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Modern Physics Letters A

Step potential problem and harmonic oscillator problem in the minimum length quantum mechanics

Soyeon Park, Byeong Hyo Woo, Min Jung, Eun Ji Jang, Won Sang Chung

Published May 19, 2015• 3 citations
Publisher

Abstract

In this paper, we use the quasi-position representation of the minimum length quantum mechanics (MLQM) to study the effects of minimum length uncertainty principle (MLUP) on the quantum mechanical system up to a first-order in β. We introduce the probability density and the probability flux to discuss two problems such as particle in a box and step potential problem. For the step potential, we compute the transmission coefficient and the reflection coefficient and compare them with those of the ordinary quantum mechanics. We also discuss the harmonic oscillator problem in MLQM.

DOI
10.1142/s0217732315500960

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In this paper, we use the quasi-position representation of the minimum length quantum mechanics (MLQM) to study the effects of minimum length uncertainty principle (MLUP) on the quantum mechanical system up to a first-order in β. We introduce the probability density and the probability flux to discuss two problems such as particle in a box and step potential problem. For the step potential, we compute the transmission coefficient and the reflection coefficient and compare them with those of the ordinary quantum mechanics. We also discuss the harmonic oscillator problem in MLQM.
In this paper, we study two generalized uncertainty principles (GUPs) including [Formula: see text] and [Formula: see text] which imply minimal measurable lengths. Using two momentum representations, for the former GUP, we find eigenvalues and eigenfunctions of the free particle and the harmonic oscillator in terms of generalized trigonometric functions. Also, for the latter GUP, we obtain quantum mechanical solutions of a particle in a box and harmonic oscillator. Finally we investigate the statistical properties of the harmonic oscillator including partition function, internal energy, and heat capacity in the context of the first GUP.
in ordinary quantum mechanics for the position uncertainty since the minimum value for the uncertainty can be tend to zero. Different theories of quantum gravity, such as string theory, loop quantum gravity, and black hole physics, confirm a minimal observable length. Kempf et al have formulated The minimal length is a fundamental length for position measurement, although there is no lower bound a generalized uncertainty principle (GUP) by modifying the Heisenberg uncertainty principle to include an appearance of the length in quantum mechanics. In this study, we have found out the influence of the GUP on the particle for its confinement in a 1D potential box and calculated energy eigenvalues. The modified Schrödinger equation for the particle is factorized to be of second order to obtain the eigenvalues in a handy way. The obtained energy levels are modified from the usual result with GUP parameter dependency. The modification in the energy spectrum due to the GUP characterized by the presence of the minimal length has been also explored graphically.
In this paper, use is made of the tools of analytical mechanics and the concept of operators to obtain the time-independent and time-dependent Schrodinger wave equations for quantum mechanical systems. Derivations are embarked upon of expressions for reflection and transmission coefficients for a particle of mass m as well as of energy E moving under different potential set-ups across step functions, barriers and well functions. The tunneling effect is then discussed. The transmission probability equation obtained in this research has been observed to be more accurate than the transmission probability expression deduced by some researchers in 2014 for a tunneling barrier. This research work finds applications in nuclear magnetic resonance imaging systems, synchrotrons, gyrators, accelerators, and in electrodynamics.
In the framework of the generalized uncertainty principle, the position and momentum operators obey the modified commutation relation X,P=iħ1+βP2 , where β is the deformation parameter. Since the validity of the uncertainty relation for the Shannon entropies proposed by Beckner, Bialynicki-Birula, and Mycielski (BBM) depends on both the algebra and the used representation, we show that using the formally self-adjoint representation, that is, X=x and P=tan⁡βp/β , where [x,p]=iħ , the BBM inequality is still valid in the form Sx+Sp≥1+ln⁡π as well as in ordinary quantum mechanics. We explicitly indicate this result for the harmonic oscillator in the presence of the minimal length.
The quantum harmonic oscillator is a quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics. Coherent state is physically interesting because it is a minimum uncertainty state at which the Heisenberg uncertainty relation reaches its minimum. If a charged particle of charge moves in an electromagnetic field, there is a Lorentz force acting on the particle. The electric and magnetic are invariant under the gauge transformation. In the presence of an external vector potential, the Hamiltonian is changed, so is the probability current operator.