Nonrelativistic Mechanics

Dirac achieved the combination of QM and relativity. Relativistic corrections are necessary when particles approach the speed of light. Electrons near heavy nuclei will achieve such velocities, and for these atoms, relativistic qnantnm treatments are necessary for accurate description of the electron density. However, for typical organic molecules, which contain only first- and second-row elements, a relativistic treatment is unnecessary. Solving the Dirac relativistic equation is much more difficult than for nonrelativistic computations. A common approximation is to utilize an effective field for the nuclei associated with heavy atoms, which corrects for the relativistic effect. This approximation is beyond the scope of this book, especially since it is unnecessary for the vast majority of organic chemistry. 

The complete nonrelativistic Hamiltonian for a molecule consisting of n electrons and N nuclei is 

---

All phenomena of classical nonrelativistic mechanics are solely based on Newton s laws of motion, which are valid in any inertial frame of reference. The natural symmetry operations of classical mechanics are the Galilean transformations, mediating the transition from one inertial coordinate system to another. The fundamental laws of classical mechanics can equally well be formulated applying the elegant Lagrangian and Hamiltonian descriptions based on Hamilton s action principle. Maxwell s equations for electric and magnetic fields are introduced as the basic laws of classical electrodynamics. 

In this chapter basic concepts and formal structures of classical nonrelativistic mechanics and electrodynamics are presented. In this context the term classical is used in order to draw the distinction with respect to the corresponding quantum theories which will be dealt with in later chapters of this book. In this chapter we will exclusively cover classical nonrelativistic mechanics or Newtonian mechanics and will thus often apply the word nonrelativistic for the sake of brevity. The focus of the discussion is on those aspects and formal structures of the theory which will be modified by the transition to the relativistic formulation in chapter 3. 

The subject of classical mechanics is the description of the motion of material bodies under the influence of given forces. All phenomena of classical nonrelativistic mechanics can be deduced from three basic axioms or laws of motion, which were first presented by Sir Isaac Newton in 1687 in his work Philosophiae Naturalis Principia Mathematica [39]. In modern language they can be formulated as ... 

The Newtonian laws of motion have been formulated for inertial frames of reference only, but no special IS has been singled out so far, since classical nonrelativistic mechanics relies on the Galilean principle of relativity ... 

The maximal invariance (symmetry) group for a free p>oint particle in nonrelativistic mechanics is shown to be a 12-parameter group instead of the 10-parameter Galilei group. This elementary but by no means trivial discussion may be of interest for the advanced reader but goes beyond the scope of this book. 

In the last chapter the basic framework of classical nonrelativistic mechanics has been developed. This theory crucially relies on the Galilean principle of relativity (cf. section 2.1.2), which does not match experimental results for high velocities and therefore has to be replaced by the more general relativity principle of Einstein. It will directly lead to classical relativistic mechanics and electrodynamics, where again the term classical is used to distinguish this theory from the corresponding relativistic quantum theory to be presented in later chapters. 

In contrast to the three-dimensional situation of nonrelativistic mechanics there are now two kinds of vectors within the four-dimensional Minkowski space. Contravariant vectors transform according to Eq. (3.36) whereas covariant vectors transform acoording to Eq. (3.38) by the transition from IS to IS. The reason for this crucial feature of Minkowski space is solely rooted in the structure of the metric g given by Eq. (3.8) which has been shown to be responsible for the central structure of Lorentz transformations as given by Eq. (3.17). As a consequence, the transposed Lorentz transformation A no longer represents the inverse transformation A . As we have seen, the inverse Lorentz transformation is now more involved and given by Eq. (3.25). 

Like the electron, the proton does not obey nonrelativistic mechanics. The proton s magnetic dipole operator is... 

Landau L D and Lifschitz E M 1977 Quantum Mechanics (Nonrelativistic Theory) (Oxford Pergamon)... 

Similarides Between Potential Ruid Dynamics and Quantum Mechanics Electrons in the Dirac Theory The Nearly Nonrelativistic Limit The Lagrangean-Density Correction Term Topological Phase for Dirac Electrons What Have We Learned About Spinor Phases ... 

