Find the energies of six lowest energy levels of a particle
Answers
The simplest form of the particle in a box model considers a one-dimensional system. Here, the particle may only move backwards and forwards along a straight line with impenetrable barriers at either end.[1] The walls of a one-dimensional box may be visualised as regions of space with an infinitely large potential energy. Conversely, the interior of the box has a constant, zero potential energy.[2] This means that no forces act upon the particle inside the box and it can move freely in that region. However, infinitely large forces repel the particle if it touches the walls of the box, preventing it from escaping. The potential energy in this model is given as
{\displaystyle V(x)={\begin{cases}0,&x_{c}-{\tfrac {L}{2}}<x<x_{c}+{\tfrac {L}{2}},\\\infty ,&{\text{otherwise,}}\end{cases}},}where L is the length of the box, xc is the location of the center of the box and x is the position of the particle within the box. Simple cases include the centered box (xc = 0 ) and the shifted box (xc = L/2 )
Position wave function[edit]In quantum mechanics, the wavefunction gives the most fundamental description of the behavior of a particle; the measurable properties of the particle (such as its position, momentum and energy) may all be derived from the wavefunction.[3] The wavefunction {\displaystyle \psi (x,t)} can be found by solving the Schrödinger equation for the system
{\displaystyle i\hbar {\frac {\partial }{\partial t}}\psi (x,t)=-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial x^{2}}}\psi (x,t)+V(x)\psi (x,t),}where {\displaystyle \hbar } is the reduced Planck constant, {\displaystyle m} is the mass of the particle, {\displaystyle i} is the imaginary unit and {\displaystyle t} is time.
Inside the box, no forces act upon the particle, which means that the part of the wavefunction inside the box oscillates through space and time with the same form as a free particle:[1][4]
{\displaystyle \psi (x,t)=[A\sin(kx)+B\cos(kx)]\mathrm {e} ^{-i\omega t},} (1)where {\displaystyle A} and {\displaystyle B} are arbitrary complex numbers. The frequency of the oscillations through space and time are given by the wavenumber {\displaystyle k} and the angular frequency {\displaystyle \omega }respectively. These are both related to the total energy of the particle by the expression
{\displaystyle E=\hbar \omega ={\frac {\hbar ^{2}k^{2}}{2m}},}which is known as the dispersion relation for a free particle.[1] Here one must notice that now, since the particle is not entirely free but under the influence of a potential (the