QM Solved Problems #1

1.      A particle moves along the x-axis in a potential field:

V(x) = 0, |x| < a

V(x) = ∞, |x| > a

(I.e. a box with closed, isolated walls at x = ± a).

a. Calculate the eigen-values of energy and define the corresponding normalized eigen-functions.

b. At t = 0, the particle is in interval |x| < c, (c < a). Inside the interval, all positions of the particle have the same chances. Assume that the corresponding wave function is real and positive. Please find the chance at energy measurement of finding the particle at:

i.   the ground state;

ii.  the first excited state;         

c. Assume that at the energy measurement, the ground state was found and then, we suddenly remove the walls of the box to x = ± b (b > a). What is then the probability of finding the particle at:

            i.   the ground state?

            ii.  the first excited state?

           

            Solution: