\newpage
\section{Particle characterization and particle tracking using interference properties}
		\label{sec:chapter2}

\subsection{Introduction}

Properties of coherent light to produce interference is widely use in metrology for a long time with, for example, the famous Fabry-Pérot  \cite{fabry_theorie_1899, perot_application_1899} and Michelson interferometers \cite{michelson_relative_1887}. The latter was initially used to measure earth's rotation and is still used today, in particular, for the recent measurement of gravitational waves
\cite{ligo_scientific_collaboration_and_virgo_collaboration_gw151226_2016}.
Since the beginning of the century, interest on tracking and characterizing colloidal particles risen thanks to the democratization of micro fluidics and lab-on-a-chip technologies. In the following I will provide some insights on the three most used :

\begin{itemize}
	\item Reflection Interference Contrast Microscopy (\gls{RICM})
	\item Lorenz-Mie fit
	\item Rayleigh-Sommerfeld back-propagation
\end{itemize}

The first one, \gls{RICM}, uses the principle of optical difference path as a Michelson interferometer. The other two, uses the interference between the light scattered by the colloid and the incident light. Generally, both of the sources  are colinear, thus, speak of in-line holography.

\subsection{Reflection Interference Contrast Microscopy}

\begin{figure}[h]
	\centering
	\includegraphics[scale=1]{02_body/chapter2/images/RICM.png}
	\caption{Figure from \cite{davies_elastohydrodynamic_2018} representing \gls{RICM} with two wavelengths. (a) Left: interference patterns created with a wavelength $\lambda_1 = 532$ nm (scale bar $ 5~\mathrm{\mu m}$).
		Right: radial intensity profile (black dots) extracted from the image, azimuthally averaged (magenta line) and fitted with Eq.\ref{Eq.RICM} to measure the height of the particle (here $h$). (b) Same as (a) with a wavelength $\lambda_2 = 635$ nm. (c) Time series of the height of a particle $h$ (green: $ \lambda_1$, magenta: $\lambda_2$) and the particle velocity measured along the flow in blue. }
	\label{fig.RICM}
\end{figure}

Reflection Interference Contrast Microscopy was first introduced in cell biology by Curtis to study embryonic chick heart fibroblast \cite{curtis_mechanism_1964} in 1964. \gls{RICM} gained in popularity 40 years after both in biology and physics \cite{filler_reflection_2000, siver_use_2000, weber_2_2003, limozin_quantitative_2009, nadal_probing_2002, raedler_measurement_1992}. It was also used recently in soft matter physics to study elastohydrodynamic lift at a soft wall \cite{davies_elastohydrodynamic_2018}.

When we illuminate a colloid with a plane wave from the bottom, a part of the light is reflected at the surface of the glass substrate and at the colloid's surface. The difference of optical path between two reflection create interference patterns. Let's take an interest at the mathematical description of this phenomenon. In the far field, we can describe two different one-dimensional electric field vectors of the same pulsation $\omega$ \cite{f_bohren_absorption_1998} as:

\begin{equation}
	\vec{E}_1(\vec{r}, t) = \vec{E}_{0_1} \cos(\vec{k}_1 \cdot \vec{r} - \omega t + \epsilon_1) ~,
\end{equation}
and
\begin{equation}
	\vec{E}_2(\vec{r}, t) = \vec{E}_{0_2} \cos (\vec{k}_2 \cdot \vec{r} - \omega t + \epsilon_2) ~.
\end{equation}

\nomenclature{$\vec{E}$}{Electrical field}
\nomenclature{$k$}{Wave number}
\nomenclature{$\omega$}{Pulsation}

Where the $k$ is the wave number $k=2\pi n_{\mathrm{m}}/\lambda$, $\lambda$ denoting the illumination wavelength, $n_\mathrm{m}$ the optical index of the medium, $\epsilon_{1,2}$ the initial phase of each wave and $\vec{r}$ the position from the source. Here, the origin ($\vec{r} = \vec{0}$) could be taken at the position of the first reflection (on the glass slide) thus at the particle, $\vec{r}$ would be given by the particle's height such that $|r| = z$ the particle-subtract distance. Experimentally, we measure the intensity of the interference patterns, those can be computed from the time averaged squared sum of the electric field $\vec{E} = \vec{E}_1 + \vec{E}_2$. The measured intensity is thus given by:

