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Journal of Physics A Mathematical and General

The distorted wave Glauber approximation

F. Riordan

Published Jun 1, 1978• 2 citations
Publisher

Abstract

A solution of the Pauli equation with non-zero potentials defines quantum scalar and vector potentials and magnetic fields and quantum trajectories. If a line integral of perturbing potentials and fields along these quantum trajectories is added to the phase of this solution, an approximate solution of the perturbed equation is found. Glauber theory is a special case and the conditions of applicability are similar. Applications given start from the harmonic oscillator and from a homogeneous magnetic field and add a perturbation.

DOI
10.1088/0305-4470/11/6/011

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•Jul 3, 1997•0 citations
A solution of the Pauli equation with non-zero potentials defines quantum scalar and vector potentials and magnetic fields and quantum trajectories. If a line integral of perturbing potentials and fields along these quantum trajectories is added to the phase of this solution, an approximate solution of the perturbed equation is found. Glauber theory is a special case and the conditions of applicability are similar. Applications given start from the harmonic oscillator and from a homogeneous magnetic field and add a perturbation.
The Dirac equation for an electron subjected to a potential A mu is shown to be equivalent to the following classical equations for a beam of non-interacting electrons subjected to A mu in addition to 'quantum' scalar and four-vector potentials together with an anomalous electromagnetic field affecting the spin: the Hamilton-Jacobi equation for the momentum, the continuity equation for the charge distribution, Frenkel's relativistic equation for the spin and a new equation giving a phase theta as an integral along the electron's path. These equations are coupled through the 'quantum' potentials and fields. In the classical limit theta is constant along the electron's path and the equations decouple. Two approximation procedures based on these coupled hydrodynamic equations are given: (1) the semiclassical to lowest order in powers of h(cross), (2) the distorted wave Glauber approximation assumes the solution of the Dirac equation and hence of the hydrodynamic equations for an unperturbed A mu and that the electrons travel along the same four-dimensional paths in the perturbed potential, but with different 'speed'.
This chapter discusses the quasi-isotropic approximation (QIA) with the semiclassical asymptotics of the Pauli equations for spin 1/2 particles in a magnetic field. The case of spin 1/2 and the Pauli equation are taken to avoid complex manipulations and demonstrate the essence of novel features brought in by the QIA to this well-studied problem. The role of quantum constraints ensuring the description of the Stern-Gerlach splitting is played by the special initial conditions. The latter would imply that particle world lines are dependent on the states of particle spin. This is exactly what occurs in the Stern-Gerlach effect. However de Broglie was unable to find the expression for that additional term. Precession of the magnetic moment in this mode comes about due to interference mainly arising from the difference in eikonals corresponding to these modes. The solutions admit all the best known solutions as particular cases, in particular, the Stern-Gerlach particle trajectories and solutions of the Pauli-Galanin-Rubinow-Keller type.
The starting point in a quantum mechanical problem is the construction of a Schrödinger equation. A Schrödinger equation is exactly solvable only for a certain number of simple potentials. In most of the cases the Schrödinger equation can be solved exactly. The solution of the Schrödinger equation for the real systems is usually complicated and often exact solutions. The perturbation theory has been proven to be useful in several fields of theoretical physics and also in chemistry. It provides considerable insight into various phenomena. The main disadvantage of this method is that in certain problems the perturbation series converges slowly or diverges. The splitting of an energy level due to the interaction of the nuclear magnetic dipole and the magnetic field Be produced by the electron orbit current is called the fine structure. The term hyperfine structure is used to describe extremely small splitting of spectral lines.
We calculate exact solutions of the Schrödinger equation for a particle constrained to move along a spherical surface and interacting with non-central potentials, namely, (i) Makarov, (ii) ring-shaped pseudo-harmonic oscillatory, and (iii) Kratzer potentials. We also study exact solutions of the Pauli equation in the same geometrical setting for a charged particle in the presence of a uniform magnetic field. In this case, the two-component spinor can adhere to the surface only if the magnetic field intensity has certain special values. The solutions of Schrödinger equations allow us to obtain exact Pauli spinors and their corresponding energy eigenvalues for the same non-central potentials.
Here a special case of perturbation in quantum harmonic oscillator is studied. Here we assume the perturbed potential to be a Harmonic Oscillator that has been shifted in the position space.We construct the new creation and annihilation operators for the new Hamiltonian to find out its energy eigenstates. What is interesting about the solution of this system is that we find out that the ground state of this system is a coherent state of the ground state of the original harmonic oscillator. We know that the Hamiltonian of electromagnetic field is similar to that of the harmonic oscillator where the electric field and magnetic field are expressed in terms of time evolution of annihilation and creation operators. According to the quantum electromagnetic field theory, photon states are expressed as the coherent states of the ground states of the unperturbed harmonic oscillator.In the previous research works,it has been established how coherent states provide a better picture of Electric field compared to the energy eigenstates. This paper mainly deals with why the coherent states provide a more complete picture of electric and magnetic field theory compared to just energy eigenstates. Here we find how our perturbation act as a coherent state generator and adds a new term in the Hamiltonian. The Perturbation is thought to arise from the only available interaction i.e. observation. The perturbation also explains how observation leads to coherence of photon states. We will find out how the new perturbation affects the time evolution of electromagnetic field. This also explains how the classical electromagnetic field theory takes into account the factor of observation, due to which the classical laws are in perfect agreement with the quantum laws.
Abstract Path integral formulae for the (p, q) system and (a†, a) of the Bose and Fermi systems can only be calculated exactly in the harmonic oscillator case. Other physical situations require some approximation schemes. In this chapter we try to make explicit calculations for the harmonic oscillator and discuss a perturbation technique and the WKB approximation that is considered to be most suited to the path integral formalism. As an illustrative example, a detailed computation is performed for the case where non-trivial solutions, called instantons, exist in the classical equation of motion.