Pauli equation

The important assumption is now made that for each group at the initial time ig we have equal a priori probabilities for the existence of each state in a group. Furthermore, the phases of the states are randomly distributed. The latter assumption is equivalent to assuming a diagonal density matrix, and the former asserts that each of the diagonal elements is equal within the group. Therefore, we have 

It should be realized that Eq. (212) puts the system in such a condition that we are then permitted to observe an irreversible return to equilibrium. 

Considering that there are a number of groups k from which there may be transitions to a specific group v, and likewise from the group v to the various groups, we discern that 

Since similar equations can be obtained for any time, -(- aAt, where a is a positive integer, one is tempted to write directly 

We want to relate f to P, of the Pauli equation. The fraction of undissociated molecules at time t 0 t oo) is 2v - where v is the level next to the highest. The rate at which this fraction is depleted by molecular losses at the highest level is 

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Figure A3.13.15 shows a scheme for such a Pauli equation treatment of energy transfer m highly excited ethane, e.g. equation (A3.13.75), fomied at energies above both tln-esholds for dissociation in chemical activation ...

Perturbation theory for the transition rates in the Pauli equation... 

Importantly, the value of the results gained in the present section is not limited to the application to actual systems. Eq. (4.2.11) for the GF in the Markov approximation and the development of the perturbation theory for the Pauli equation which describes many physical systems satisfactorily have a rather general character. An effective use of the approaches proposed could be exemplified by tackling the problem on the rates of transitions of a particle between locally bound subsystems. The description of the spectrum of the latter considered in Ref. 135 by means of quantum-mechanical GF can easily be reformulated in terms of the GF of the Pauli equation. 

The program also provides a facility to correct the calculated values for relativistic effects, starting from the Pauli equation ... 

If the modified Pauli equation is used, i.e. spin-orbit coupling is dropped, the spin-up and spin-down calculations are uncoupled. One may make separate band structure calculations for spin-up and spin-down bands which are shifted with respect to one another. The system in filled to the Fermi level and the net magnetic moment is the difference between spin-up and spin-down occupation numbers. 

Self-consistent energy band calculations for the actinide metals have been made by Skriver et al. for the metals Ac-Am. The modified Pauli equation was used for this series of calculations but the corrections arising from use of the Dirac equation have recently been incorporated An fee structure was assumed for all the metals in both series of calculations. 

Eq. (36) may also be expressed as a system of two first-order equations, i.e. as the radial Dirac equation in the representation of Biedenharn. Let us rewrite the radial Dirac-Pauli equation (18) with V = —Zjr in the form... 

Figure 1. Variational ground state energy of a Z = 90 hydrogen-like atom obtained from the Dirac-Pauli equation as a function of a (abscissa) and (5 (ordinate) while a = b = s (left figure) and as a function of a (abscissa) and b while a = (5 = Z (right figure). The arrows are proportional to the gradient of . The saddle points correspond to the exact eigenvalues of the Dirac Hamiltonian.

On the nonrelativistic quantum level, both the time-independent and time-dependent Schrodinger equations can be used to demonstrate the existence of RFR. As shown by Sakurai [68], the time-independent Schrodinger-Pauli equation can be used to demonstrate ordinary ESR and NMR in the nonrelativistic quantum limit. This method is adopted here to demonstrate RFR in nonrelativistic quantum mechanics with the time-independent Schrodinger-Pauli equation [68] ... 

In a static magnetic field, the minimal prescription shows that the time-independent Schrodinger-Pauli equation of a fermion in a classical field is... 

Electron motion is more generally formulated in a form of the Schrodinger equation, including the spin in the presence of external fields known as the Pauli equation. This equation is gauge invariant in the sense that a transformation as in (5) also changes the quantum wavefunction as... 

Equation (39) corresponds to the Boltzmann approximation of statistical physics (or the so-called Pauli equation). We shall discuss it in more detail in Section VII. 

This equation is, of course, well known and often called the Pauli equation. We recognize on the right-hand side the familiar gain and loss terms. The transition probabilities which appear in the Pauli equation correspond to the Born approximation for one-photon processes. For further reference let us summarize the main properties of this weakly coupled approximation. 

A. Kyprianidis and J. P. Vigier, Theoretical implications of neutron interferometry derived from the causal interpretation of the Pauli equation, Hadronic J. Suppl. 2(3), 534—556 (1986). 

Attempts to formulate a causal description of electron spin have not been completely successful. Two approaches were to model the motion on either a rigid sphere with the Pauli equation [102] as basis, or a point particle using Dirac s equation, which is pursued here no further. The methodology is nevertheless of interest and consistent with the spherical rotation model. The basic problem is to formulate a wave function in polar form E = RetS h as a spinor, by expressing each complex component in spinor form... 

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