# Treating propagators like unit vectors

I recently taught myself a trick that I want to hold onto. Most people know propagators can be treated like matrices, since these objects also satisfy some sort of non-communative multiplication rule. I need a way to treat a propagator like a vector object (don't ask why, but I have a very good reason). Let $\{A_i\}$ be a complete basis for the unitary matrices. What I mean is that for any unitary $U$ in the $N$ -dimensional Hilbert space,

Further, I am going to introduce the inner product $\langle \alpha, \beta \rangle = \mathrm{tr} (\alpha^\dagger \beta)$ and require that $\langle A_i, A_j \rangle = N \delta_{ij}$ . Then the components $x_i(t)$ form a vector. It's easy to show that for any unitary,

Therefore the vector $\vec{x}(t) = (x_1(t), x_2(t), \dots )$ represents a unitary if and only if it a unit vector.

How should matrix multiplication be represented in terms of vectors? Since $\{A_i\}$ is a complete basis for the unitary matrices, any product between unitary operators may be written in terms of the basis elements. To aid multiplications, introduce a rank-3 tensor defined by $A_i A_j = \sum_k g_{ijk} A_k$ . The individual elements of the tensor can be computed using $\langle A_k, A_i A_j \rangle = N g_{ijk}$ . Then for any two unitaries $U = \sum_i x_i(t) A_i$ and $V = \sum_j y_j(t) A_j$ their product is

Further if $\vec{z}$ is a vector associated with the unitary $W = UV$ , their vectors are related by $z_k(t) = \sum_{ij} x_i(t) y_j(t) g_{ijk}$ .

What about dynamics? A propagator is a solution to a Schrödinger equation. Similarly, we can produce a simple Schrödinger equation for the vector,

where $\tilde{H}_{ij} = \langle A_i, H(t) A_j \rangle / N$ is a pseudo-Hamiltonian.

# GNU Make + Inkscape

Producing figures for scientific papers can be a time consuming task. When I make a figure, I usually follow the following steps:

1. Draw figure in a graphing program (matplotlib, Igor Pro, Mathematica, etc.) and export to a vector-compatible format
2. Touch up figure in Inkscape
3. Convert from SVG to PDF
4. Compile a $\LaTeX$ document using the PDF figure

Frequently however, I have to return to Inkscape to make minor modifications to figure colours, fonts, line weights, and other attributes.  I now use Make + Inkscape to automate steps (3) and (4).  Make is smart enough to detect whether I modified an SVG file, and automatically converts the modified SVG images to PDFs before the document is compiled.  Here is how it works:

I organize the project like this ...

project/
Makefile
master.tex
figures/
Makefile
pdf/
figure1.pdf
figure2.pdf
...
svg/
figure1.svg
figure2.svg
...


The main Makefile has a build target for the $\LaTeX$ document that depends on a PHONY target for the figures. Here is how I do it:

LATEXFILE = master
FIGURES = figures
TEX = pdflatex -interaction=batchmode
BIBTEX = bibtex

#
# Make targets
#
all: $(LATEXFILE).pdf # Builds figures, latex, bibtex, and reruns latex until linked$(LATEXFILE).pdf: $(LATEXFILE).bbl while ($(TEX) $(LATEXFILE); \ grep -q "Rerun to get cross"$(LATEXFILE).log) do true; \
done; \
echo "********************************************"; \
echo "* REPORT: summary of compilation warnings  *"; \
echo "********************************************"; \
cat $(LATEXFILE).log | grep "LaTeX Warning:"; # Reconstructs PDF figures using SVG sources .PHONY: figures figures: cd$(FIGURES) && $(MAKE) # Builds main document$(LATEXFILE).aux: figures
$(TEX)$(LATEXFILE)
$(TEX)$(LATEXFILE)

$(LATEXFILE).bbl:$(LATEXFILE).aux
$(BIBTEX)$(LATEXFILE)


Before running the first $\LaTeX$ pass, Make switches to the figures directory and runs the following Makefile.

#
# MAKEFILE
#
INKSCAPE = inkscape
SOURCE = svg
TARGET = pdf

#
# Construct lists
#
SOURCES = $(wildcard$(SOURCE)/*.svg)
TARGETS = $(patsubst$(SOURCE)/%.svg, $(TARGET)/%.pdf,$(SOURCES))

all: $(TARGETS)$(TARGET)/%.pdf: $(SOURCE)/%.svg$(INKSCAPE) -z --file=$< --export-pdf=$@

.PHONY: clean
clean:
rm -f \$(TARGETS)


# An atom says...

Here is a little animation of a hydrogen-like atom being driven along the $^2\mathrm{S}_{1/2} \rightarrow {}^2\mathrm{P}_{1/2}$ transition.  The image you see is a density plot of the probability.  Used Mathematica.

# Operations for quantum operators

A unary operation is an operation with only one operand.  Specifically it is a function of the form $f: \mathcal{A} \mapsto \mathcal{A}$ where $\mathcal{A}$ is a set.  Here we consider unary maps where $\mathcal{A}$ corresponds to the set of quantum operators in a Hilbert space.  Let $A \in \mathcal{A}$ be a quantum operator.  We may represent $A$ in any particular basis by a matrix of the form $A = \sum_{ij} A_{ij} |i \rangle \langle j |$ .  Then

Operation $f(A)$
Negation $-A$ $\sum_{ij} (- A_{ij}) |i \rangle \langle j|$
Complex conjugate $A^*$ $\sum_{ij} A_{ij}^* |i \rangle \langle j |$
Transpose $A^T$ $\sum_{ij} A_{ij} |j \rangle \langle i |$
Conjugate transpose $A^\dagger$ $\sum_{ij} A_{ij}^* |j \rangle \langle i |$