Representation Dirac

In the Dirac representation, the position operator has the following representation... 

Ponderomotive force, 382 Position operator, 492 in Dirac representation, 537 in Foldy-Wouthuysen representation, 537 spectrum of, 492 Power, average, 100 Power density spectrum, 183 Prather, J. L., 768 Predictability, 100 Pressure tensor, 21 Probabilities addition of, 267 conditional, 267 Probability, 106... 

The relations 04,0 1 + 0.104- = 2Ski, Clifford algebra, for which we choose the Dirac representation. In a phase space language, employing Weyl quantisation, this Hamiltonian can be written as (see, e.g., (Dimassi and Sjostrand, 1999))... 

The Foldy-Wouthuysen and Dirac representations for a free particle [1 85... 

In performing the similarity transformation above, we have changed our representation of the particle from a Dirac representation to the so-called Foldy-Wouthuysen representation. This new representation provides a very simple link with the non-relativistic Schrodinger-Pauli representation. The latter, which is a two-component representation, just corresponds to the two upper components of the Foldy-Wouthuysen representation (3.125). It must be noted that under a similarity transformation such as (3.105), the operators which represent physical observables are also transformed. For example, the position observable whose operator is R in the Dirac representation is transformed to R " in the Foldy-Wouthuysen representation where... 

We must therefore ask what is the significance of the operator R in the Foldy-Wouthuysen representation. This new observable, which is called the mean position, has an operator representative in the Dirac representation R where... 

R = mean position operator in the Dirac representation R = position operator in the F-W representation (confusion)... 

X is therefore a perfectly sensible position operator in the Dirac representation its properties were studied in detail by Newton and Wigner [60]. For more information see, for example, [61, 1.6]. 

We use the standard Dirac representation with the 4x4 Dirac matrices... 

Here, we used a KS orbital basis Xn) and the resulting density matrix In the nonrelativistic SchrOdinger picture, [Xn] is the orbital basis set [20], but in a consistent relativistic description, the proper density constructed from four-component spinors, has to be used. In the DK formalism, they can be obtained by back-transforming to the Dirac representation Xn = U Xn -... 

Finally, the result of Eq. (4.99) has to be arranged so that to satisfy the limit (4.91) as well. For that we use the delta-Dirac representation ... 

The Schrodinger field equation of the ath nucleus is conventionally put onto the curved spacetime. The procedure is (1) ignore the spin connection in D (g) in Equation 12.7, (2) use the Dirac representation with /j and approximate the... 

J. Autschbach, W. H. E. Schwarz. Relativistic electron densities in the four-component Dirac representation and in the two-component picture — Hydrogenlike systems. Theor. Chem. Acc., 104 (2000) 82-88. 

The key issue here is the accuracy of the representation of p. If the basis set is too small, there could be a serious loss of accuracy. However, for a reasonably large primitive basis, the same basis could be used for the representation of p as for the molecular calculations. Even more usefully, since the kinematic factors do not change the symmetry of the atomic basis functions, they can be used to redefine the contraction coefficients. This redefinition essentially generates a contracted basis set in the modified Dirac representation. 

We now turn to the Gaunt interaction, and use the terms from the modified Dirac representation in (15.54) to derive the Breit-Pauli operators. These terms need no renormalization, because they are all of order 1/c. The three classes of operators defined in (15.54) are considered in turn. 

Substitution of Xq for X in (19.14) gives the ZORA equation in the modified Dirac representation... 

This is essentially the nonrelativistic operator multiplied by fi. We must transform this operator to the modified Dirac representation, in which we have... 

Before we can start with the discussion of time-dependent perturbation theory in the form of response theory, we need to introduce an alternative formulation of quantum mechanics, called the interaction or Dirac representation. In general, several representations of the wavefunctions or state vectors and of the operators of quantum mechanics are equivalent, i.e. valid, as long as the expectation values of operators ( 0 I d I o) or inner products of the wavefunctions ( o n) are always the same. Measurable quantities and thus the physics are contained in the expectation values or inner products, whereas operators and wavefunctions are mathematical constructs used in a particular formulation of the theory. One example of this was already discussed in Section 2.9 on gauge transformations of the vector and scalar potentials. In the present section we want to look at a transformation that is related to the time dependence of the wavefunctions and operators. 

The interaction or Dirac representation becomes, on the other hand, useful, if one deals with a system that is described by a time-dependent Hamiltonian such as... 

There exist many different but essentially equivalent approaches (Dirac, 1958 Langhoff et al, 1972 Zubarev, 1974 Olsen and Jprgensen, 1985 Pickup, 1992) for obtaining the time-dependent wavefunction. Here, we will use the interaction or Dirac representation of the time-dependent wavefunction... 

It is convenient to adopt the Dirac representation of the operators ov and p, in which... 

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