Foldy-Wouthuysen

Morrison, J., and Moss, R., 1980, Approximate solution of the Dirac equation using the Foldy-Wouthuysen Hamiltonian , A/o/. Phys. 41 491. 

Equation (9-392) together with (9-394) and (9-395) are the proofs of the assertions that x is the position operator in the Foldy-Wouthuysen representation.16 (Note also that x commutes with /J the sign of the energy.) We further note that in the FTP-representation the operators x x p and Z commute with SFW separately and, hence, are constants of the motion. In the F W-representation the orbital and spin angular momentum operators are thus separately constants of the motion. The fact that... 

Foldy, L, L., 497,498,536,539 Foldy-Wouthuysen representation, 537 "polarization operator in, 538 position operator in, 537 Ford, L. R., 259 Four-color problem, 256 Fourier transforms of Schrodinger operators, 564... 

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... 

Our approach is based on a systematic semiclassical study of the Dirac equation. After separating particles and anti-particles to arbitrary powers in h, a semiclassical expansion of the quantum dynamics in the Heisenberg picture is developed. To leading order this method produces classical spin-orbit dynamics for particles and anti-particles, respectively, that coincide with the findings of Rubinow and Keller Hamiltonian relativistic (anti-) particles drive a spin precession along their trajectories. A modification of that method leads to a semiclassical equivalent of the Foldy-Wouthuysen transformation resulting in relativistic quantum Hamiltonians with spin-orbit coupling. 

In the previous discussion the semiclassical separation of particles and antiparticles employed projection operators and the associated subspaces of the Hilbert space. By suitable choices of bases such a separation can also be constructed with the help of unitary operators rotating the Hamiltonian into a block-diagonal form. Such a procedure is closely analogous to the Foldy-Wouthuysen transformation that provides a similar separation in a non-relati-vistic limit. A (unitary) semiclassical Foldy-Wouthuysen transformation Usc rotates the Dirac-Hamiltonian Hd into... 

The problem of nonrelativistic limit description for fundamental particles and their interactions may be solved in different ways. Although in all methods of nonrelativistic expansion the first terms of the Hamiltonians coincide, however the difference begins to arise at transition to the higher orders of expansion. The method of Foldy-Wouthuysen... 

Foldy-Wouthuysen transformation the spin-same orbit h and spin-other orbit... 

Semiclassical methods from quantum mechanics with first-order relativistic corrections obtained from the Foldy-Wouthuysen transformation match with the weak relativistic limit of functionals obtained from quantum electrodynamics, neglecting the (spurious) Darwin terms. 

An explicit expression for the Breit potential was derived in [2] from the one-photon exchange amplitude with the help of the Foldy-Wouthuysen transformation ... 

Hence, at low momenta the photon-nucleus interaction vertex (after the Foldy-Wouthuysen transformation and transition to the two-component nuclear spinors) is described by the expression... 

The Breit-Pauli Hamiltonian is an approximation up to 1/c2 to the Dirac-Coulomb-Breit Hamiltonian obtained from a free-particle Foldy-Wouthuysen transformation. Because of the convergence issues mentioned in the preceding section, the Breit-Pauli Hamiltonian may only be employed in perturbation theory and not in a variational procedure. The derivation of the Breit-Pauli Hamiltonian is tedious (21). 

In contrast to the one-electron terms, the reduction of the 4x4 Dirac-Breit Hamiltonian to the 2x2 Breit-Pauli Hamiltonian is very tedious for the two-electron terms as each interaction term has to be transformed according to the Foldy-Wouthuysen protocol. As the derivation can be found for example in Refs. (56-58) and in detail in Ref. (21), we only present here the transformed terms and discuss their dimension. The two-electron Breit-Pauli operator gBP (i, j) reads... 

Relativity becomes important for elements heavier than the first row transition elements. Most methods applicable on molecules are derived from the Dirac equation. The Dirac equation itself is difficult to use, since it involves a description of the wave function as a four component spinor. The Dirac equation can be approximately brought to a two-component form using e.g. the Foldy-Wouthuysen (FW) transformational,12]. Unfortunately the FW transformation, as originally proposed, is both quite complicated and also divergent in the expansion in the momentum (for large momenta), and it can thus only be carried out approximately (usually to low orders). The resulting equations are not variationally stable, and they are used only in first order perturbation theory. 

From the four-component Dirac-Coulomb-Breit equation, the terms [99]—[102] can be deduced without assuming external fields. A Foldy-Wouthuysen transformation23 of the electron-nuclear Coulomb attraction and collecting terms to order v1 /c1 yields the one-electron part [99], Similarly, the two-electron part [100] of the spin-same-orbit operator stems from the transformation of the two-electron Coulomb interaction. The spin-other-orbit terms [101] and [102] have a different origin. They result, among other terms, from the reduction of the Gaunt interaction. 

We have therefore achieved our objective in that equation (3.83), which is correct to order 1 /c2, contains even operators only. It would, of course, be possible to proceed further with the Foldy Wouthuysen transformation but there is little point in doing so, since the theory is inaccurate in other respects. For example, we have treated the electromagnetic field classically, instead of using quantum field theory. Furthermore, we shall ultimately be interested in many-electron diatomic molecules, for which it will be necessary to make a number of assumptions and approximations. 

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

One can gain some insight into the nature of the Dirac wave equation and the spin angular momentum of the electron by considering the Foldy Wouthuysen transformation for a free particle. In the absence of electric and magnetic interactions, the Dirac Hamiltonian is... 

In field-free, four-dimensional space, the sequence of Foldy-Wouthuysen transformations can be summed to infinity and written in closed form,... 

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... 

The explicit form of the position operator R " in the Foldy-Wouthuysen representation can be obtained by substituting S from equations (3.104) and (3.120) into (3.126). After some manipulation, the final result is... 

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... 

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