HW 8. With solutions.

Solutions below.
Homework 8 includes spin problems and problems using spin matrices. Basically, what are called two level systems.


Problem 3 could be really difficult and is completely optional. While you can do one or two iterations by hand, it is mostly intended to be done using a computer. People taking computational physics might find this interesting. If you do get results for problem 3, I would be interesting in hearing about them by email, especially the comparison of the two methods for modeling time-dependence.

For problem 1, please begin each section by writing the matrix.( Please ask questions if you are not sure what to do. In my experience, using determinants can be a waste of time and it doesn't teach you much about eigenvectors. It can be better to guess eigenvectors and find their eigenvalues by matrix /vector multiplication. Eigenvectors belong to particular matrices. The definition of an eigenvector is that when you multiply that matrix times the vector, the result is the same vector multiplied by a scalar. If that happens, then the scalar is the eigenvalue of that eigenvector. They "belong" to each other.
In addition to guessing, you can also use wolfram alpha (with specific numbers) to gain insight and eigenvectors. For a) and b), I would just try to guess. For c) I would use WA and then think about what it is telling me and how the states are getting "unbalanced".
1. 2x2 matrices:
a) What are the eigenvectors and eigenvalues of the 2x2 matrix  \(\sigma_x\).
b) What are the eigenvectors and eigenvalues of the 2x2 matrix
\(I - \Delta \sigma_x\), where \(I\) is the 2x2 identity matrix and \(\Delta\) is real.
c) What are the eigenvectors and eigenvalues of the 2x2 matrix
\(2 I + 2 \sigma_z - \sigma_x\), where \(I\) is the 2x2 identity matrix.
extra credit: In what sense would you called these eigenvectors "unbalanced" whereas the ones from a and b were "balanced". What caused that unbalancing?
d) What is the trace of each matrix? How does it compare with the sum of its eigenvalues?
e) Note that for each matrix, its eigenvectors are mutually orthogonal and that these matrices are Hermitian!  (your answer: "noted!")

2. (this problem is triple credit)  Two-level systems are popular in physics as a way to simplify and capture the essence of particular phenomena. Pauli spin matrices, like \(\sigma_x\), get used a lot, including in the study of qubits in GaAs. One can make quantum wells at the interface of GaAs and GaAlAs. (GaAs and GaAlAs are \(sp^3\)- bonded semiconductors that can be made with a high degree of purity (nearly perfect) using molecular beam epitaxy. A 2-dimensional electron metal (sometimes known as a 2DEG) forms at the interface. Small dots in which a single electron can be localized can be made by etching.)
For a single electron in a double dot quantum well (see link), one can write the approximate Hamiltonian in the form:
\( H=\frac{\epsilon}{2} \sigma_z - \Delta \sigma_x \). This is a two-state approximation and is written using the two state basis \(L = |L\rangle = (1,0) \) 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.
a) The most interesting case is \(\epsilon=0\). That is when quantum computing takes place. What are the eigenvalues and eigenvectors of  H in this case? (The case \(\epsilon=0\).) In this context an eigenvector correspond to an eigenstates and its eigenvalue is the energy of an electron in that state.
b) As discussed in the linked paper, suppose you manipulate the system so that the initial state at t=0 is L, but for t>0, (\(\epsilon=0\))
What is the state as a function of time for t > 0?
What would be the best time to try to capture the electron in the right hand well?
c) Suppose the quantum dots are 10 nm  apart and are each 10 nm in diameter. Let's say the left-hand dot is centered at x=0 and the right-hand dot is centered at x=20 nm. Make an argument as to why \(x= 10 \,nm - 10 \,nm\: cos(2\pi t/T)\) is a pretty good approximation for the expectation value of x for \(t > 0\)
d) What is T? What is T for \(\Delta=0.2\, meV\)
e) At t=0 the probability density(PD) is 100% in the left well and 0% in the right well. How long does it take for the PD in the right well to grow to 20%?
f) extra credit: Discuss and analyze about how good the above approximate Hamiltonian would be for our 1D two-well system. Think about where and how would it differ from what you would get using our numerically obtained ground state and 1st ES. (I mean, I think it is pretty good and the differences are just a few percent here and there, but one could elucidate and examine that a bit...)

 3. Extra credit, optional. This problem presents a matrix approach to time dependence.
Consider the above Hamiltonian with \(\epsilon=0\) and \(\Delta=0.2\, meV\). (We need a specific value in order top set appropriate time increments for this iterative approach to time dependence.)
The full Schrodinger equation can be written as :
\(H \Psi(t) = i \hbar \frac{\partial \Psi(t)} {\partial t}\).
I believe the solution can be written as:
\(\Psi(t) = e^{-iHt/\hbar} \Psi(0)\), where H is a 2x2 matrix.
a) Suppose the electron starts off entirely in the left well, that is \(\Psi(0) = (1,0)\).
b) What is \(\Psi(t)\) a short time later?  Short enough that you can expand the exponential.  That is, use \(e^{-iHt/\hbar} \approx 1 - iHt/\hbar\), where 1 is the 2x2 identity matrix, H is a 2x2 matrix...
c) About how long does it take to get to 20% PD in the right hand well? (How accurate is this approximation compared to your result from problem 4e? Does this depend on the time interval you choose and the number of iteration steps you need?)
d) Using this matrix method, can you find the period of oscillation? If so, what is it and how does it compare with that from problem 2?

4. There is no problem 4.

5. a) What are the eigenvectors and eigenvalues of the matrix \(\sigma_y\).
b) What are the eigenvectors and eigenvalues of the matrix \(\sigma_z\).
c) By matrix muliplication, show work, evaluate \(\sigma_y \sigma_z\). evaluate \(\sigma_z \sigma_y\). Are the different? What is the commutation relationship that they suggest?