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Journal of Mathematical Physics

Exact Stationary States in Local Relativistic Field Theories

Gerald Rosen

Published Feb 1, 1970• 1 citation
Publisher

Abstract

For theories featuring n scalar fields and an interaction energy density that converges rapidly for large field magnitudes to the form representative of a linear theory, we present exact physical solutions to the Schrödinger stationary state equation. These solutions show that such quantum field theories are effectively linear with no interaction and scattering between the quanta.

DOI
10.1063/1.1665168

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For theories featuring n scalar fields and an interaction energy density that converges rapidly for large field magnitudes to the form representative of a linear theory, we present exact physical solutions to the Schrödinger stationary state equation. These solutions show that such quantum field theories are effectively linear with no interaction and scattering between the quanta.
We seek to introduce a mathematical method to derive the Klein-Gordon equation and a set of relevant laws strictly, which combines the relativistic wave functions in two inertial frames of reference. If we define the stationary state wave functions as special solutions like Ψ(r,t)=ψ(r)e−iEt/ℏΨ(\mathbf{r},t)=ψ(\mathbf{r})e^{-iEt/\hbar}Ψ(r,t)=ψ(r)e−iEt/ℏ, and define m=E/c2m=E/c^2m=E/c2, which is called the mass of the system, then the Klein-Gordon equation can clearly be expressed in a better form when compared with the non-relativistic limit, which not only allows us to transplant the solving approach of the Schrödinger equation into the relativistic wave equations, but also proves that the stationary solutions of the Klein-Gordon equation in a potential field have the probability significance. For comparison, we have also discussed the Dirac equation. By introducing the concept of system mass into the Klein-Gordon equation with the scalar and vector potentials, we prove that if the Schr\"{o dinger equation in a certain potential field can be solved exactly, then under the condition that the scalar and vector potentials are equal, the Klein-Gordon equation in the same potential field can also be solved exactly by using the same method.
The energy-momentum tensor T μ ν for the scalar field configurations of the relativistic Schrodinger theory is constructed. Its time component Too is identified with the energy functional E[Ψ] for the stationary solutionsΨ. The confidence into the proposed functional is supported by the fact that it reproduces exactly the relativistic mass eigenvalues of the one-particle systems. For many-particle systems, the non-relativistic limit of relativistic Schrodinger theory is found to coincide exactly with the Hartree-Fock approach.
Many nonlinear quantum optical physics phenomena need more accurate wave functions and corresponding energy or quasienergy levels to account for. An analytic expression of wave functions with corresponding energy levels for an atomic electron interacting with a photon field is presented as an exact solution to the Schrödinger-like equation involved with both atomic Coulomb interaction and electron-photon interaction. The solution is a natural generalization of the quantum-field Volkov states for an otherwise free electron interacting with a photon field. The solution shows that an N level atom in light form stationary states without extra energy splitting in addition to the Floquet mechanism. The treatment developed here with computing codes can be conveniently transferred to quantum optics in classical-field version as research tools to benefit the whole physics community.
Stationary states General properties We shall consider first some general characteristics of the ‘stationary states’. These are eigenfunctions of the Hamiltonian operator: The requirement that ψ satisfies the Schrödinger equation fixes its time dependence: Notice that (4.1.2) can only be a solution if the external potential V is independent of the time: V = V ( x ). The initial function ψ 0 is not arbitrary but must satisfy the time-independent Schrödinger equation (4.1.1). Writing ψ = R e iS /ħ one thus has The following deductions may be made from (4.1.3): ( a ) The probability density is independent of the time: |ψ| 2 = R 2 0 ( X ). ( b ) The quantum potential Q = (- ħ2/2 mR 0 )∇ 2 R 0 is time-independent. Therefore so is the total effective potential: ( c ) The velocity field v = (l/ m )∇ S 0 is independent of time and hence the trajectories, found by solving m = v ( x )| x = x ( t ) , are solutions of Moreover, ψ 0 ( x ) is often a real function for many bound state problems of interest (although not for scattering states) which implies that the velocity is zero. The particle is at rest where one would classically expect it to move since the quantum force (- ∇ Q ) cancels the classical force (- ∇ K ). ( d ) The energy of all particles in the ensemble, -∂ S /∂ t , is a constant of the motion and equal to the energy eigenvalue: with (4.1.4), we have an example of the conservation of energy (cf. §3.9.3). ( e ) The total field energy is conserved: 〈Ĥ〉 = E . […]
We obtain a complete series solution of stationary Schrödinger's equation in the general quantum systems. It is exact in the sense that any approximation means is not used, or that the whole corrections or contributions from all order perturbations are involved if the perturbation concept is introduced. Furthermore, the useful forms of our exact solution and a new expression the complete Green operator are given out. As a universal and analytical solution, it is helpful for the theoretical derivations and practical calculations in quantum theory.
We consider two kinds of solutions of a recently proposed field theory leading to a nonlinear Schr\"odinger equation exhibiting solitonlike solutions of the power-law form ${e}_{q}^{i(kx\ensuremath{-}wt)}$, involving the qqq exponential function naturally arising within nonextensive thermostatistics [${e}_{q}^{z}\ensuremath{\equiv}{[1+(1\ensuremath{-}q)z]}^{1/(1\ensuremath{-}q)}$, with e1z=ez{e}_{1}^{z}={e}^{z}e1z​=ez]. These fundamental solutions behave like free particles, satisfying $p=\ensuremath{\hbar}k$, $E=\ensuremath{\hbar}\ensuremath{\omega}$, and E=p2/2mE={p}^{2}/2mE=p2/2m $(1\ensuremath{\le}q<2)$. Here we introduce two additional types of exact, analytical solutions of the aforementioned field theory. As a first step we extend the theory to situations involving a potential energy term, thus going beyond the previous treatment concerning solely the free-particle dynamics. Then we consider both bound, stationary states associated with a confining potential and also time-evolving states corresponding to a linear potential function. These types of solutions might be relevant for physical applications of the present nonlinear generalized Schr\"odinger equation. In particular, the stationary solution obtained shows an increase in the probability for finding the particle localized around a certain position of the well as one increases qqq in the interval $1\ensuremath{\le}q<2$, which should be appropriate for physical systems where one finds a low-energy particle localized inside a confining potential.