Dirac equation time-dependent

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... 

P.A.M. Dirac, who shared the 1933 Nobel prize for physics with Schrodinger (the 1932 price went to Heisenberg), was one of the greatest pioneers of quantum mechanics. Most of his achievements entered textbooks so fast that his original papers are hardly cited. Nobody, who uses Dirac s bra-ket notation or his function would cite the original references [1]. The same is true of Dirac s time-dependent perturbation theory [2] or of the Dirac equation [3], the basis of relativistic quantum mechanics or of his subsequent work on positrons and holes [4]. 

Assuming the simplest choice for the constraints, i.e 3 = 0, the MCTDH equations of motion can be derived by inserting the MCTDH ansatz of Eq. (4.22) into the Dirac-Frenkel time-dependent variational principle of Eq. (4.20). Other choices are possible for the constraints, see Ref. [28] for details. We first introduce the projector on the space spanned by the SPFs of the k particle... 

The equations of motion for a, derived from the Dirac-Frenkel time-dependent variational principle ... 

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] ... 

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

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. 

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... 

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. 

The time-dependent Schrodinger equation (2.43) presents a serious problem from the point of view of relativity theory it does not treat space and time in a symmetric way, because second-order derivatives of the wavefunction with respect to spatial coordinates are accompanied by a first-order derivative with respect to time. One way out, as actually proposed by Schrodinger and later known as the Klein-Gordon equation, would be to have also second-order derivatives with respect to time. However, that would lead to a total probability for the particle under consideration which would be a function of time, and to a variation of the number of particles of the universe (which, at the time, was completely unacceptable). In 1928, Dirac sought the solution for this problem, by accepting first-order derivation in the case of time and forcing the spatial derivatives to also be first order. This requires the wavefunction to have four components (functions of the spatial coordinates alone), often called a four-component spinor . 

On the level of quantum mechanics we are faced with the problem of solving numerically the Dirac equation governing the time-evolution of an electron state V (f)) under the influence of a space-time-dependent (classical) electromagnetic field AgXt(r, t) including the binding nuclear potential AnUC(r) ... 

In the time concept of the pre-relativistic mechanics, the observable quantities, time t and energy E, have to be considered as another canonically conjugate pair, as in classical mechanics. The dynamic law (time-dependent energy term) of the Schrodinger equation will then completely disappear [19]. A good occasion for Weyl to introduce the relativistic view would have been his contributions to Dirac s electron theory. His other colleagues developed the method of the so-called second quantization that seemed easier for the entire community of physicists and chemists to accept. 

The derivation by Dirac [13] of TDHF can be expressed in terms of orbital functional derivatives [12]. This derivation can be extended directly to a formally exact time-dependent orbital-functional theory (TDOFT). The Hartree-Fock operator H is replaced by the OFT operator Q = TL + vc throughout Dirac s derivation. In the linear-response limit, this generalizes the RPA equations [14] to include correlation response [17]. 

Dirac proved, for Hermitian Q, that the time-dependent equation... 

Thus, one has to solve the time-dependent Dirac-equation... 

We prefer to write the time-dependent free Dirac equation as a quantum-mechanical evolution equation (that is, in the familiar Schrodinger form ) in the following way... 

The behavior of solutions of the time-dependent Dirac equation is rather strange. This can be shown explicitly for the Dirac equation in one space dimension. In this case two spinor components are sufficient, because in one space dimension the linearization E = cap + (3mc of the energy momentum relation 2 = -t- requires just one a matrix (there is only one component of... 

For a given value of E the solution tl) E,x) of equation (28) may or may not be square-integrable. If ip E,x) is square-integrable (that is, if E is an eigenvalue of H), then the corresponding solution 4> E,x,t) of the time-dependent Dirac equation is a bound state with stationary position and momentum densities (according to our tentative interpretation). Bound states occur in the presence of an external force that attracts the particle to some region of space. 

The time-dependence of wave packets moving according to the Dirac equation usually cannot be determined explicitly. In order to get a qualitative description of the relativistic kinematics of a free particle, we investigate the temporal behavior of the standard position operator. With Ho being the free Dirac operator, we consider (assuming, for simplicity, h = l from now on)... 

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