Since the inception of quantum mechanics, the classical limit has been an area of unease. Explanations proposed on this matter include the usage of Plank's constant, the decoherence parameter of the master's equation, or the quantum potential of Bohm's theory. An equally obscure related topic is the mixing of classical and quantum quantities in the dynamical equations. In this thesis, the subject of classical-quantum mixing will be studied. A method will be explored which will demonstrate one of several candidate methods whereby mixing may be removed and replaced by fully quantum non-relativistic equations. Through use of the quantum potential concept and the Feynman path integral, state-dependent forms of the classical scalar and vector potentials will be derived by assuming the existence of separate component wave functions for the particle and potential. The specific wave component | e attributed to the potential, will hold the effects of an environment. The resulting equations are devoid of classical potentials and can therefore be considered as purely quantum--no mixing takes place. The standard classical potentials emerge from the state-dependent equations by a condition which will be referred to as state-dependence reduction (SDR). Through SDR, the semi-quantum and purely quantum equations are qualitatively and quantitatively equivalent. The new purely quantum equations will be used both to interpret gauge symmetry and to clarify the reasoning behind the Aharonov-Bohm effect. It will be argued that the freedom of gauge is related to the freedom to choose from many different and distinct environments which reproduce the same experimental outcome. Likewise, the AB-effect is understood through the environment from which the topological structure of the electromagnetic field is represented. Several aspects of the environment responsible for the effective potentials will be calculated numerically for well-known physical situations including one-dimensional scattering, two-dimensional double-slit setup, two-dimensional Aharonov-Bohm effect, and the two-dimensional Dirac equation. The proposal of using state-dependent potentials to replace the semi-quantum equation represents a consistent generalization and unification of the quantum and classical potentials, and may offer some experimental results yet untapped.
Since the inception of quantum mechanics, the classical limit has been an area of unease. Explanations proposed on this matter include the usage of Plank's constant, the decoherence parameter of the master's equation, or the quantum potential of Bohm's theory. An equally obscure related topic is the mixing of classical and quantum quantities in the dynamical equations. In this thesis, the subject of classical-quantum mixing will be studied. A method will be explored which will demonstrate one of several candidate methods whereby mixing may be removed and replaced by fully quantum non-relativistic equations. Through use of the quantum potential concept and the Feynman path integral, state-dependent forms of the classical scalar and vector potentials will be derived by assuming the existence of separate component wave functions for the particle and potential. The specific wave component | e attributed to the potential, will hold the effects of an environment. The resulting equations are devoid of classical potentials and can therefore be considered as purely quantum--no mixing takes place. The standard classical potentials emerge from the state-dependent equations by a condition which will be referred to as state-dependence reduction (SDR). Through SDR, the semi-quantum and purely quantum equations are qualitatively and quantitatively equivalent. The new purely quantum equations will be used both to interpret gauge symmetry and to clarify the reasoning behind the Aharonov-Bohm effect. It will be argued that the freedom of gauge is related to the freedom to choose from many different and distinct environments which reproduce the same experimental outcome. Likewise, the AB-effect is understood through the environment from which the topological structure of the electromagnetic field is represented. Several aspects of the environment responsible for the effective potentials will be calculated numerically for well-known physical situations including one-dimensional scattering, two-dimensional double-slit setup, two-dimensional Aharonov-Bohm effect, and the two-dimensional Dirac equation. The proposal of using state-dependent potentials to replace the semi-quantum equation represents a consistent generalization and unification of the quantum and classical potentials, and may offer some experimental results yet untapped.
We show that the standard Dirac phase factor is not the only solution of the usual gauge transformation equations. The full form of a general gauge function (that connects systems that move in different sets of scalar and vector potentials), apart from Dirac phases (spatial or temporal integrals over potentials), also contains terms of classical fields that act nonlocally (in spacetime) on the local solutions of the time-dependent Schrödinger equation. As a result, the phases of wavefunctions in the Schrödinger picture are affected nonlocally by spatially and temporally remote magnetic and electric fields, in specific ways that are fully explored. These contributions go beyond the usual Aharonov–Bohm effects (magnetic or electric). (i) Application to cases of particles passing through full static magnetic or electric fields leads to cancellations of Aharonov–Bohm phases at the observation point; these cancellations are linked to behaviors at the semiclassical level (i.e. the old Werner and Brill experimental observations, or their ‘electric analogs’—or to more recent reports of Batelaan and Tonomura) but are shown to be far more general (true not only for narrow wavepackets but also for completely delocalized (spread-out) quantum states). By using these cancellations, certain previously unnoticed sign-errors in the literature are corrected. (ii) Application to time-dependent situations provides a remedy for erroneous results in the literature (concerning an uncritical use of Dirac phase factors) and leads to phases that contain an Aharonov–Bohm part and a field-nonlocal part: their competition is shown to recover relativistic causality in earlier ‘paradoxes’ (such as the van Kampen thought-experiment), while a more general consideration indicates that the temporal nonlocalities found here demonstrate in part a causal propagation of phases of quantum mechanical wavefunctions in the Schrödinger picture. This may open a new and direct way to address time-dependent double-slit experiments and the associated causal issues.
