Precession, Thomas

Let us now consider a special case, namely the electron in an electrostatic potential. 

The nonrelativistic limit of the Dirac operator H c) Schrodinger operator 

It describes particles with spin (i.e., two-component wavefianctions), but the spin does not interact with the electrostatic field (the matrix-structure of is just given by the unit matrix). 

In order to avoid any technicalities in the following computations, we assume that ( el(x) is a twice continuously differentiable function of x. Let be a nondegenerate bound state of i.e., = Eq and = 1. 

Then a first approximation to the corresponding eigenvalue of the Dirac operator ca-p + 0 — )m( -f (/fell is given by j , where 

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This expression is not quite correct, however, because of a relativistic effect in changing from the perspective of the electron to the perspective of the nucleus. The correction, known as the Thomas precession, introduces the factor on the right-hand side to give... 

Spin-orbit coupling problems are of a genuine quantum nature since a priori spin is a quantity that only occurs in quantum mechanics. However, already Thomas (Thomas, 1927) had introduced a classical model for spin precession. Later, Rubinow and Keller (Rubinow and Keller, 1963) derived the Thomas precession from a WKB-like approach to the Dirac equation. They found that although the spin motion only occurs in the first semiclassical correction to the relativistic classical electron motion, it can be expressed in merely classical terms. 

This equation describes the Thomas precession of a classical spin (Thomas, 1927), which is driven by the underlying Hamiltonian dynamics in phase space. In combination the two types of dynamics, Hamiltonian on phase space and driven precession on the sphere, yield the following picture There is a combined phase space It3 x It3 x S2 with two dynamical systems,... 

Thomas Precession. The magnetic field BN due to the nucleus will cause the spin magnetic moment of the electron to precess around it with a Larmor frequency o>n. ... 

However, a special relativistic, or kinematical, correction, is necessary it is the Thomas precession. The electron orbiting around the nucleus with speed v (where v is a reasonably large fraction of the speed of light c) causes the period of one full rotation around the nucleus to be T in the fast-moving electron rest frame, but a longer time T (time dilatation) in the stationary rest frame of nucleus [see Eq. (2.13.11)] ... 

The Thomas precession frequency coj is defined as the difference between 2%/T and 2%/T, where we keep only the first two terms of a Maclaurin series ... 

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... 

With —eE = V(/>ei this expression is twice as large as the spin-orbit term in (99). But there is still another contribution coming from an effect in relativistic kinematics the Thomas precession. As the composition of two boosts with nonparallel velocities contains a rotation, one finds that an accelerated frame of reference performs an additional precession with the frequency... 

The result (146) overestimates the correct spin-orbit interaction (see section 4.6) by a factor 2. This can be explained by noting [7], that in the just-given derivation one ignores the Thomas precession, which has to do with relativistic kinematics - and is ignored in the nrl of quantum mechanics, and which compensates half of (146). In addition there are also spin-independent effects of relativistic kinematics (see section 4.6). 

We have abstracted so far from the so-called Thomas precession. This originates in the relativistic transformations which account for the fact that the electron is moving in a curved path around a fixed nucleus. If an axis of the gyroscope obeys an own dynamical precession with the Larmor angular velocity coL = (e/m)B, then the corrected precession in the inertial system associated with the fixed nucleus is (o = (oL + o>r, the Thomas precession being... 

The quoted papers still form the basis of our understanding of (J and J in molecules studied by condensed-phase NMR. No new terms were found later. Ramsey [2] had cautiously thought that his terms at least partially and perhaps completely explained the chemical shifts. For freely rotating molecules in the gas phase, the small Thomas precession chemical-shift term of Ramsey [8] was later improved by Reid and Chu [10] and by Rebane and Volodicheva [11]. For the latest references to it, see Ref. [12]. 

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