Midterm solutions and commentary.

      A midterm is an opportunity to evaluate, but, more than that, it is also an opportunity to provide emphasis. Realistically, many things are taught in physics classes that are neither learned nor remembered.
      One could ask oneself, what does this midterm communicate regarding what are points of emphasis from the 1st half of this class? What does it communicate about what the teacher may think is particularly important to learn and remember? (more on this below the break)

https://drive.google.com/file/d/0B_GIlXrjJVn4OXgyY2dhOTMzWnM/edit?usp=sharing

What I see here is a strong emphasis on the relationship between confinement and kinetic energy. Not that the kinetic energy uncertainty grows, as suggested in some emails I have received, but rather that the kinetic energy itself increases in response to confinement. (Consider the example of the infinite square well. For an electron confined in that system there is momentum uncertainty, but the kinetic energy is precisely defined and not uncertain at all.)

Additionally, there is also emphasis on familiarity with wave-functions in general, and on those of the hydrogen atom in particular --what they look like, how they can be combined (hybridization) and the specialness and importance of the unusual state, \(\psi_{200}\).

Also, the last problem, 7, provides a window into what we will do next. This expectation value is actually time dependent. If you calculated it without time dependence, what you calculated is actually the t=0 value of something that will change dramatically with time. This time dependence comes from the \( e^{-i E_n t/\hbar}\) part of the wave-function, which we have been ignoring so far (but no longer). This kind of adding of states is different from the adding of states that we called hybridization for the H-atom 1st-excited state, and it enables true electron motion in quantum physics. The motion is wavelike, of course -- the motion of something represented as a superposition state or wave-packet involving states of different energies. Can you see how to include the time dependence and how the result wil change in time? Can you figure out how to make a wave combination that will enable an electron to move back and forth in a H-atom? ... or go in circles?