In this thesis we are investigating a different formalism of non-relativistic quantum mechanics called the path integral formalism.It is a generalization of the classical least action principle.The introduction to this subject begins with the construction of the path integral in terms of the idea of probability amplitudes whose absolute square gives the probability of finding a system in a particular state.Then we show that if the Lagrangian is a quadratic form one needs only to calculate the classical action besides from a time-dependent normalization constant to find the explicit expression of the path integral.We look in to the subject of two kinds of slit-experiments: The square slit, the single-and the double-Gaussian slit.Also, the propagator for constrained paths is calculated and applied to the Aharonov-Bohm effect, which shows that the vector potential defined in classical electrodynamics have a physical meaning in quantum mechanics.It is also shown that the path integral formulation is equivalent to the Schrödinger description of quantum mechanics, by deriving the Schrödinger equation from the path integral.Further applications of the path integral are discussed.
In this thesis we are investigating a different formalism of non-relativistic quantum mechanics called the path integral formalism.It is a generalization of the classical least action principle.The introduction to this subject begins with the construction of the path integral in terms of the idea of probability amplitudes whose absolute square gives the probability of finding a system in a particular state.Then we show that if the Lagrangian is a quadratic form one needs only to calculate the classical action besides from a time-dependent normalization constant to find the explicit expression of the path integral.We look in to the subject of two kinds of slit-experiments: The square slit, the single-and the double-Gaussian slit.Also, the propagator for constrained paths is calculated and applied to the Aharonov-Bohm effect, which shows that the vector potential defined in classical electrodynamics have a physical meaning in quantum mechanics.It is also shown that the path integral formulation is equivalent to the Schrödinger description of quantum mechanics, by deriving the Schrödinger equation from the path integral.Further applications of the path integral are discussed.
The Aharonov-Bohm (AB) effect is a pure quantum effect that implies a measurable phase shift in the wave function for a charged particle that encircles a magnetic flux located in a region inaccessible to the particle. Classically, such a nonlocal effect appears to be impossible since the Lorentz force depends on only the magnetic field at the location of the particle. In quantum mechanics, the Hamiltonian, and thus the Schr\"odinger equation, has a local coupling between the current due to the particle and the electromagnetic vector potential A, which extends to the entire space beyond the region with finite magnetic field. This has sometimes been interpreted as meaning that in quantum mechanics A is in some sense more ``fundamental'' than B in spite of the former being gauge dependent and thus unobservable. Here we shall, with a general proof followed by a few examples, demonstrate that the AB effect can be fully accounted for by considering only the gauge-invariant B field, as long as it is included as part of the quantum action of the entire isolated system. The price for the gauge-invariant formulation is that we must give up locality---the AB phase for the particle will arise from the change in the action for the B field in the region inaccessible to the particle.
Generalized solutions of the standard gauge transformation equations are presented and discussed in physical terms.They go beyond the usual Dirac phase factors and they exhibit nonlocal quantal behavior, with the well-known Relativistic Causality of classical fields affecting directly the phases of wavefunctions in the Schrödinger Picture.These nonlocal phase behaviors, apparently overlooked in path-integral approaches, give a natural account of the dynamical nonlocality character of the various (even static) Aharonov-Bohm phenomena, while at the same time they seem to respect Causality.For particles passing through nonvanishing magnetic or electric fields they lead to cancellations of Aharonov-Bohm phases at the observation point, generalizing earlier semiclassical experimental observations (of Werner & Brill) to delocalized (spread-out) quantum states.This leads to a correction of previously unnoticed sign-errors in the literature, and to a natural explanation of the deeper reason why certain time-dependent semiclassical arguments are consistent with static results in purely quantal Aharonov-Bohm configurations.These nonlocalities also provide a remedy for misleading results propagating in the literature (concerning an uncritical use of Dirac phase factors, that persists since the time of Feynman's work on path integrals).They are shown to conspire in such a way as to exactly cancel the instantaneous Aharonov-Bohm phase and recover Relativistic Causality in earlier "paradoxes" (such as the van Kampen thought-experiment), and to also complete Peshkin's discussion of the electric Aharonov-Bohm effect in a causal manner.The present formulation offers a direct way to address time-dependent singlevs double-slit experiments and the associated causal issues-issues that have recently attracted attention, with respect to the inability of current theories to address them.