PARTICLE IN A Box
In this chapter we want to learn how to find wave functions and energy levels for various systems. As often happens, the simplest problems may not correspond exactly to any situation found in nature, but they often serve as enlightening first models. Our first example fits that description, serving as a first approximation to the behavior of an electron that is free to move within a long. straight molecule or along a very thin wire. Our system consists of a particle confined between two rigid walls separated by a distance L (Fig. 42-1). We’ll make it a one-dimensional problem, with the particle moving always along the x-axis and the walls located at x = 0 and x = L. The particle never gains or loses energy; both its energy E and the magnitude of its momentum p art; constant. The potential energy corresponding the rigid walls is infinite, and the particle cannot escape. The situation’ often described succinctly as a ”particle in a box.” We begin with some assumptions. Because the particle is confined the region 0 :5 X :5 L, we expect the particle’s wave function to be z outside that region. Also, it seems physically reasonable that the wa function should be a continuous function of x. If it is, then it must be z at the region’s boundaries, x = 0 and x = L. These two conditions
boundary conditions for the problem. They should look familiar, since these are the same conditions that we used to find normal modes of a vibrating string in Section 20-4 (Fig. 42-2) and the electric field in a standing electromagnetic wave in Section 33-7; you may want to review those discussions. The energy and the magnitude of momentum are constant for our particle. Thus we consider a one-dimensional wave function ‘If that does not depend on time. Through the de Broglie relation A= hip, a definite wavelength Acorresponds to a definite magnitude of momentum p, so it’s reasonable to assume that ‘If is a sinusoidal function of x. If so, a possible form is ow we are only two short steps from determining the possible energy levels. From A we can find the momentum p using the de Broglie relation p = hlA; the energy E is all ic and thus equals il2m. For each value of n there are corresponding values of p, E; let’s call them P., A., and En’ Putting the pieces together, we get A particle trapped in a box is rather different from an electron bound in an atom, but it is reassuring that this energy is of the same order of magnitude as actual atomic energy levels. You should be able to show that replacing the electron with a proton or a neutron (m = 1.67 x 10-27 kg) in a box the width of a medium-sized nucleus (L = 1.1 x 10-‘4 m) gives E, = 1.7 MeV. This shows us that the energies for particles in the nucleus are about a million times larger than the energies of electrons in atoms, giving us a clue as to why each nuclear fission and fusion.