Nonrelativistic quantum mechanics

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... 

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. 

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])... 

On the nonrelativistic quantum level, both the time-independent and time-dependent Schrodinger equations can be used to demonstrate the existence of RFR. As shown by Sakurai [68], the time-independent Schrodinger-Pauli equation can be used to demonstrate ordinary ESR and NMR in the nonrelativistic quantum limit. This method is adopted here to demonstrate RFR in nonrelativistic quantum mechanics with the time-independent Schrodinger-Pauli equation [68] ... 

Nonrelativistic quantum mechanics, extended by the theory of electron spin and by the Pauli exclusion principle, provides a reliable theory for the computation of atomic spectral frequencies and intensities, of cross sections for scattering or capture of electrons by atomic systems, of chemical bonds and many properties of solids, including magnetic properties, although with much more complicated systems it has not always proved possible to develop with adequate accuracy the consequences of the theory. Quantum mechanics has also had a limited success in nuclear theory although m this field it is possible that a more fundamental system of mechanics is required. 

The effect of using relativistic rather than nonrelativistic quantum mechanics to obtain the predicted atomic orbitals is threefold (a) a contraction and stabilization of the s, /2 and p1 /2 shells, (b) the splitting of the energy levels due to the spin-orbit coupling, and (c) an expansion (and destabilization) of the outer d and all f shells. These effects are of approximately equal magnitude and all increase as Z2. In Figure 15.14, we show the magnitude of these effects for uranium. The chemical... 

This article deals with QBSs in nonrelativistic quantum mechanics. Suppose that a QBS satisfies the time-independent Schrodinger equation with a Hamiltonian H. One may extract a normalizable wavefunction out of T such that in the large-amplitude region, by writing... 

Feynman, R.R (1948). Space-time approach to nonrelativistic quantum mechanics, Rev. Mod. Phys. 20, 367-387. 

The operator H is intended to describe the energy of a particle in a given external field. As in nonrelativistic quantum mechanics, the influence of an external field is described by a potential-energy V (x) that is added to the kinetic energy Hq. Hence the Dirac operator with an external field reads... 

For particles with spin-1/2 we would expect (on the basis of nonrelativistic quantum mechanics) that spinors with two components would be sufficient. But the Dirac spinors have to be (at least) four-dimensional. A mathematical reason lies in the nature of the algebraic properties that have to be satisfied by the Dirac matrices a and 0 if the Dirac equation should satisfy the relativistic energy-momentum relation in the sense described above, see (6). 

As in nonrelativistic quantum mechanics, this interpretation leads to the choice of the operator p = —i iV as an operator for the momentum of the particle, and the multiplication operator x as an operator representing the position observable. The symbols p and x in fact denote collections of three operators, the components of the position and momentum with respect to the directions in a cartesian coordinate system. All these operators act component-wise, that is, the differentiation resp. multiplication is applied to each component of the wave function. 

The velocity operator is usually defined as the time-derivative of the position operator. In nonrelativistic quantum mechanics, the velocity is equal to (mass times) momentum and hence a constant of motion - in agreement with Newton s second law which characterizes the free motion by a constant velocity. In relativistic quantum mechanics, however, we find... 

As in nonrelativistic quantum mechanics, the generators of the rotations are the angular momentum operators. But here the angular momentum is a sum of an orbital angular momentum and the spin,... 

Here the unitary operator generated by L acts only on the argument of the wave function (as in nonrelativistic quantum mechanics), while the unitary matrix exp(—i(pn S) only affects the spinor components. Hence... 

To one who is familiar with nonrelativistic quantum mechanic it may appear quite clear what is meant by a spherically symmetric potential— any potential that actually only depends on x, so that it is invariant under any rotation applied to the system. It is indeed true that a scalar function

R ) for all rotation matrices R, if and only if it is a function of r = [x]. But a general potential in quantum mechanics is given by a Hermitian matrix, and the unitary operators (86) representing the rotations in the Hilbert space of the Dirac equation can also affect the spinor-components. Hence it is not quite straightforward to tell, which potentials are spherically symmetric. 

In nonrelativistic quantum mechanics, the degeneracy due to spherical symmetry of the Schrodinger equation is 2i +1, where f is the orbital angular quantum number. The spin just doubles the number of states, and if there is no magnetic field, the two spin-states are energetically indistinguishable. Hence, a nonrelativistic particle with spin has a 2(2f -I- l)-fold degeneracy due to spherical symmetry. 

In nonrelativistic quantum mechanics, the angular momentum barrier prevents the collaps to the center. The angular momentum barrier is an effective potential of the form + l)/r that appears if one writes the kinetic energy in polar coordinates. For the Dirac equation the role of the angular momentum barrier is obviously played by the term /c/r in the radial Dirac operator. This term is effectively repulsive for both signs of k, because it appears only off-diagonal. The point is, that the repulsive angular momentum barrier k jr cannot balance the attractive Coulomb potential 7/r for r —> 0, as soon as I7I > k. ... 

Evidently, there is a problem with Dirac Hamiltonians of the sort we have been discussing because the spectrum goes from — to -l-< there is no global lower bound. It is conventional to assume that this is the end of the matter, and that variational methods cannot be applied to Dirac Hamiltonians. This is false. The bound state spectrum of an atom is indeed bounded below, more or less where one wants it to be, so that provided due care is taken with the choice of trial functions, we can proceed exactly as in nonrelativistic quantum mechanics. We can then extend this in the usual way to molecules and solids. Here, we shall merely summarize the argument leading to this conclusion. 

The result can easily be extended to all one-body Dirac problems encountered in atomic and molecular structure, providing a firm basis for variational calculations just as it does in nonrelativistic quantum mechanics. 

The low-energy contribution can be written in the frame of nonrelativistic quantum mechanics as [41] ... 

Our decision in favor of combining nonrelativistic quantum mechanics with the nrl of electrodynamics becomes very important, when we consider the interaction between moving electrons (section 7). In the nrl there is only a nonretarded (instantaneous) Coulomb interaction, while both the magnetic interaction and the retardation of the Coulomb interaction are relativistic corrections and are therefore neglected. One needs to consider them only if one also includes relativistic corrections to the kinematics. 

In nonrelativistic quantum mechanics the formulation of an n-electron Hamiltonian is rather straightforward, because in the nrl of electrodynamics the electrostatic potential satisfies the Poisson equation (95), which implies (in atomic units) that the potential due to a point charge is equal to 1/r and the interaction between a pair of electrons is simply... 

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