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European Journal of Physics

Uncertainty relations for a particle confined in a finite square well potential

Akira Tamura, Yohhei Tatsumi

Published Aug 4, 2017• 4 citations
Publisher

Abstract

Abstract In order to specify the uncertainty relation for a particle confined in a finite square well potential we derived the standard deviation of momentum Δ p and that of position Δ x . It is shown that Δ p has a peak as a function of the well width and both Δ x and Δ p ⋅ Δ x have a dip in contrast to the case of the particle confined in an infinite square well potential. We clarified that Δ p 0 ⋅ Δ x 0 of the particle occupying the ground state exists in the finite range as a function of the well width and the potential-barrier height, but a particle confined in an infinite square well potential has a constant Δ p 0 ⋅ Δ x 0 . In addition, we found that the values of Δ p n ⋅ Δ x n for excited states diverge in a certain range of well widths, where the energy level of the confined particle exceeds the potential barrier.

DOI
10.1088/1361-6404/aa8417

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Abstract We have presented the thermodynamic properties of a particle moving freely both in an infinite and finite square well in contact with a heat reservoir at constant temperature. In both cases, we have computed the temperature dependence of the partition function, entropy, average energy and specific heat of the particle in an exact way. The exact results for various thermodynamic parameters deviate considerably from its known theoretical values and reveal the importance of such calculations. The effect of the width of the finite well on the different thermodynamic quantities have also been examined critically.
The quantum model of finite potential well is a most subject in the book of quantum mechanics. The solutions maybe derive from one dimension Schrödinger’s equation for describe a bound state energy of confined particle in the finite potential well. Sometime, the problem of confined particle in the finite potential well does not have an exact solution yet, but there are in fact exact solution.In this work a numerical technique is adopted to find the exact solution for the quantum model of finite potential well. To achieve this numerical solution, I use Fortran code computer program by the Newton-Raphson method for calculating the Eigen state energy for a particle (electron) confined in the finite potential well. Five values of potential width and depth have studied, the results showed that large potential width and large potential depth led me to more bound states, while smaller potential width and depth led me to fewer bound states.A comparison between infinite potential and finite potential has been presented. The results bound state energy and penetration. The results of comparison showed that the wave function of particle confined in the infinite potential cannot penetrate in the well, while the wave function of particle in the finite potential was penetrate in the well; the energy levels of an infinite well are much higher than that the corresponding energy levels for finite potential well. For confined particle in 2 or 3-dimension, we can reduce from result of 1-dimension.
The quantum Finite Square Well (FSW) model is a well-known topic in most quantum mechanics (QM) books.A couple of equations can be derived from one dimensional Schrodinger equation for a finite potential well for describing the bound Eigen states within the well.Sometimes the FSW problem does not have an exact solution, yet, there are in fact exact solutions.In this work a computational techniques is adopted to find the exact solution for FSW.To achieve this computational solution, a computer program has been written in Basic language for calculating the Eigen state energy for a particle confined in finite potential well using the iterative method (IM).Six values of potential width have been studied with potential depth.The results showed that wider potential width led us to more bound states, while narrower potential width led us to less bound states.Six values of potential depth have been studied with potential width.The results showed that larger potential depth led us to more bound states, while smaller potential width led us to less bound states.A Comparison Between Finite and Infinite Potential Well has been also presented.The result of comparison showed that energy levels of an infinite well are much higher than that the corresponding energy levels for finite potential well.In general, the matching between the results of the iterative method and the graphical method (GM) proving that the iteration method can be regarded as a useful tool for describing the solutions of the 1-dimensional FSW problem.