Schrdinger equation

Problem 2. Previously we studied the time-independent Schrodinger equation. Now, we can take a look at the time-dependent version given by HV(x, t ) = in- V(x, t). where H is the Hamiltonian operator. Consider the situation for the free particle in the 1-dimensional box of length L so that V(x) = 0 and V(0, () = 0 = V(L, t). (a) Take a separation of variables ansatz and find a set of solutions (one for every positive integer n) to the time-dependent equation. (b) Show that a super position of solutions is also a solution. (c) For a single solution ,(x, t), show that is independent of t. This shows that the states un are stationary since their total probability does not depend on time