A general method to treat the quantal two-body problem relativistically is developed on the line of Dirac's many-time theory. The case where two charged fermions interact with each other through the electromagnetic field is investigated as an example. We construct a canonical transformation which eliminates the electromagnetic field variables in the original Schrödinger equations, and derive the potential energy between two charged particles. Finally, some properties of the simultaneous wave equations for the system of two particles are discussed, and a method to treat the stationary state of that system is indicated.
For describing the pair correlations of the electrons in the many- electron systems a new method of reducing the many-particle problem to the two- particle problem is developed. Instead of the wave function, one- and twoparticle density matrices are used. In the process of deriving an equation for the two-particle wave function, which describes a pair of electrons in the field of the others, a transition to the second quantization representation is accomplished. The solution of the obtained equation is carried out with the help of perturbation theory. (auth)
The Hamiltonian of a system of charged particles interacting with the electromagnetic field is investigated. For an arbitrary system the multipole expansion of the interaction between the system and the field is derived by means of a suitable canonical transformation. The transformed Hamiltonian is obtained from the Hamiltonian of the system by replacing the momenta by the transformed kinetic momenta and by adding to the Hamiltonian a term representing the interaction of the system with the electric component of the field. By expanding this interaction term, as well as the transformed momenta, in powers of the dimension of the system over the wavelength, the multipole expansion of the Hamiltonian is obtained. For a system interacting with a classical field the multipole form of the Hamiltonian is exactly equivalent to the original Hamiltonian. For a quantized field this is not true, and the multipole form of the transformed Hamiltonian is shown to be equivalent to the original Hamiltonian only for first-order radiation processes.
By a canonical transformation on the Dirac Hamiltonian for a free particle, a representation of the Dirac theory is obtained in which positive and negative energy states are separately represented by two-component wave functions. Playing an important role in the new representation are new operators for position and spin of the particle which are physically distinct from these operators in the conventional representation. The components of the time derivative of the new position operator all commute and have for eigenvalues all values between $\ensuremath{-}c$ and
c. The new spin operator is a constant of the motion unlike the spin operator in the conventional representation. By a comparison of the new Hamiltonian with the non-relativistic Pauli-Hamiltonian for particles of spin \textonehalf{}, one finds that it is these new operators rather than the conventional ones which pass over into the position and spin operators in the Pauli theory in the non-relativistic limit. The transformation of the new representation is also made in the case of interaction of the particle with an external electromagnetic field. In this way the proper non-relativistic Hamiltonian (essentially the Pauli-Hamiltonian) is obtained in the non-relativistic limit. The same methods may be applied to a Dirac particle interacting with any type of external field (various meson fields, for example) and this allows one to find the proper non-relativistic Hamiltonian in each such case. Some light is cast on the question of why a Dirac electron shows some properties characteristic of a particle of finite extension by an examination of the relationship between the new and the conventional position operators.
We show that the interaction between a movable mirror with a quantized field that interacts with a two-level atom may be simplified via a transformation that involves Susskind-Glogower operators (SGO). By using this transformation it is easy to show that we can cast the Hamiltonian, after a series of small rotations, into an effective Hamiltonian that may be solved. We would like to stress that the transformation in terms of SGO already simplifies enough the Hamiltonian in the sense that, in an exact way, it "eliminates" one of the three-subsystems, namely the quantized field.
A simple prescription for quantizing fields is developed and applied to the sytsem of interacting spins in an electromagnetic field. The classical equations of motion are written in the standard form $\ensuremath{-}i(\frac{d}{\mathrm{dt}})|V〉+\mathfrak{M}|V〉=0$, where the state vector
∣V〉 has the state variables, such as the generalized coordinates and momenta, as its components, and the time development operator
M contains derivative, integral, and matrix operators in general. By diagonalizing
M and replacing the resulting classical variables with operators obeying boson commutation relations, the system is quantized, and the Hamiltonian and Heisenberg equations of motion are automatically cast into the harmonic-oscillator form. Thus the quantization is reduced to the simple problem of diagonalizing a matrix. This avoids the laborious process of finding a Lagrangian density (by trial and error in general), determining the canonical variables, forming the Hamiltonian density, and inventing transformations to creation and annihilation operators. Problems with zero-frequency modes, which can be annoying in the standard treatment, are automatically eliminated by the prescription. The bookkeeping of forward and backward waves, different branches of dispersion curves, etc., is simplified. In addition to affording a straightforward and simple method of quantizing fields, the method also gives the necessary and sufficient conditions for a Hamiltonian to be transformable into harmonic-oscillator form, gives a direct relation between the classical equations of motion and the Heisenberg quantum-mechanical equations, and gives a simple method of determining the constants of the motion.