HW 8 guide. More notes on problem 2. Using overlap integrals in problem 2.

(see also the post: on Matrices, eigenvectors ...

Problem 2 involves a single electron in a double dot quantum well.  There is a paper entitled:
"Quantum Coherence in a One-Electron Semiconductor Charge Qubit" that provides some pictures and background for this problem. Even though this problem is focused on the position of an electron (not its spin), it uses a notation involving spin matrices, which we covered recently.  This sort of notation is common in physics as a way to model what are called two-level systems. That means systems which have only a total of two energy-eigenstates, or, more often, systems where we ignore (project out) all states except the lowest two. (The two of lowest energy.) Treating the double dot well as a two-level system tends to be a very good approximation because there are two states close to each other in energy and all other states have much higher energies.

     An appealing thing about this paper is that, although it is high-quality research at an active frontier, the theoretical part involves very basic quantum physics that we can understand with the background you have in wells and spin matrices.
    They present the theory part in terms of a Hamiltonian.  A Hamiltonian is very analogous to a Schrodinger equation. It has (energy) eigenstates. It is essentially a short-hand, simpler way to do things which avoids the messiness of boundary conditions (in this case) and the mathematical details of the wave function. What they use is \( H=\frac{\epsilon}{2} \sigma_z - \Delta \sigma_x \). These are the 2x2 spin matrices we talked about two weeks ago, but they are using them for a problem with no spin. The parameters epsilon and Delta are just real numbers with units of energy. The parameter epsilon is included to represent tilting of the well toward one side; the parameter Delta is proportional to the energy splitting of the two eigenstates. The time to tunnel from one well to the other will thus be inversely proportional to Delta, as you will show.
    Whenever you present things in terms of a matrix, there has to have been a basis chosen. In this case that basis is: \(L = |L\rangle = (1,0) \) and \(R=|R\rangle = (0,1) \).  L is a state where the electron is in the left-hand well; R is the state where the electron is in the right hand well. This is a convenient, simple choice, but these are not energy eigenstates.
    I will add the part about overlap integrals later. It pertains to the part of the problem where one expresses the state L as a superposition of the two energy-eigenstates. The overlap integrals are where the 1/sqrt(2) factors come from. The overlap integrals (or inner products) are 1/sqrt(2).

PS. Please let me know if you need more time for problem 2 or for anything on this weeks homework. I am a little unsure as to how difficult it is for you and how confusing it might be as to what you are are expected to do. Is it easy? Is it confusing? Would you like more time so that we can cover this in class on Tuesday?