanned by the spatially symmetrical and antisymmetrical wave functions (Psiri, V2) and (Phiri, V2) containing functions a(ri) and hiri), we have ... 

In this equation VJ (U) is the matrix made up from the elements of U in exactly the same way that VJ is made up from the elements of C. A precise account of how this is to be done is given in Section 6.19 of (12). Should it turn out that U is a constant matrix then T>J (XJ) is a constant matrix and (33) simply represents a linear combination. If U is a unit matrix then JMk > is invariant. It should be noted here that this coupling of rotations by the permutations can mean that certain rotational states are not allowed by the Pauli principle and this is important in assigning statistical weights to rotational states. 

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