Particle in the box  The following images shows a particle of mass m that is trapped in the second region of the potential shown that is from x=0 t x=L. In this same region the potential is zero. The particle in the box has solutions of the form:  (a) Show your work, of finding the expected value of the kinetic energy between points 0 and L. (b) Suppose the particle can absorb energy and change from state  to state  + 3. Draw a possible spectrum of the first three transitions, using Energy instead of frequency on the horizontal axis. If you suppose the particle can absorb energy and change from state  to state  + 3. (c) What would be the parameters to model a benzene ring using the particle in the box? And what would be the boundary condition? (d) If we double the length of the box in the first image, so that the limits now go from  =  to  = +, what will be the wave function x() in this case ? (e) Derive an approximate expression for the uncertainty minimum at the moment for the extended box of question (d). Approximate what this value would be if the box measured 1 meter.

Vi = 0 = V = 0 V3 = 0 Regin 1 Regin 2 Region 3 I = 0 T =L 2 4x(x) = Vi sin(x). L