Relativistic Free Particle

According to Eq. (3.107) the trajectory of a relativistic particle may equally well be described by the functions r(f) or x r). Consequently, also the Lagrangian can be expressed in terms of either r t) or x r) and its corresponding derivatives with respect to t or t, respectively. We thus have to consider the Lagrangians 

Note that L2 does not explicitly depend on proper time t, since according to Eq. (3.92) r is uniquely determined by the space-time vector x and the 4-velocity u. For a better comparison between the three-dimensional formulation (Li) and the explicitly covariant formulation (L2) we have employed the velocity v = r (instead of r itself) in Eq. (3.139). Both Lagrangians Li and L2 do not represent physical observables and are therefore not uniquely determined. According to the Hamiltonian principle of least action given by Eq. (2.48), 5S = 0, they only have to yield the same equation of motion. This is in particular guaranteed if even the actions themselves are identical, i.e.. 

We first consider the three-dimensional formulation employing Li. For a free particle we require the usual space-time symmetries and conservation laws to be valid (cf. section 2.2.3) due to spatial translational invariance (i.e., momentum conservation) Li must not explicitly depend on r, due to temporal translational invariance (i.e., energy conservation) it must not explicitly depend on f, and due to rotational invariance it must not depend on the direction of velocity, but only on its magnitude v = v, i.e., L = Ti( ) = here 

These three equations have to equal the spatial components of the elementary equation of motion as given by Eq. (3.124), which for a free particle with vanishing Minkowski force f = 0 read 

Now we repeat our considerations for the explicitly covariant Lagrangian L2. Due to the homogeneity of space-time, i.e., translational invariance in space and time L2 must not depend on x, and due to rotational invariance it must not depend on the 4-velocity u directly, but only on its four-dimensional length i.e., L2 = L2(m) = Analogously to the nonrelativistic 

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The commutators of the relativistic and non-relativistic free-particle operators with orhital angular momentum look rather different, hut can he recast in the same mathematical form using the velocity operators of respective domains... 

External fields are introduced in the relativistic free-particle operator hy the minimal substitutions (17). One should at this point carefully note that the principle of minimal electromagnetic coupling requires the specification of particle charge. This becomes particularly important for the Dirac equation which describes not only the electron, but also its antiparticle, the positron. We are interested in electrons and therefore choose q = — 1 in atomic units which gives the Hamiltonian... 

It is clear from Ho that the Douglas-Kroll transformation makes use of a model space of relativistic free-particle spinors, and that it is defined by a perturbative expansion with the external potential as perturbation. Indeed, using the formulas given above, we get the familiar expressions for the second-order Douglas-Kroll-transformed Dirac operator, which is often dubbed Douglas-Kroll-Hess (DKH) operator... 

The action of a relativistic free particle can be written on manifestly covariant form[41]... 

The details of implementation of scalar relativity in GTOFF were presented in [41] and reviewed in [75], so we summarize the essential assumptions and methodological features here. First, all practical DFT implementations of relativistic corrections of which we are aware assume the validity (either explicitly or implicitly) of an underlying Dirac-Kohn-Sham four-component equation. We do also. The Hamiltonian is therefore a relativistic free particle Hamiltonian augmented by the usual non-relativistic potentials... 

Equation [44] is clearly not Lorentz invariant the left- and right-hand sides transform differently under Lorentz transformations. To develop a strictly Lorentz-invariant theory, we must therefore proceed along a different route. From Eq. [40] it is natural to take as the energy of a relativistic free particle... 

It is instructive to compare the 2-component Hamiltonian T-Cl with the Pauli Hamiltonian, which was derived in the previous section. The term Ep contains the relativistic free-particle energy, which is well-behaved for all values of the momentum. Ep, which is the kinetic energy operator for the positive-energy states, is a positive definite operator. In the Pauli Hamiltonian we see that this operator is expanded in powers of p/mc, which does not converge if p/mc > 1—a situation that will occur in any potential if the electron is sufficiently close to the nucleus. As mentioned above, the mass-velocity term is not bounded from below and so cannot be used variationally. 

The classical, non-relativistic energy E for a free particle, i.e., a particle in the absence of an external force, is expressed as the sum of the kinetic and potential energies and is given by... 

TABLE 1.3 Comparison of Relativistic and Classical Expressions for a Free Particle Moving in x Direction... 

L. de Broglie, F. Halbwachs, P. Hillion, T. Takabayasi, and J. P. Vigier, Space-time model of relativistic extended particles in Minkowski space II, free particle and interaction theory, Phys. Rev. 129 (1963). 

The variational formalism makes it possible to postulate a relativistic Lagrangian that is Lorentz invariant and reduces to Newtonian mechanics in the classical limit. Introducing a parameter m, the proper mass of a particle, or mass as measured in its own instantaneous rest frame, the Lagrangian for a free particle can be postulated to have the invariant form A = mulxiilx = — mc2. The canonical momentum is pf, = iiiuj, and the Lagrangian equation of motion is... 

A Volkov state is obtained from the solution of the time-dependent Schrodinger equation for a free particle in an external plane-wave laser field. Such states were first derived by Volkov, in a relativistic context [21]. 

Fig. 2.9 Energy diagrams for a free particle in an one-dimensional potential well, obtained with and without the relativistic correction (see also ref. 25).

We now apply these formulae to the case of a free particle with velocity v. Writing j8 for v/c, and employing the relativistic formulae for energy and momentum (see p. 271), we have here... 

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)... 

In this section we shall discuss the interactions that arise upon the introduction of electromagnetic fields in the relativistic electronic Hamiltonian, and we shall also consider the form of electromagnetic interactions in the non-relativistic limit. To simplify matters, we shall first limit our attention to one-electron systems. Consider the time-independent Dirac equation for a free particle... 

The above considerations leads to the somewhat troubling question of whether (128) represents the true non-relativistic limit of the Dirac equation in the presence of external fields. Referring back to (110) we have certainly obtained the non-relativistic limit of the free-particle part Lm, but we have in fact retained the interaction term as well as the Lagrangian of the free field. In order to obtain the proper non-relativistic limit, we must consider what is the non-relativistic limit of classical electrodynamics. This task is not facilitated by the fact that, contrary to purely mechanical systems, the laws of electrodynamics appear in different unit systems in which the speed of light appears differently. In the Gaussian system Maxwell s laws are given as... 

These problems can be solved if one starts from the (untruncated ) Foldy-Wouthuysen transformation for a free particle, the only case for which the transformation is known anal3d ically, and incorporates the effects of the external potential on top. Along these lines, the so-called Douglas-Kroll-HeB (DKH) method [61-64] is constructed which is probably the most successful quasi-relativistic method in wave function based quantum chemistry. No details will be given here since this topic has been extensively discussed in volume 1 [34] of this series. Meanwhile several density functional implementations exist based on the Douglas-Kroll-HeJ3 approach [39-45]. In recent years. 

These equations are modified when we turn to relativistic mechanics. The Lagrangian for a free particle is now given 1 ... 

Under the absence of the external forces (equivalently with free motion anyway) the above Lagrangian has to recover in the non-relativistic limit v c the classical Lagrangian limit (the kinetic energy in fact) of the free particle... 

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