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.
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 .
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 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.
The most basic example of a quantum mechanical problem, the one-dimensional particle in a box, is revisited. Three aspects are treated. 1.It is shown that a number of textbooks give an incorrect result for the expectation value of x2, and, therefore, do not obtain the correct spread Δx. Consequently, the verification of the Heisenberg relation is also in error. 2. It is found that textbooks do not mention the origin dependence of expectation values, which is a nuisance, as the problem is evidently treated in the literature with two choices of origin. 3. Superposition of an infinite number of even (cosine) solutions of the one-dimensional particle in the box allows for an analytical demonstration of complete localization of the particle (Δx = 0).
Schrodinger equations are widely used in solving a variety of problems, one of them is particles in a box in especially of potential wells bound state. Particle in a box is a discrete circumstances, so we must examined whether or not the degeneration of these particles. The Schrodinger equation analysed to get a form of modeling of energy levels and wave functions in potential wells of the circumstance used numerical method. We also can show energy level of each state from the model. The numerical approaches is finnite difference method using Matlab R2010 software. Numerical calculation shows k value. k 1 and k 3 gets from solution of cos equation, whereas k 2 and k 4 gets from solution of sin equation. The k value is used to know the energy level in potential wells. energy level model show wave fucntion. In potential well bound state, Energy level and wave functions be smaller when potential energy is high and it is high when potential when potential energy is small. When wide of potential well is smaller, energy level and wave function be smaller too. But there isn’t show energy level and wave function when wide of potential well is high. The shape of energy level and and wave function of particle in box.
This chapter exclusively deals with the application of Schrödinger equation on various systems such as the particle in a box; the particle in a rectangular three-dimensional box; energy levels for a cubic potential box; the tunnel effect; the final equation for tunnel effect, that is, transparency factor which gives a clue to understand that the probability of transmission decreases rapidly as the width of the barrier increases; quantum mechanical explanation of emission of α -particle; the particle on a ring considering spherical polar co-ordinates; and the particle on sphere. This chapter also comprises the Legendre polynomials, associated Legendre equation, associated Legendre function and spherical harmonics, Hermite polynomials and its applications, and orthogonal properties of Hermite polynomials. It also provides the standard topics like simple linear harmonic oscillator, and its classical and quantum mechanical treatment which also includes the asymptotic and series solutions and energy levels of a linear harmonic oscillator. The Schrödinger equation also plays with rigid rotors comprising its rigorous treatment and also the F and T equations which are very important. The chapter ends with the references, standard several computational problems, questions on concepts, and also standard numerical problems for the benefits of the readers.