WarmUp Exercise #4, 9/2  (Assignment Due 9/7)

We will talk about the general time dependence of states Tuesday.  These questions will help prepare you for the next steps we will be taking with time-dependent phenomena in quantum mechanics.

Assume that Psi(x,t) = Psi_REAL + Psi _IMAGINARY let's think about how to write this in MAGNITUDE and DIRECTION form: A(x, t) and θ(x, t) so that we have:

Psi(x,t) = A(x, t) e^{-i θ(x, t)}. 

1. The magnitude of the wave function, A, is what in terms of the real and imaginary parts of the wave function?  How does it relate to something you know about?
2. The direction (angle or phase angle), θ, of the wave function is what in terms of the real and imaginary parts of the wave function?
3. Given the form, A(x, t) e^{-i θ(x, t)}, for the wave function, write down the expression for the probability current density in this notation.  The probability current density is J = hbar/2mi [ psi* (d psi/dx) - (d psi*/dx) psi ].

Answer in the form below, a short answer for each is fine. 
This report is due by 8:00 am Tuesday morning.