Update week7.do.txt

Author: mhjensen

doc/src/week7/week7.do.txt

@@ -371,7 +371,7 @@ plt.show()
 !split
 ===== Two-qubit Hamiltonian =====
 
-We end this lecture with a discussion on how to rewrite the two-qubit Hamiltonian rom last week (and project 1)
+We end this review from last week with a discussion on how to rewrite the two-qubit Hamiltonian rom last week (and project 1)
 !bt
 \[
 \mathcal{H}=\begin{bmatrix} \epsilon_{1}+V_z & 0 & 0 & V_x \\
@@ -434,7 +434,7 @@ and
 !split
 ===== How do we perform measurements? =====
 
-The above tensor products need to rewritten in terms of specific
+The above tensor products have to be rewritten in terms of specific
 transformations so that we can perform the measurements in the basis of
 the Pauli-$\bm{Z}$ matrices. As we discussed earlier, we need to find
 a transformation of the form
@@ -448,12 +448,11 @@ the identity matrix, $\bm{U}$ is a unitary matrix and $\bm{M}$
 represents the gate/matrix which performs the measurements, often
 represented by a Pauli-$\bm{Z}$ gate/matrix.
 
-The implementation of these measurements will be discussed next week.
 
 
 !split
 ===== Explicit expressions =====
-In order to perform our measurements, will then need the following operators $\bm{U}$
+In order to perform our measurements we  need the following operators $\bm{U}$
 !bt
 \begin{align*}
 \bm{Z}\otimes\bm{I}\hspace{1cm} & \bm{U}=\bm{I}\otimes\bm{I}\\
@@ -474,6 +473,10 @@ where we have
 \]
 !et
 
+!split
+===== More complete list and derivations of expressions for strings of operators =====
+
+For a two qubit system we list here the possible transformations 
 
 
 !split
@@ -484,9 +487,9 @@ Physics _62_ (1965) 188), for the interaction among  $2$ and more
 fermions that can occupy two different energy levels.
 
 
+In project 1 we consider a two-fermion case and a four-fermion case.
 
-
-For four fermions, the case we consider first here, each levels has
+For four fermions, the case we consider in the examples  here, each levels has
 degeneration $d=4$, leading to different total spin values.  The two
 levels have quantum numbers $\sigma=\pm 1$, with the upper level
 having $2\sigma=+1$ and energy $\varepsilon_{1}= \varepsilon/2$. The
@@ -497,7 +500,8 @@ level has spin up.  In addition, the substates of each level are
 characterized by the quantum numbers $p=1,2,3,4$.
 
 
-
+!split
+===== Four fermion case =====
 
 We define the single-particle states (for the four fermion case which we will work on here)
 !bt
@@ -508,6 +512,10 @@ We define the single-particle states (for the four fermion case which we will wo
 \]
 !et
 The single-particle states span an orthonormal basis.
+
+!split
+===== Hamiltonian =====
+
 The Hamiltonian of the system is given by
 
 !bt
@@ -535,6 +543,8 @@ while $H_{2}$ is a spin-exchange term. The latter
 moves a pair of fermions from a state $(p\sigma ,p' -\sigma)$ to a state
 $(p-\sigma ,p'\sigma)$.
 
+!split
+===== Quasispin operators =====
 
 We are going to rewrite the above Hamiltonian in terms of so-called  quasispin operators
 !bt