3-6. [8 Points Total] Many books only present the time-independent form of the Schrodinger equation. This is misleading, because even for cases where you can use the time-independent formulation, there is STILL A TIME DEPENDENT term needed in the full expression for the wave function. In these problems, we will work from the time-dependent Schrodinger equation and see how, for many cases, we can solve for the time-dependence and the position-dependence of the wave function separately. 3. [2 Points] Starting with Schrodinger's equation: 2m ax -4( x, t ) + V (x, 1 ) Y ( x, 1 ) = in - 4 (x, 1) Show that, if you re-express the total wave function as a function of position - lower case w(x)- times a function of time - lower case o(t): Y(x, ! ) = 4(x)p(1) ...you can rearrange Schrodinger's equation so that all of the terms involving position are on one side and all of the terms involving time are on the other side (This is called separation of variables: we are separating x and t): 2my(x) day(x)+V(x) =in d p(t) di (). 4. [1 Point] Explain why you could not separate the variables in this way if the potential energy function was changing with time.5. [1 Point] Explain why the only possible solution for this equation is for both sides to be equal to a constant (We'll call it C). That is: 2 my(x)day(x)+V (x) = in d p(t) dt p(1) = C ...such that we end up with 2 expressions: 2 m w(x ) day ( x ) + V ( x )=C & in I d -p(1) = c p(t) dt (HINT: It's very similar to the logic that says that x = t?, is not a possible solution for m . d'x / dr = -kx) 6. [4 Points] To find o(t), we must now solve the time equation: in 1 d - 9(1) =C p(t) dt 1. Show that o(() = ed* is a solution 2. Express this solution also in terms of sines and cosines. 3. What are the angular frequency (@) and the frequency (f) for this solution? 4. Given that the relationship between energy and frequency is the same for matter particles and for photons, what is the relationship between C and the energy (E) of the electron (or matter particle) in this case? What is q(t) in terms of E