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Journal of Physics Conference Series

Determination of energy levels, probabilities, and expectation values of particles in the three-dimensional box at quantum numbers ≤5

Bambang Supriadi, Zainur Rasyid Ridlo, Nur Aida, V Rizqiyah, Rahma Wati, Andi Mulawakkan Firdaus

Published Apr 1, 2019• 3 citations
Publisher

Abstract

Static potential boxes represent an area that is limited by an infinite potential wall, where there is no external potential in it. The solution of the Schrodinger equation gives a wave function that can explain the energy level and determine the probability of finding a particle at any point. The main focus of the research in this paperwas to solve the solution to the energy levels, probabilities, and expectation values of a particle in a three-dimensional box, where control limits are given for the potential box width, namely and L for each situation. In this research, the results showed that probability meetings depend on changes in box width and quantum number of particles, while particle energy levels depend on the length of the box and the quantum number of particles. Based on the relation between probabilities and expectation values, the most stable size of the box to find the particles in ground state to the fourth excitation state is at the width of and L .

DOI
10.1088/1742-6596/1211/1/012035

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Static potential boxes represent an area that is limited by an infinite potential wall, where there is no external potential in it. The solution of the Schrodinger equation gives a wave function that can explain the energy level and determine the probability of finding a particle at any point. The main focus of the research in this paperwas to solve the solution to the energy levels, probabilities, and expectation values of a particle in a three-dimensional box, where control limits are given for the potential box width, namely and L for each situation. In this research, the results showed that probability meetings depend on changes in box width and quantum number of particles, while particle energy levels depend on the length of the box and the quantum number of particles. Based on the relation between probabilities and expectation values, the most stable size of the box to find the particles in ground state to the fourth excitation state is at the width of and L .
Abstract Particle in a box is one of the applications of the Schrodinger equation. Schrodinger equation gives wave function which is used to determine the probability and expectation value of particle in a box. It is also can explain the energy levels of the particle. This study aimed to determine the probability, expectation values, and energy levels of a particle in a three-dimensional box with variations in main quantum numbers in each coordinate axis. The magnitude of the wave function and energy levels are influenced by the main quantum number and the width of the box. The results show that variations in the main quantum number influence the probability of particle in the three-dimensional box except for the width of the box <?CDATA L2\frac{L}{2}2L​?> L 2 and L the probability of particle showing the same value in all variations of the main quantum number. Variations in the main quantum number also influence the expectation value of particle in a three-dimensional box except for the width of the box L the expectation value shows the same value in all variations of the main quantum number. All variations of the main quantum number also influence the magnitude of the energy level of the particle.
This paper is illustrated about the behavior of quantum mechanics looked like the that of classical mechanics when the principle quantum number approaches large values. This phenomenon was explained based on the energy quantization and probability density. When the energy levels of Bohr hydrogen atom for large quantum states are determined, these energy values are continuous values. It has been stated that the one- dimensional particle in a box wave function and probability density by changing the quantum numbers. The probabilities of finding the particle in a box for two sections are calculated with the aid of computer programming. It is found that the calculated probability density results depend not only on the quantum number but also on particle’s location.
This chapter discusses the problem of one particle in a one-dimensional box free of the potential energy inside the box, that is, no external force is exerted on the particle inside the box. The corresponding Schrödinger equation is given along with the boundary conditions which yields an elliptic second-order differential equation and the solutions are given next: the wave function, quantum numbers and corresponding energy. The normalized wave function is also given which is similar to that of the one-particle harmonic oscillator. Afterwards, the Hamiltonian function of the particle in a bidimensional box is given along with its wave function that is the product of two wave function of the particle in a one-dimensional box and the energy is the sum of two energy components, x and y, of the particle in a one-dimensional box. Finally, there is a couple of exercises to plot the wave functions of a rectangular box to find the probability density of the wave function for a symmetrical box. (164 words).
The goals of this research were to determine the particle probability density in three-dimensional box not perturbation on interval 0 until L/4 in quantum number n≤5. Type of this research was pure literature review research. The research method of Simpson rule at Matlab program. The result of the research were the probability to find particle (electron) in three dimensions box had determined by width size box (the position of particle) in position (x, y, z) so founded probability density depend on quantum number (nx, ny, nz)
In this paper we have solved analytically the Schrodinger equation for a microscopic particle in a one-dimensional box with two infinite walls, which the potential function inside it, has a linear form. Based on the solutions of this special quantum mechanical system, we have shown that as the quantum number approaches infinity the expectation values of microscopic particle position and square of the linear momentum are equal to the classical time average of particle position and its square of the linear momentum, respectively.
The effect of a δδδ distribution potential on a quantum mechanical particle in a box
We study the effect of a δδδ distribution potential placed at x0≥0x_0\geq 0x0​≥0 and multiplied by a parameter ααα on a quantum mechanical particle in an infinite square well over the segment [− L2,L2]\left[-\,\frac{L}{2},\frac{L}{2}\right][−2L​,2L​]. We obtain the limit of the eigenfunctions of the time independent Schrödinger equation as α↗+∞α\nearrow+\inftyα↗+∞ and as α↘−∞α\searrow-\inftyα↘−∞. We see how each solution of the Schrödinger equation corresponding to α=0α=0α=0 changes as ααα runs through the real line. When x0x_0x0​ is a rational multiple of LLL, there exist solutions of the Schrödinger equation which vanish at x0x_0x0​ and are unaffected by the value of ααα. We show that each one of these has an energy that coincides with the energy of a certain limiting eigenfunction obtained by taking ∣α∣→∞|α|\to\infty∣α∣→∞. The expectation value of the position of a particle with wave function equal to the limiting eigenfunction is x0x_0x0​.