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Birkhäuser Boston eBooks

Construction of a Quantum Field Linked to the Coulomb Potential

Nicolas Privault

Published Jan 1, 1998• 0 citations
Publisher

Abstract

We construct a non-Gaussian field and its associated creation and annihilation operators, describing non-interacting bound states with no momentum in a Coulomb potential. The system is shown to behave quantum mechanically as a collection of coupled harmonic oscillators, allowing us to construct the field by injection into the quantum mechanical space of a free Gaussian field, and to study its time development. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

DOI
10.1007/978-1-4612-2022-0_15

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We construct a non-Gaussian field and its associated creation and annihilation operators, describing non-interacting bound states with no momentum in a Coulomb potential. The system is shown to behave quantum mechanically as a collection of coupled harmonic oscillators, allowing us to construct the field by injection into the quantum mechanical space of a free Gaussian field, and to study its time development. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
The most realistic situations in quantum mechanics involve the interaction between two or more systems. In the most of reliable models, the form and structure of the interactions generate differential equations which are, in the most of cases, almost impossible to solve exactly. In this paper, using the Schwinger Quantum Action Principle, we found the time transformation function that solves exactly the harmonic oscillator interacting with a set of other harmonic coupled oscillators. In order to do it, we have introduced a new special set of creation and annihilation operators which leads directly to the \emph{dressed states} associated to the system, which are the real quantum states of the interacting \emph{\textquotedblleft field-particle\textquotedblright} system. To obtain the closed solution, it is introduced in the same foot a set of \emph{normal mode} creation and annihilation operators of the system related to the first ones by an orthogonal transformation. We find the eigenstates, amplitude transitions and the system spectra without any approximation. At last, we show that the Schwinger Variational Principle provides the solutions in a free representation way.
Abstract We consider the time-independent Schrödinger equation for the harmonic oscillator potential and determine its bound states and energy levels in one spatial dimension using two approaches: the algebraic and the analytic method. Using the former, we introduce ladder operators: the annihilation and creation operators, as well as the occupation-number operator, and determine the ground state of the harmonic oscillator to be a Gaussian wave packet, while the latter method yields the general bound state solutions in terms of the Hermite polynomials. We further discuss the zero-point energy and uncertainty relation for the quantum harmonic oscillator and make a comparison to the classical harmonic oscillator. Finally, we examine the three-dimensional harmonic oscillator, which leads us to the description of systems with multiple degrees of freedom via the tensor product
It is well known that the analysis of a relativistic n -body problem invariant under the transformations of the Poincaré group and involving one time has only been done for n =1. For n >1 one uses the second quantization formalism of field theory. In this paper we state it in the ordinary space time coordinates associated with the n-bodies as Dirac did it in the one body case. We apply the formalism first to the two body problem if the interaction is of an harmonic oscillator form and then extend it to n particles. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.