Study guide for final.

Points of emphasis on the final will include understanding how to model motion using superposition of eigenstates, understanding how to use degenerate states to create off-center states, understanding the cost of confinement and localization in quantum physics, understanding the nature of kinetic energy, and understanding tunneling in bound state systems. There will be no problems on scattering on the final.

Compared to the last midterm, there will be much less graphing of wave-functions.

You can bring the pages of notes you had for each of the two midterms (2 pages total) and add to that anything that we have covered since the 2nd midterm. Be prepared to calculate expectation values.

Added Monday morning:   The final will encompass everything we have studied this quarter, however, there will be points of emphasis and de-emphasis that can help you focus your study. Mixed states and modeling time dependence using mixed states is a point of emphasis. In quantum it can be difficult to model motion. Making a particle go in a straight line was mathematically difficult because it involved a superposition of an infinite number of energy eigenstates. You won't be asked about that superposition. The only thing you could be asked about from our study of a free electron is how a moving electron has two kinds of kinetic energy, one associated with motion and another associated with confinement. I think this was covered in problem 4 of HW5.
     In two and 3 dimensions, it is possible to model circular motion. Intriguingly the most elementary modeling of circular motion is easier modeling linear motion in quantum physics. Using just two appropriately chosen energy eigenstates, one can model an electron going back-and-forth or moving in a circle or even an ellipse. There are HW problems for the 1DHO, 2DHO, infinite square well and hydrogen that relate to this. Really understanding deeply how mixed states work and how they can relate to motion is an important skill in introductory quantum physics. We covered a number of examples of electrons that move back and forth. One of those involves tunneling, that is motion through a region where the energy of the electron, which does not have a specific value due to the electron being in a superposition state, is nevertheless definitely less that that of the potential energy in a region through which it moves.  Understanding tunneling, how an electron can move from one well to another is important. If we look back at the difference between a  finite square well and an infinite square well, we see that the main difference is that the electron wave function extends outside the finite square well. The reason for this is that the electron can lower its KE by becoming less localized (less confined). It is this tendency of an electron to do whatever it can reasonably do to become less localized that leads ultimately to tunneling. For example, the non-zero value of the ground state wave-function between two wells in a double well system is intimately related to the difference in energy between the GS and 1st excited state, which is directly related to tunneling times.
     One of our main themes this quarter was the energy cost of localizing an electron. This is something that has no analogue in non-quantum physics. This energy cost comes directly from the \(-\hbar^2 \frac{\partial^2}{\partial x^2} \frac{1}{2m}\) term of the Schro Wave Equation. Solutions to the Wave Equaton are called wave-functions. They have to start from zero and end up and zero, so that they are normalizable, and the bending associated with getting them to do that yields a second derivative the integral of which is always positive. So there is an inherent, unavoidable cost to localizing an electron, which is the essence of what is called x,p uncertainty. Really it is the expectation values of x^2 and p^2 that are relevant to what we call x,p uncertainty. The cost of localization manifested itself in every bound state system we studied, and even for the free electron wave-packet. Anyway, a deep understanding of how localizing and/or confining an electron has a cost and how that depends on the electron's mass and \(\hbar\) is a good thing.
   There will be a question about angular momentum similar to problem 8 from HW 7. A key thing for that problem is to know that \(L_z = \frac{\hbar}{i} \frac{\partial}{\partial \phi}\) and to understand the relationship between cartesian and spherical coordinates.
   You will also be asked to calculate expectation values involving 1st excited states of hydrogen. Reviewing the post and video on sp2 states and Dirac notation again will be helpful for that because it helps teach the skill of organizing and evaluating expectation values and matrix elements (using Dirac notation).
    Basically, if you understand localization, tunneling, hydrogen atom states, expectation value calculations and mixed states, you should be good.