# Worked Problems in Physics

## LaTeX wordpress problems solved

Posted in Nielsen/Chuang by rpmuller on March 5, 2010

Since my earlier post about problems with LaTeX appears to be getting a lot of hits, I thought I would follow up with my solution, so that at least there wouldn’t be misinformation out there about LaTeX. I made a few minor modifications to my workflow, and now both the latex/wordpress plugin and the latex2wp program are working without any problems. Here’s what I had to do.

• No fancy classes of LaTeX: I’m a huge fan of the tufte-latex class, since it makes everything look so dang pretty. Haven’t tried revtex, my second favorite, but I assume that’s out as well.
• No macros. This was a huge sacrifice, since I normally define special macros for everything in quantum, especially bras and kets. However, I found that the extra typing of “\left|i\right>” wasn’t that awful when I got down to it.
• No eqnarray environments. I’ve learned to use normal equations plus the array environment, which doesn’t work quite as well.

What I typically do is write the post in LaTeX, convert it with latex2wp, open the html in emacs, and post do the blog in the HTML mode of the wordpress window. If I stick to the above rules, everything seems to be working well.

## Nielsen and Chuang Exercise 2.30

Posted in Nielsen/Chuang by rpmuller on March 5, 2010

Exercise 2.30 Show that ${A\otimes B}$ is Hermitian if ${A,B}$ are Hermitian.

$\displaystyle (A\otimes B)^\dagger = (A^\dagger\otimes B^\dagger) = A\otimes B.$

Thus ${A\otimes B}$ is Hermitian.

## Nielsen and Chuang Exercise 2.29

Posted in Nielsen/Chuang by rpmuller on March 5, 2010

Exercise 2.29 Show that ${A\otimes B}$ is unitary if ${A,B}$ are unitary.

$\displaystyle (A\otimes B)^\dagger(A\otimes B) = (A^\dagger\otimes B^\dagger)(A\otimes B) = (A^\dagger A)\otimes(B^\dagger B)=I\otimes I.$

Thus ${A\otimes B}$ is unitary.

## Nielsen and Chuang Exercise 2.28

Posted in Nielsen/Chuang by rpmuller on March 5, 2010

Exercise 2.28 Show that the transpose, complex conjugate, and adjoint operations distribute over the tensor product.

• Transpose

$\displaystyle (A\otimes B)^T = \left[ \begin{array}{rrr} a_{11}B & a_{12}B & \dots \\ a_{21}B & a_{22}B & \dots \\ \vdots & & \ddots \end{array} \right]^T$

You take the transpose of a block matrix of this form by first transposing the blocks, and then transposing within a block. Thus,

$\displaystyle = \left[ \begin{array}{rrr} a_{11}B^T & a_{21}B^T & \dots \\ a_{12}B^T & a_{22}B^T & \dots \\ \vdots & & \ddots \end{array} \right] = A^T \otimes B^T$

• Complex conjugate

$\displaystyle (A\otimes B)^* = \left[ \begin{array}{rrr} a_{11}B & a_{12}B & \dots \\ a_{21}B & a_{22}B & \dots \\ \vdots & & \ddots \end{array} \right]^* = \left[ \begin{array}{rrr} a_{11}^*B^* & a_{12}^*B^* & \dots \\ a_{21}^*B^* & a_{22}^*B^* & \dots \\ \vdots & & \ddots \end{array} \right] = A^*\otimes B^*$

• Adjoint Since the adjoint is composed of both transpose and the complex conjugate, and both of these operations distributes, the adjoint distributes as well.
• ## Nielsen and Chuang Exercise 2.27

Posted in Nielsen/Chuang by rpmuller on March 5, 2010

Exercise 2.27 Calculate the matrix representation of the tensor products of the Pauli operators ${X\otimes Z}$, ${I\otimes X}$, ${X\otimes I}$. Is the tensor product commutative?

$\displaystyle X \otimes Z = \left[ \begin{array}{rrrr} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 \\ 0 & 0 & -1 & 0 \end{array} \right]$

$\displaystyle I \otimes Z = \left[ \begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \end{array} \right]$

$\displaystyle Z \otimes I = \left[ \begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{array} \right]$

The tensor product does not commute.