\begin{equation}
	\begin{aligned}
		I & = \langle \vec{E}^2 \rangle = \langle \vec{E}_1^2 + \vec{E}_2^2 + 2\vec{E}_1 \cdot \vec{E}_2 \rangle
		= \langle \vec{E}_1^2 \rangle + \langle \vec{E}_2^2 \rangle  + 2 \langle \vec{E}_1 \cdot \vec{E}_2 \rangle \\
	\end{aligned}
\end{equation}

where $ \langle \vec{E}_1^2 \rangle $ and  $\langle \vec{E}_2^2 \rangle$ are respectively given by $I_1$ and $I_2$, the incident light intensities. Using the trigonometric formula $2 \cos (a)\cos (b) = \cos (a+b) + \cos (a-b) $ we have:

\begin{equation}
	\langle
	\vec{E}_1 \cdot \vec{E}_2 \rangle =
	\langle
	\frac{1}{2} \vec{E}_{0_1}  \vec{E}_{0_2}
	\left[
		\cos
		\left(
			\vec{k}_1 \cdot \vec{r} - \vec{k}_1 \cdot \vec{r} + \phi
		\right)
		+
		\cos
		\left(
			2\omega t + \phi'
		\right)
	\right]
	\rangle~.
\end{equation}

As we average over the time, the second $\cos$ will vanish since in general $\langle \cos(at + b) \rangle_ t = 0$ thus:

\begin{equation}
	\langle \vec{E}_1 \cdot \vec{E}_2 \rangle = \frac{1}{2} \langle  \vec{E}_{0_1}  \vec{E}_{0_2} \rangle
	\cos
	\left(
	\vec{k}_1 \cdot \vec{r} - \vec{k}_2 \cdot \vec{r} + \phi
	\right)
\end{equation}

with $\phi$ the phase difference between the two fields, which is generally equal to $\pi$ due to the reflection properties on a higher index. Indeed, a colloid has generally a greater optical index than the dilution medium.  Finally, the total intensity can be read as:

\begin{equation}
	I = I_1 + I_2 + 2 \sqrt{I_1 I_2}
	\cos
	\left(
	\vec{k}_1 \cdot \vec{r} - \vec{k}_2 \cdot \vec{r} + \phi
	\right)
\end{equation}

By taking $k_1 = - k_2$ due to the reflection properties, we have:

\begin{equation}
	I = I_1 + I_2 + 2 \sqrt{I_1 I_2}
	\cos
	\left(
	\frac{4 \pi n_{\mathrm{m}}}{\lambda} z + \phi
	\right)
\end{equation}

If we now suppose that we have a spherical particle at a height $z$ we can develop the radial interference intensity $I(x)$ as \cite{ raedler_measurement_1992}:

\begin{equation}
	I(x) = A_0 + A_1 \mathrm{e}^{-b_1 x^2} + A_2^{-b_2 x^2} \cos \left[ \frac{4\pi n_m}{\lambda}\left( g(x) + z \right) + \phi \right]
	\label{Eq.RICM}
\end{equation}

Where $A_i$ and $b_i$ are fit parameters and $g(x)$ denotes the contour of the sphere.
Finally, this method is great because the equation is computationally light and permits to have a quick tracking of particles. However, as we can see on Eq.\ref{Eq.RICM}, due to the periodicity of the cosinus, the interference pattern will be the same for all heights $z$ separated by a distance $\lambda / 2n_\mathrm{m} \approx 200 $ nm (for $\lambda = 532$ nm and $n_{\mathrm{m}} = 1.33$). It is possible to extend this limitation by using 2 different wavelength to $\approx 1.2 ~ \mathrm{\mu m}$ as used in \cite{davies_elastohydrodynamic_2018}. Despite the precision of this method which can reach the $10$ nm spatial resolution; the measurement ambiguity is not compatible with the study of  micro-particle Brownian motion, hence, \gls{RICM} is not usable for our context. As a matter of fact, we experimentally reach height span of a few microns.

