The Non-Relativistic Limit

In the context of attempts to use only units based on the SI system an alternative definition of atomic units was proposed. One starts from the SI system (with 4 basic units), and measures then mass, electric charge, action and quantities of the dimension of the dielectric constant of the vacuum in units of m, e, h, and 47reo respectively. On this way one arrives at the same Hamiltonian in atomic units, as following Hartree - as long as no magnetic quantities are involved. We must therefore reconsider the system of units, when we come to electrons in the presence of magnetic fields (section 2,10). 

When we do not specify V and do not express p as —ihV, only the constants m and c are explicitly present, and we keep m in the expansion in powers of c. If we specify the attraction potential of a nucleus as —Zjr, (for either definition of atomic units) we automatically set also m 

starting from the Gaussian system of units, there are 3 basic units, but 4 fundamental constants, there are more possibilities to define atomic units, in addition to that proposed by Hartree. Rather than basing this system on h, m, and e, one may choose h, m, and c as the basic units, such that in this natural or relativistic system gets the value a and is then called the coupling strength. 

While the Hartree system is appropriate fore the discussion of relativistic effects, because the limit o — 0 ( or c — oo) corresponds to non-relativistic quantum mechanics, in the natural system the limit a — 0 (or — 0) 

---

In the non-relativistic limit the small component of the wave function (eq. (8.14)) is... 

Kutzelnigg, W. (1989) Perturbation theory of relativistic corrections 1. The non-relativistic limit of the Dirac equation and a direct perturbation expansion. Zeitschrifi fur Physik D, 11, 15-28. 

To understand the physical implications of these equations it is customary to consider the non-relativistic limit of their stationary solutions, assuming... 

In the non-relativistic limit both E and e(j> are small compared to me2 and hence... 

In order to develop a fully consistent non-relativistic theory for particles and fields we must first determine the non-relativistic limit of Maxwell s equations. This is to some extent hampered by the fact that the equations, and in particular the occurrence of the speed of light c, change according to the unit system chosen, a situation that has been analyzed by Jackson [25] and Kutzelnigg [15,35]. In STbased atomic units, the non-relativistic limit is obtained directly as the limit c 00 and leads to the equations... 

In the non-relativistic domain the Coulomb term is chosen as the two-electron operator which is fully consistent with the non-relativistic limit of... 

The second term on the right-hand side of the equation gives for point nuclei directly the one-electron spin-orhit operator (2) of the Breit-Pauli Hamiltonian and can he eliminated to give a spin-free equation that becomes equivalent to the Schrddinger equation in the non-relativistic limit. In a quaternion formulation of the Dirac equation the elimination becomes particularly simple. The algebra of the quaternion units is that of the Pauli spin matrices... 

Table II gives calculated results for copper porphine and experimental values for copper tetraphenylporphine. Results at the non-relativistic limit (c = >) were obtained from the same program by setting the speed of light to a very large number (10 a.u.) ...

Another important feature of the Dirac-Pauli representation is its natural adaptation to the non-relativistic limit. If V —E l << x( then Eq. (2) transforms directly to its non-relativistic counterpart known as the Levy-Leblond equation ... 

All these expressions may be deduced, if we start with the relativistic wave functions and relativistic Hamiltonian and look for the non-relativistic limit, retaining in the corresponding expansions terms of the order a2. Notice that all terms in Eq. (1.17) contain small parameter 1/c2, which is why they usually cause small corrections. 

Relativistic quantities rQ-q and vQ-, in the non-relativistic limit, correspond to the A>transition operators, leading, for k = 1, to the well-known length and velocity forms. However, in the case of relativistic transition probabilities (4.3) and (4.4) this limit depends on the K-value chosen. 

Let us consider the non-relativistic limit of the relativistic operators describing radiation. Expressing the small components of the four-component wave functions (bispinors) in terms of the large ones and expanding the spherical Bessel functions in a power series in cor/c, we obtain, in the non-relativistic limit, the following two alternative expressions for the probability of electric multipole radiation ... 

---