Dirac time-dependent

The Dirac (time-dependent) equation for an electron in the presence of an external electromagnetic field can be arranged into the form... 

Note that the derivation of Eq. (141) was based on the Dirac time-dependent perturbation theory. We show in Section IV-B (3) how it may be obtained using another scheme. 

The earliest appearance of the nonrelativistic continuity equation is due to Schrodinger himself [2,319], obtained from his time-dependent wave equation. A relativistic continuity equation (appropriate to a scalar field and formulated in terms of the field amplitudes) was found by Gordon [320]. The continuity equation for an electron in the relativistic Dirac theory [134,321] has the well-known form [322] ... 

The time-dependent Hartree-Fock theory was first discussed by Dirac (1930b) and subsequently in perturbative form by Dalgamo and Victor (1966). Its relationship to time-dependent perturbation theory has been discussed by Langhoff, Epstein and Karplus (1972). 

For a free electron Dirac proposed that the (time-dependent) Schrodinger equation should be replaced by... 

An expansion in powers of 1 /c is a standard approach for deriving relativistic correction terms. Taking into account electron (s) and nuclear spins (1), and indicating explicitly an external electric potential by means of the field (F = —V0, or —— dAjdt if time dependent), an expansion up to order 1/c of the Dirac Hamiltonian including the... 

Its poles are determined to any order of by expansion of M. However, even in the lowest order in the inverse Laplace transformation, which restores the time kinetics of Kemni, keeps all powers to Jf (t/xj. This is why the theory expounded in the preceding section described the long-time kinetics of the process, while the conventional time-dependent perturbation theory of Dirac [121] holds only in a short time interval after interaction has been switched on. By keeping terms of higher order in i, we describe the whole time evolution to a better accuracy. 

Computational strategies can be based on variational procedures using the Dirac-Frenkel time-dependent variational principle (TDVP). Introducing a shorthand notation so that... 

Abstract. The relativistic periodically driven classical and quantum rotor problems are studied. Kinetical properties of the relativistic standard map is discussed. Quantum rotor is treated by solving the Dirac equation in the presence of the periodic -function potential. The relativistic quantum mapping which describes the evolution of the wave function is derived. The time-dependence of the energy are calculated. 

Due to the high barrier, it is safe to assume that the induction time is much shorter (by a factor of e P ) than the reaction time (1/T) so that the time dependence on the right hand side of Eq. 13 may be ignored. Then, noting that the derivative of a step function is a Dirac delta function, and using detailed balance one finds the desired formula ... 

Equations of motion for the time-dependent coefficients Aj time-dependent single particle functions, and time-dependent Gaussian parameters A K s) = aj c s), f r]jK s can be derived via the Dirac-Frenkel variational principle [1], leading to... 

All that remains to be done for determining the fluctuation spectrum is to compute the conditional average, Eq. (31). However, this involves the full equations of motion of the many-body system and one can at best hope for a suitable approximate method. There are two such methods available. The first method is the Master Equation approach described above. Relying on the fact that the operator Q represents a macroscopic observable quantity, one assumes that on a coarse-grained level it constitutes a Markov process. The microscopic equations are then only required for computing the transition probabilities per unit time, W(q q ), for example by means of Dirac s time-dependent perturbation theory. Subsequently, one has to solve the Master Equation, as described in Section TV, to find both the spectral density of equilibrium fluctuations and the macroscopic phenomenological equation. 

The relativistic or non-relativistic random-phase approximation (RRPA or RPA)t is a generalized self-consistent field procedure which may be derived making the Dirac/Hartree-Fock equations time-dependent. Therefore, the approach is often called time-dependent Dirac/Hartree-Fock. The name random phase comes from the original application of this method to very large systems where it was argued that terms due to interactions between many alternative pairs of excited particles, so-called two-particle-two-hole interactions ((2p-2h) see below) tend to... 

The time-dependent formulation of Raman scattering has been introduced by Lee and Heller (1979), Heller, Sundberg, and Tannor (1982), Tannor and Heller (1982), and Myers, Mathies, Tannor, and Heller (1982). Its derivation is strikingly simple. We start from the Kramers-Heisenberg-Dirac formula (14.1) and (14.2) without the nonresonant term and transform it into an integral over time by using the identity... 

As shown by Dirac [79], the corresponding time-dependent equation takes the... 

When extended to include electronic correlation, for which an exact but implicit orbital functional was derived above, the TDHF formalism becomes a formally exact theory of linear response. In practice, some simplified orbital functional Ec[ 4>i ] must be used, and the accuracy of results is limited by this choice. The Hartree-Fock operator Ti is replaced by G = Ti + vc. Dirac defines an idempo-tent density operator p whose kernel is JA i(r) i (r/)- The Did. equations are equivalent to [0, p] = 0. The corresponding time-dependent equations are itijtP = [Q(t), p(t)]. Dirac proved, for Hermitian G, (hat the time-dependent equation ih i(rt) implies that p(l) is idempotent. Hence pit) corresponds to a normalized time-dependent reference state. 

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