Paths into the Schrodinger Equation via Classical and Quantum Field Theories

Manu B. Paranjape

C. Xu, Feng Feng, Yingjun Li

As is known to all, the electron scattering in classical electromagnetic potential is one of the most widespread applications of quantum theory. Nevertheless, many discussions about electron scattering are based upon single-particle Schrodinger equation or Dirac equation in quantum mechanics rather than the method of quantum field theory. In this paper, by using the path integral approach of quantum field theory, we perturbatively evaluate the scattering amplitude up to the second order for the electron scattering by the classical electromagnetic potential. The results we derive are convenient to apply to all sorts of potential forms. Furthermore, by means of the obtained results, we give explicit calculations for the one-dimensional electric potential.

J. D. Franson

The Schrodinger equation is not covariant. Nevertheless, quantum field theory is often formulated using the Schrodinger equation to describe the time evolution of the system, which is equivalent to using Feynman path integrals. It is well known that scattering theory gives covariant results provided that the interaction vanishes in the asymptotic limit of $t \rightarrow \pm \infty$, but there are other situations of interest that do not satisfy those conditions. As an example, the Aharonov-Bohm effect is derived here using second-quantized field theory to describe all the electrons as well as the electromagnetic field, which allows the effects of retarded vector potentials to be calculated in a straightforward way using the Feynman propagator. The results are in agreement with the usual expression for the Aharonov-Bohm effect when the retardation of the electromagnetic field is negligible, but they predict a fractional phase shift, a lack of covariance, and violations of causality when retardation effects are significant. These results suggest that quantum field theory based on the Schrodinger equation or Feynman path integrals is incomplete in the sense that it cannot give a correct description of all observable phenomena.

Mark S. Swanson

Yulei Feng

In this paper, we try to give a new approach to the quantum mechanics(QM) on the framework of quantum field theory(QFT). Firstly, we make a detail study on the (non-relativistic) Schrödinger field theory, obtaining the Schrödinger equation as a field equation, after field quantization, the Heisenberg equations for the momentum and position operators of the particles excited from the (Schrödinger) field and the Feynman path integral formula of QM are also obtained. We then give the probability concepts of quantum mechanics in terms of a statistical ensemble, realizing the ensemble(or statistical) interpretation. With these, we make a series of conceptual modifications to the standard quantum mechanics, especially propose a new assumption about the quantum measurement theory which can solve the EPR paradox from the view of the QFT. Besides, a field theoretical description to the double-slit interference experiment is developed, obtaining the required particle number distribution. In the end, we extend all the above concepts to the relativistic case so that the ensemble interpretation is still proper. Two extra topics are added, in the first one, an operable experiment is proposed to distinguish the Copenhagen interpretation from the ensemble one via very different experimental results. While the second topic concerns with the extensions of the concept of coherent state to both the Bosonic and Fermionic field cases, to obtain the corresponding classical fields. And in the concluding section, we make some general comparisons between the standard QM and the one derived from the QFT, from which we claim that the QFT is the fundamental theory.