electron_in_a_box1

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previous index next Electron in a Box Michael Fowler, University of Virginia 9/1/08 Plane Wave Solutions The best way to gain understanding of Schrödinger’s equation is to solve it for various potentials. The simplest is a one-dimensional ―particle in a box‖ problem. The appropriate potential is V ( x ) = 0 for x between 0, L and V ( x ) = infinity otherwise—that is to say, there are infinitely high walls at x = 0 and x = L , and the particle is trapped between them. This turns out to be quite a good approximation for electrons in a long molecule, and the three-dimensional version is a reasonable picture for electrons in metals. Between x = 0 and x = L we have V = 0, so the wave equation is just 22 2 ( , ) ( , ) 2 x t x t i t m x    . A possible plane wave solution is ( , ) ( ) x t Ae i px Et . On inserting this into the zero-potential Schrödinger equation above we find E = p 2 /2 m , as we expect. It is very important to notice that the complex conjugate, proportional to () i px Et e  , is not a solution to the Schrödinger equation! If we blindly put it into the equation we get E = – p 2 /2 m , an unphysical result. However, a wave function proportional to i px Et e  gives E = p 2 /2 m , so this plane wave is a solution to the equation. Therefore, the two allowed plane-wave solutions to the zero-potential Schrödinger equation are proportional to e i px Et ( ) and i px Et e respectively. Note that these two solutions have the same time dependence iEt e .
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2 To decide on the appropriate solution for our problem of an electron in a box, of course we have to bring in the walls—what they mean is that ψ = 0 for x < 0 and for x > L because remember | ψ | 2 tells us the probability of finding the particle anywhere, and, since it’s in the box, it’s trapped between the walls, so there’s zero probability of finding it outside. The condition ψ = 0 at x = 0 and x = L reminds us of the vibrating string with two fixed ends—the solution of the string wave equation is standing waves of sine form. In fact, taking the difference of the two permitted plane-wave forms above gives a solution of this type: ( , )
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