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Information Transfer in Quantum Mechanics

Tibor Blazsó

Published Apr 15, 2015• 0 citations
OpenAlex

Abstract

This paper deals with the most elementary information transmission in Quantum Mechanics. A simple quantum mechanical system under Coulomb-type potential is investigated. Like the classical case, a deep relation between the potential energy and the rate of information transfer is established for quantum-mechanical situation. The corresponding equation is presented. The article shows the circumstance for the free particle, too.

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This paper deals with the most elementary information transmission in Quantum Mechanics. A simple quantum mechanical system under Coulomb-type potential is investigated. Like the classical case, a deep relation between the potential energy and the rate of information transfer is established for quantum-mechanical situation. The corresponding equation is presented. The article shows the circumstance for the free particle, too.
Quantum parallelism implies a spread of information over the space in contradistinction to the classical mechanical situation where the information is "centered" on a fixed trajectory of a classical particle. This means that a quantum state becomes specified by more indefinite data. The above spread resembles, without being an exact analogy, a transfer of energy to smaller and smaller scales observed in the hydrodynamical turbulence. There, in spite of the presence of dissipation (in a form of kinematic viscosity), energy is still conserved. The analogy with the information spread in classical to quantum transition means that in this process the information is also conserved. To illustrate that, we show (using as an example a specific case of a coherent quantum oscillator) how the Shannon information density continuously changes in the above transition . In a more general scheme of things, such an analogy allows us to introduce a "dissipative" term (connected with the information spread) in the Hamilton-Jacobi equation and arrive in an elementary fashion at the equations of classical quantum mechanics (ranging from the Schrödinger to Klein-Gordon equations). We also show that the principle of least action in quantum mechanics is actually the requirement for the energy to be bounded from below.
The laws of quantum mechanics differ significantly from the laws of classical mechanics. The mathematical apparatus of quantum mechanics is based on the amazing Schrödinger equation. This is a specific partial differential equation with respect to a complex-valued function called the wave function. This characteristic allows, in particular, estimating the probability of finding a quantum mechanical particle in a given volume. We establish the Schrödinger equation for a free particle and for a particle moving in an external field. Then the problem of passing a potential barrier is investigated. It is shown that, in contrast to the classical case, even a relatively slow quantum mechanical particle is capable of overcoming a sufficiently high potential barrier. The Appendix considers the concept of a normalized wave function and analyzes the movement of a particle in a potential well with infinite walls.
This chapter discusses several applications of the formalism of theory of quantum mechanics. The time-independent Schrodinger equation is applied to understand simple quantum systems such as a free particle, a particle in an infinite square well, step potentials, potential barrier and finite square well. The chapter deals with the simplest case, a free particle. It is the most elementary application of the formalism. A free particle means that no forces are acting on the particle, and hence, there is no interaction between the particle and its environment. According to classical mechanics, a free particle moves with a constant velocity along a straight line. The energy of the wave associated with the free quantum particle is same as that of the classical free particle. The chapter describes a quantum free particle as a wave packet traveling in space and time. It discusses a quantum particle under the influence of a force.
In the information interpretation of quantum mechanics, information is the most fundamental, basic entity. Every quantized system is associated with a definite discrete amount of information (cf. Zeilinger). This information content remains constant at all times and is permutated one-to-one throughout the system evolution. What is interpreted as measurement is a particular type of information transfer over a fictitious interface. The concept of a many-to-one state reduction is not a fundamental one but results from the practical impossibility to reconstruct the original state after the measurement.
This chapter deals with some solutions of Schrödinger's equation in situations that are relatively simple, but which nevertheless have important applications in chemistry. Most of these deal with the quantization of energy. However, there is another remarkable prediction of quantum mechanics, known as tunnelling. In classical physics, kinetic energy cannot be negative; in other words, the total energy cannot be less than the potential energy. Schrödinger's equation can, however, be solved when this is the case. In some situations, these solutions are physically significant. The chapter then looks at the harmonic oscillator, which is one of the most important elementary models in mechanics, and is especially relevant in chemistry in connection with the vibration of molecules. It also considers rotations and angular momentum.