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).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).

CRC Press eBooksMar 21, 20220 citations

Particles in Boxes

James E. House

Elsevier eBooksApr 28, 20170 citations

Particle in a Box

WORLD SCIENTIFIC eBooksJun 1, 20010 citations

Complete solutions of particle in three dimensional box with variations in main quantum number

Bambang Supriadi, Lailatul Nuraini, A S R Maulani +3 more

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 $\frac{L}{2}$ ?> 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.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 $\frac{L}{2}$?> 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.

Journal of Physics Conference SeriesMay 1, 20200 citations

Particle in a Box

James M. Feagin

Springer eBooksJan 1, 19940 citations

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 +3 more

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 .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 .

Journal of Physics Conference SeriesApr 1, 20193 citations

\frac{L}{2}

?> 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.