In this exercise, we will apply the material in Sect. 4.4.4 (page144) to calculate the factor of confinement of a particle in a finite well. For convenience we consider symmetric case, we will translate the x-axis so that the potential equals to 0 in the region: a/2 < x < a/2.

(a) Rewrite the Eq. (4.57) in this new coordinate system. Use the boundary condition to eliminate some trivial constants. By symmetry, we search for solutions in two families of functions: even and odd function. Show that the even solutions satisfy two equations:

while the odd solutions satisfy:

How can you resolve these equations graphically?

(b) The particle is in the ground state, which is even, of energy E. Find the probability for the particle to stay in the well. This quantity is defined as the confinement factor (or coefficient of confinement). Student can find a simulation of this problem at http://www.sgi.com/fun/ java/john/wave-sim.html.

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