In this paper we use Dirac's relativity quantum mechanics to derive the well-known Kramers-Heisenberg dispersion formula for an atom with one electron. The treatment is not limited to the case of a central field, but is quite general. An expression is also obtained for the dipole moment to which is due the incoherent scattering. The formulae obtained are similar to those obtained by O. Klein for the case of a central field. We find in this way an explicit expression for f, the number of dispersion electrons for any line of the optical spectrum (being a measure of the intensity of the line), in terms of the solutions of the four wave equations of Dirac's theory. It is further shown that the sum of the number of dispersion electrons for any state of the atom is not exactly unity, but differs from it by an amount of the order of 10−4. The result Σf = 1 has been shown by London to hold exactly for the simple wave equation as originally given by Schrödinger. It is here shown that the exact relativity treatment has a very small effect.
A quantum theory of nonlinear phenomena is considered in which account is taken of all contributions from electric and magnetic multipoles, without any multipolar approximation. Exact closed analytical expressions for the multipoles are first obtained from quantum electrodynamics by applying a generalization of the Power-Zienau transformation. The standard procedure is invoked to calculate general electromagnetic susceptibility tensors ${\ensuremath{\chi}}_{\ensuremath{\mu}\ensuremath{\nu}{\ensuremath{\alpha}}_{1}{\ensuremath{\beta}}_{1}\ensuremath{\cdots}{\ensuremath{\alpha}}_{n}{\ensuremath{\beta}}_{n}}^{[n]}$ (${\ensuremath{\omega}}_{1}, \dots{}, {\ensuremath{\omega}}_{n}$). Such a tensor is a shorthand for the electric and magnetic susceptibilities as well as mixed susceptibilities. By considering a relativistic theory based on the Dirac equation, the spin susceptibilities are automatically included. The formalism in this way allows for the computation of a wide range of electro- and magneto-optical effects to any order of perturbation and to any order of multipolar approximation.
This is the first in a series of papers. It deals with the linked-cluster summation of the perturbation series in quantum mechanics and quantum statistical physics. In particular, a formalism is developed for calculating the matrix element of the operator
exp(+sH), where
H is the quantum-mechanical Hamiltonian operator. It is found that the matrix element may be written as an ordinary
c-number integral of a
c-number function over the phase space. This function is found by doing a linked-cluster summation of the usual perturbation series. It is in the form of an exponential of an effective Hamiltonian,
Heff, which is a function of the parameter
s and the
c-number momentum and position coordinates. We have assumed that the Hamiltonian has no spin-dependent terms. The matrix element is written in such a form that it may be used for calculating the transition matrix elements in time-dependent perturbation theory as well as for calculating the canonical partition function
ZN in quantum statistical mechanics. The former is obtained by substituting $s=\ensuremath{-}\frac{\mathrm{it}}{\ensuremath{\hbar}}$ and the latter by substituting $s=\ensuremath{-}\ensuremath{\beta}$. The series for
Heff depends only on the potential and is independent of the statistics. The effects of the statistics (Bose-Einstein, Fermi-Dirac, or Maxwell-Boltzmann) are all included in a certain factor multiplying
exp(sHeff). Thus we have succeeded in separating the statistical and quantum corrections. These latter arise from the noncommutativity of the position and momentum operators. The effective Hamiltonian is an infinite series. A diagrammatic technique is given for evaluating this series. It is an expansion in powers of the coupling constant and can also be expanded in powers of Planck's constant. In general, it is not possible to give a closed-form expression for
Heff. But for most practical cases it is sufficient to keep the first few powers of Planck's constant $\ensuremath{\hbar}$, because it is very small. We therefore have an entirely new method of approximation. The statistical effect may always be included exactly. No numerical estimates are made in this paper.
We describe a unified approach for the determination of ac-Stark correction and Kramers-Heisenberg dispersion formula. In both cases the contribution from infinite intermediate summation appearing in the expression for the corresponding matrix elements are evaluated exactly in the dipole approximation for the ground state of hydrogen atom using a variation of the Dalgarno-Lewis method. The analytical expressions obtained can be efficiently used for the numerical evaluation of matrix element for all values of incident photon energy and comparison is made with results obtained by different methods.
The perturbation expansions of the wavefunction, under the action of external monochromatic radiation are obtained, which have no secular or divergent terms, in presence of resonance (without any damping factor). The expectation values of the electric dipole moment operator with respect to these characteristic states are calculated. The dispersion formulae are obtained in the neighbourhood of resonance; they are free from any divergences. The dispersion curves obtained from them have all the basic features of the usual dispersion curves in the neighbourhood of resonance.