The aim of this section is to show how the modulus-phase formulation, which is the keytone of our chapter, leads very directly to the equation of continuity and to the Hamilton-Jacobi equation. These equations have formed the basic building blocks in Bohm s formulation of non-relativistic quantum mechanics [318]. We begin with the nonrelativistic case, for which the simplicity of the derivation has... 

The first consistent attempt to unify quantum theory and relativity came after Schrddinger s and Heisenberg s work in 1925 and 1926 produced the rules for the quantum mechanical description of nonrelativistic systems of point particles. Mention should be made of the fact that in these developments de Broglie s hypothesis attributing wave-corpuscular properties to all matter played an important role. Central to this hypothesis are the relations between particle and wave properties E — hv and p = Ilk, which de Broglie advanced on the basis of relativistic dynamics. 

Quantum Mechanical Generalities.—It will be recalled that in nonrelativistic quantum mechanics the state of a particle at a given instant t is represented by a vector in Hilbert space (f)>. The evolution of the system in time is governed by the Schrodinger equation... 

Relativistic quantum mechanics yields the same type of expressions for the isomer shift as the classical approach described earlier. Relativistic effects have to be considered for the calculation of the electron density. The corresponding contributions to i/ (0)p may amount to about 30% for iron, but much more for heavier atoms. In Appendix D, a few examples of correction factors for nonrelativistically calculated charge densities are collected. Even the nonrelativistically calculated p(0) values accurately follow the chemical variations and provide a reliable tool for the prediction of Mossbauer properties [16]. 

Since the cross section for nonrelativistic Coulomb scattering is the same in classical and quantum mechanics, equation (2) must contain much of the essential physics in the slowing-down process. However, it also contains an undetermined minimum energy transfer rmin which is nominally zero and hence leads to an infinite stopping force. 

Even in the framework of nonrelativistic quantum mechanics one can achieve a much better description of the hydrogen spectrum by taking into account the finite mass of the Coulomb center. Due to the nonrelativistic nature of the bound system under consideration, finiteness of the nucleus mass leads to substitution of the reduced mass instead of the electron mass in the formulae above. The finiteness of the nucleus mass introduces the largest energy scale in the bound system problem - the heavy particle mass. 

A weakly bound state is necessarily nonrelativistic, v Za (see discussion of the electron in the field of a Coulomb center above). Hence, there are two small parameters in a weakly bound state, namely, the fine structure constant a. and nonrelativistic velocity v Za. In the leading approximation weakly bound states are essentially quantum mechanical systems, and do not require quantum field theory for their description. But a nonrelativistic quantum mechanical description does not provide an unambiguous way for calculation of higher order corrections, when recoil and many particle effects become important. On the other hand the Bethe-Salpeter equation provides an explicit quantum field theory framework for discussion of bound states, both weakly and strongly bound. Just due to generality of the Bethe-Salpeter formalism separation of the basic nonrelativistic dynamics for weakly bound states becomes difficult, and systematic extraction of high order corrections over a and V Za becomes prohibitively complicated. 

Nonrelativistic quantum electrodynamics (NRQED) [11] is an attempt to combine the simplicity of the quantum mechanical description with the power and rigor of field theory. The idea is to write ordinary relativistic quantum electrodynamics in the form of a nonrelativistic expansion with a Lagrangian containing vertices with arbitrary powers of fields. This is useful if we want to consider essentially nonrelativistic processes, like nonrelativistic bound states and threshold phenomena. In such a physical situation the dominant dynamics is nonrelativistic, and the calculations could be in principle simplified if... 

Unlike the Lamb shift, the hyperfine splitting (see Fig. 8.1) can be readily understood in the framework of nonrelativistic quantum mechanics. It originates from the interaction of the magnetic moments of the electron and the nucleus. The classical interaction energy between two magnetic dipoles is given by the expression (see, e.g., [1, 4])... 

---