Velocity Dirac operator

In Dirac s theory the classical expression for E is replaced by the Dirac operator Hq. Translating the classical expression for the velocity into quantum mechanics would thus lead to the operator... 

A physical system is close to the nonrelativistic limit, if all velocities of the system are small compared to the velocity of light. Hence the nonrelativistic limit of a relativistic theory is obtained if we let c, the velocity of light, tend to infinity. In the nonrelativistic theory, there is no limit to the propagation speed of signals. For the Dirac equation, the nonrelativistic limit turns out to be rather singular. If we simply set c = oo, we would just obtain infinity in all matrix elements of the Dirac operator H. We must therefore look for cancellations. 

Note that the subscript on the a matrices refers to the particle, and a here includes all of the tlx, tty and components in eq. (8.4). The first correction term in the square brackets is called the Gaunt interaction, and the whole term in the square brackets is the Breit interaction. The Dirac matiices appear since they represent the velocity operators in a relativistic description. The Gaunt term is a magnetic interaction (spin) while the other term represents a retardation effect. Eq. (8.27) is more often written in the form... 

For the non-relativistic case (Schrbdinger equation), T = -V. For relativistic case (Dirac equation), T = c a p + 3mc where m is the rest mass of the electron, c is the velocity of light. We have preferred to write the T operator in a general form, covering both cases, given the importance of the relativistic approach in band calculations for actinide solids - see Chap. F... 

The quantum mechanical expression for the charge-weighted current density is obtained from Eq. (26) when we replace the classical velocity r (f) by the Dirac velocity operator caL and evaluate its expectation value (21),... 

Relativistic corrections of order v2/c2 to the non-relativistic transition operators may be found either by expanding the relativistic expression of the electron multipole radiation probability in powers of v/c, or semiclas-sically, by replacing p in the Dirac-Breit Hamiltonian by p — (l/c)A (here A is the vector-potential of the radiation field) and retaining the terms linear in A. Calculations show that in the general case the corresponding corrections have very complicated expressions, therefore we shall restrict ourselves to the particular case of electric dipole radiation and to the main corrections to the length and velocity forms of this operator. 

For further discussion see Refs. [60-62]. In the above, E has been replaced by the gradient of the total Coulomb potential, and the electronic velocity (v), has been replaced by the momentum operator (p). The factor of two in the denominator, also derivable from the Dirac equation, is due to the Thomas precession. Given a spherically symmetric Coulomb potential and some simple algebraic reductions, the above expression is usually rewritten according to... 

The deficiencies of this procedure have been carefully analysed by Boring and Wood [62] who worked with an approximate treatment of the Dirac equations, due to Cowan and Griffith [63]. In this method the spin-orbit operator is omitted from the one-electron Hamiltonian but the mass-velocity... 

After the development of the Dirac equation one might have guessed that, within a framework in which the state of a many-electron system is described by a multi-Dirac spinor, the velocity factors Vj in (1.2) should simply be replaced by their formal counterparts in Dirac theory, viz, coj. This yields the operator... 

For materials containing atoms with large atomic number Z, accelerating the electrons to relativistic velocities, one must include relativistic effects by solving Dirac s equation or an approximation to it. In this case the kinetic energy operator takes a different form. 

In a strict sense, the time-evolution generated by the operator (2) is acausal A wavepacket that is initially strictly localized in a finite region of space instantaneously spreads over the whole space. Even for the Dirac equation there are some problems with causality and localization (see, e.g., [5]), but since the propagator of the Dirac equation (the time-evolution kernel) has support in the light-cone, distortions of wave functions and wave fronts can at most propagate with the velocity of light. 

The operator aj t) is unitarily equivalent to the Dirac matrix aj and hence it has the eigenvalues 1. A measurement of the (component j of the) velocity at any time t therefore can only give the results c. It appears as if Dirac particles can only move with the velocity of light. 

Let us define the operator which describes the difference between the Dirac velocity ca and the classical velocity ... 

The Breit-Pauli (BP) approximation [140] is obtained truncating the Taylor expansion of the Foldy-Wouthuysen (FW) transformed Dirac Hamiltonian [141] up to the (p/mc) term. The BP equation has the well-known mass-velocity, Darwin, and spin-orbit operators. Although the BP equation gives reasonable results in the first-order perturbation calculation, it cannot be used in the variational treatment. 

Going to an operator form, let us remember that in the Dirac theory the operator corresponding to a velocity v is... 

The paradox was elucidated through the trembling motion (Zillerbewegung) discovered by Schrodinger [8] while investigating the velocity operators cik introduced by Dirac to linearize his equation. The equation of motion of a velocity component, Vk = coik, can be written as... 

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