1. The following questions regard the quantum mechanical particlein a box.
a) Explain why the n=0 solution is not allowed for the particle ina box but is allowed for the free particle. Cast your explanation in terms of the underlying physicalmeaning of the solution in both contexts. Hint: substitute n = 0 into the respectivewavefunctions.
b) Show that the normalization constant of the particle in a boxwavefunction is 2L . What would be the normalization constant if the box length were 2L?
c) Graph the wavefunction for the n = 3 solution to the particle ina box and indicate the corresponding energy.
d) Calculate the probability that a particle in the n = 3 statewill be found between x = 0 and x = L/3. Show that this value can be expected from the symmetry of thewavefunction.
e) Show that the quantum mechanical probability distributionapproaches the classical probability distribution as the energy of the particle approaches infinity fora fixed box length, L. Hint: it is more intuitive to do this graphically though it is easy to doanalytically.