The quantum motion of one particle which is in a one-dimensional infinite well which exists a potential barrier have been studied.From the statistical analysis of energy spectra,the nonintegrable phenomenon is in quantum mechanics,but this model in the classical is integrable.The reason for that is the tunneling in quantum mechanics.The degree of noninteger is related to the form of the potential barrier.The more complex the potential barrier is,the more clearly the chaos is.
The quantum motion of one particle which is in a one-dimensional infinite well which exists a potential barrier have been studied.From the statistical analysis of energy spectra,the nonintegrable phenomenon is in quantum mechanics,but this model in the classical is integrable.The reason for that is the tunneling in quantum mechanics.The degree of noninteger is related to the form of the potential barrier.The more complex the potential barrier is,the more clearly the chaos is.
Abstract A distinguishing feature of quantum mechanics, as compared with classical mechanics, is the phenomenon of matter exhibiting wave-like properties. A direct consequence of this wave property of particles is the tunneling of particles through potential energy barriers.-3 The classical particle is forbidden from penetrating the barrier region. In this chapter, the problem of one-dimensional (1D) quantum mechanical tunneling of particles through potential barriers of arbitrary shape is treated. We show that tunneling is a mixed boundary value problem, and obtain the action integral that leads to Schrodinger’s equation and at the same time gives the equations defining the boundary conditions (BCs).
Classical mechanics is a singular theory in that real-energy classical particles can never enter classically forbidden regions. However, if one regulates classical mechanics by allowing the energy E of a particle to be complex, the particle exhibits quantum-like behavior: complex-energy classical particles can travel between classically allowed regions separated by potential barriers. When Im E → 0, the classical tunneling probabilities persist. Hence, one can interpret quantum tunneling as an anomaly. A numerical comparison of complex classical tunneling probabilities with quantum tunneling probabilities leads to the conjecture that as Re E increases, complex classical tunneling probabilities approach the corresponding quantum probabilities. Thus, this work attempts to generalize the Bohr correspondence principle from classically allowed to classically forbidden regions.
We use an one dimensional model of a square barrier embedded in an infinite potential well to demonstrate that tunneling leads to a complex behavior of the wave function and that the degree of complexity may be quantified by use of the spatial entropy function defined by S = -\int |Ψ(x,t)|^2 ln |Ψ(x,t)|^2 dx. There is no classical counterpart to tunneling, but a decrease in the tunneling in a short time interval may be interpreted as an approach of a quantum system to a classical system. We show that changing the square barrier by increasing the height/width do not only decrease the tunneling but also slows down the rapid rise of the entropy function, indicating that the entropy growth is an essentially quantum effect.
In generic Hamiltonian systems that are neither completely integrable nor fully chaotic, phase space consists of a mixture of regular and chaotic components. In classical dynamics, transitions between different invariant sets in phase space are strictly forbidden, and these sets act as dynamical barriers to one another. In quantum mechanics, in contrast, wave effects allow transitions through such dynamical barriers. This process, known as dynamical tunneling, refers to penetration through dynamical barriers in phase space and was first recognized in the early 1980s. Since then, various aspects of dynamical tunneling have been elucidated, significantly advancing our understanding of such a novel quantum phenomenon. In this article, we provide an overview of several phenomenological perspectives of dynamical tunneling, including chaos-assisted and resonance-assisted tunneling, and also introduce approaches based on classical mechanics extended into the complex domain. In particular, we seek to clarify what is meant by the common claim that "chaos leads to an enhancement of the tunneling probability", which is often made when dynamical tunneling is dressed. We discuss what regime this refers to and, if such an enhancement occurs, what its likely origin is.
Based on the theory of Schrdinger equation for the free particle,the solution of that equation was discussed in detail. It is found that the equation only exist an insignificant solution. And the mediocre solution can be also obtained from the one-dimensional infinitely deep potential well in a potential well width tending to infinite and from one-dimensional square potential barrier in the barrier height tending to zero. Although the solution is insignificant,but it describes some ideal that is consistent with that of classical mechanics. It explains the connection between the classical mechanics and quantum mechanics. The mediocre solutions show that 1) For infinite free particles,it is followed by Newton' s first motion law,2) For infinite free particle,it does not exist matter wave characteristics. Only if there is some potential( i. e. interactions),the Wave-particle duality be embodied.
This chapter explores several cases of one-dimensional quantum motion, emphasizing the physical insights and analytical results. The particle in a rectangular potential well is examined, including the derivation of its quantized energy spectrum and normalization constant, particularly for the case of an infinitely deep well. Comparisons are drawn between the quantum and classical descriptions of motion. The analysis extends to a rectangular potential barrier, where reflection and transmission coefficients are calculated, and the phenomenon of quantum tunneling is explored. The tunnel effect paradox is addressed, along with its applications to α-decay and the cold emission of electrons from metals. Finally, the linear harmonic oscillator is treated in detail, including its energy eigenvalues, eigenstates, parity properties, and zero-point energy — the latter discussed as a consequence of the uncertainty principle and the wave-like behavior of microscopic particles. Metadata/Keywords: one-dimensional motion, rectangular potential well, α-decay, cold emission, quantum tunneling, linear harmonic oscillator, zero-point energy, parity, uncertainty principle, energy eigenstates