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Journal of Modern Physics

How Does Wave Packet of a Free Particle Yield?

Tian-Hai Zeng

Published Jan 1, 2015• 0 citations
Publisher

Abstract

Preparing a particle in a superposition or a wave packet of eigenstates of a physical quantity is to let it interact with a large object. The composite system composed of the particle and the large object evolves into an entangled state. When the state of the large object is considered to be approximately unchanged, the entangled state can be approximately considered as a product state, and then the particle is prepared in an approximate superposed state. We consider the Schrodinger equation for a composite system with interactions between subsystems as a fundamental postulate and a single particle’s Schrodinger equation must be approximately obtained from it. We argue that superposition of states exists only in composite systems. Interaction exchanging some quantities between subsystems makes conservation laws strictly hold, and no wave packet of a free particle yields. With this point, we can also understand the double-slit experiment and the tunnel phenomenon.

DOI
10.4236/jmp.2015.67090

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Preparing a particle in a superposition or a wave packet of eigenstates of a physical quantity is to let it interact with a large object. The composite system composed of the particle and the large object evolves into an entangled state. When the state of the large object is considered to be approximately unchanged, the entangled state can be approximately considered as a product state, and then the particle is prepared in an approximate superposed state. We consider the Schrodinger equation for a composite system with interactions between subsystems as a fundamental postulate and a single particle’s Schrodinger equation must be approximately obtained from it. We argue that superposition of states exists only in composite systems. Interaction exchanging some quantities between subsystems makes conservation laws strictly hold, and no wave packet of a free particle yields. With this point, we can also understand the double-slit experiment and the tunnel phenomenon.
The one-particle Schrödinger and Dirac equations are derived, without making any quantization hypothesis, from the postulate that an elementary particle is a physical wave packet of dynamically varying form and size, rather than a pointlike object located somewhere within the support of a wavefunction that serves only as a mathematical probability amplitude. An isolated multiparticle system can, similarly, be described in principle as a set of interacting wave packets governed by coupled one-particle equations, a picture from which, in the nonrelativistic limit, the multiparticle Schrödinger equation can be obtained as a sufficiency condition for the system to have a conserved total energy equal to a particular value E .
Abstract This chapter continues with a study of the time independent Schrödinger equation and seeks to contrast the quantum behavior of a free particle with that of a particle localized in a potential quantum well. A free particle can exist over all space or can be localized in a wave package. The wave packet is a coherent superposition of the plane waves that make up the wave function that localizes the particle because of constructive and destructive interference. The wave packet spreads out in time because the waves leading to constructive interference get out of phase. In Chapter 2, the particle was localized by a quadratic potential energy. Here, the potentials are described as piecewise constant. The approach is based on assuming a one-dimensional space, x, which is relevant to many problems in the laboratory. The solution is easily generalized to higher dimensions (x-y or x-y-z), but the physics remains the same. The objective is to understand the shape of the eigenfunctions in space and to be able to relate this to the probability density of locating the particle, as well as understanding the relevance of these systems to today’s technology.
There is evidence that superpositions of states of non-composite quantum systems A exist and can be interpreted as the simultaneous existence of two or more physically distinct quantum states of A. This paper shows, however, that entangled superposition states have a quite different character from simple superposition states. Whereas a simple superposition superposes two coherent states of a non-composite object A, entangled states superpose correlations between states of two incoherent subsystems A and B of a composite system AB. Thus an entangled state is best conceptualized as a coherent or phase-dependent superposition of correlations between incoherent or phase-independent subsystem states.
Many wave phenomena are related to interactions. When preparing a particle in a superposed state starting from an eigenstate, we must let it interact with an apparatus or environment. Considering neglected interactions, states of apparatus and Newtonic idea about measurement, we attempt to modify some concepts of quantum mechanics. Our modifications include wave-particle duality, the principle of superposition of states, the meaning of the Schrodinger equation, the postulate of quantum measurement, and the uncertainty relations, without changing past correct results. These may help us to understand some typical quantum phenomena with near classical and intuitive ideas, to mediate some contradictions logically, and then to go a step further toward perfecting quantum mechanics. We also review the argument between Einstien and Bohr about quantum mechanics, and conclude that the result is a draw.
We propose and analyze a scheme by which a many-particle system can be prepared in highly entangled wave-packet states. One of the particles is prepared initially in a quantum superposition of multiple coherent states and then coupled via a quadratic interaction Hamiltonian to a number of other particles. The system evolves into a highly entangled wave-packet state. An appropriate measure of this time-dependent entanglement is given. This scheme is applicable to a number of systems of interest in quantum-information science.
One of the fundamental conceptual difficulties in quantum mechanics arises from the fact that two of the axioms of the theory concerning the evolution of a microscopic system are contradictory. The first axiom assumes that the state of a quantum system evolving undisturbed by any observer obeys a linear deterministic law expressed by the SchrSdinger equation. The second one (wave-packet reduction principle) states that, when the quantum system interacts with an apparatus devised to measure some of its dynamical variables, its state changes abruptly and stochastically, jumping into a normalized eigenstate of the measured variable. This evolution is not deterministic nor linear and cannot be accounted for by the SchrSdinger equation. As a consequence the internal coherence of the theory becomes questionable. On one side the influence of the apparatus on the quantum system cannot be described as an interaction of the quantum constituents of the measuring macroscopic apparatus with the measured quantum system. On the other side, the impossibility of considering a macroscopic object as made by quantum constituents, i .e . the difficulty of quantum mechanics in describing the behaviour of macroscopic systems, is even more dramatic if we consider the evolution of the closed system consisting of the macroscopic apparatus and of the microscopic system. In such a case the appearance of superpositions of macr0scopically distinguishable states related to the apparatus cannot be avoided. These superpositions contradict both our classical experience and wave-packet reduction principle, which provides for statistical mixtures.