\subsection{Lorenz-Mie Fit}
\label{chap:LM_fit}

When a colloid is illuminated with a plane wave, a part of the light is scattered. In consequence, the superimposition of the incident field $\vec{E}_0$ and scattered field $\vec{E}_s$ interferes. The interference patterns thus obtained are called holograms. If the particle size is at the same order of magnitude or greater than the illumination wavelength, it is not possible to use Rayleigh approximations \cite{strutt_lviii_1871}. Indeed, we would need to use what we call  the Lorenz-Mie theory which describes the scattering of dielectric spheres; this theory was found by Lorenz and independently by Mie in 1880 and 1908 \cite{lorenz_lysbevaegelsen_1890, mie_beitrage_1908}.

It is in the early 2000 that the Lorenz-Mie theory was first used in order to track and characterize particles \cite{ovryn_imaging_2000, lee_characterizing_2007}. Since then, a lot of studies has been realized with this method \cite{katz_applications_2010}. In the following I will describe the Lorenz-Mie Fit method. In in this part, the height of the particle $z$ is the distance between the particle's center and the focal plane of the objective lens.

Let the incident field be a plane wave uniformly polarized along the axis $ \hat{e}$, with an amplitude $E_0$ and propagating along the $\hat{z}$ direction :
\begin{equation}
	\vec{E}_0(\vec{r},z) = E_0(\vec{r}) \mathrm{e}^{ikz}\hat{e}
\end{equation}

Let's consider a particle of radius $a$ at a position $\vec{r}_p $, the scattered field can be written using the Lorenz-Mie theory \cite{f_bohren_absorption_1998} as:

\begin{equation}
	\vec{E}_s(\vec{r}, z) =  \vec{f}_s(k(\vec{r} - \vec{r}_p))E_0(\vec{r}) \exp \left(-ikz\right)
\end{equation}

With $\vec{f}_s$ the Lorenz-Mie scattering function \cite{f_bohren_absorption_1998}. The intensity $I$ that we measure at $\vec{r}$ is given by the superimposition of incident and scattered waves. Since the measurements are done at the focal plane, $I$ is given by:

\begin{equation}
	\begin{aligned}
	I(\vec{r}) & = |\vec{E}_s(\vec{r}, 0) + \vec{E}_0(\vec{r}, 0)|^2 \\
	& = E_0^2(\vec{r}) + 2 E_0^2\operatorname{Re} \left(\vec{f}_s(k(\vec{r}- \vec{r}_p)) \hat{e}\right) + | \vec{f}_s(k(\vec{r}- \vec{r}_p)) |^2
	\end{aligned}
\end{equation}

The most of the experimental defects on the images are due to spacial illumination variation caused by dust particle and such. It can be corrected by normalizing the image by the background. In another word, we normalize  $I(\vec{r})$ by the intensity of the incident field $I_0 = E_0(\vec{r})^2$ which is the experimental background. It can be measured by different methods, one is to have an empty field of view and the other one, which is more convenient is to take the median of a stack of images. Naturally, for having the latter to work, the movie should be long enough to have the particle diffuse enough, if not a ghost of the particle will appear on the background. This process also permits getting rid of the immobile particle that could generate any additional noise. An example of hologram before and after the normalization is shown in Fig.\ref{fig.Lorenz_mie_demo} a-c). We write the normalized intensity $I/I_0$:

\begin{equation}
	\frac{I(\vec{r})}{I_0(\vec{r})} = 1 + 2 \operatorname{Re}
	\left(
		\vec{f}_s(k(\vec{r}- \vec{r}_p)) \hat{e}
	\right)
	+
	|
		\vec{f}_s(k(\vec{r}- \vec{r}_p))
	|^2
	\label{Eq.normalized_Mie}
\end{equation}

Now that we have the analytical form of the holograms' intensity, it is possible to fit an experimental one to Eq.\ref{Eq.normalized_Mie} as shown in Fig.\ref{fig.Lorenz_mie_demo} d-e). For the sake of completeness, I will detail the Lorenz-Mie scattering function, $\vec{f}_s(k\vec{r})$ which is given by the series:

\begin{equation}
	\vec{f}_s(k \vec{r}) = \sum _{n=1} ^{n_c}
	\frac
	{
		i^n (2n +1)
	}
	{
		n(n+1)
	}
	\left(
		i a_n \vec{N}^{(3)}_{eln}(k\vec{r})
		-
		b_n \vec{M}^{(3)}_{oln}(k\vec{r})
	\right)
	\label{Eq.Lorenz-Mie-function}
\end{equation}

where $\vec{N}^{(3)}_{eln}(k\vec{r})$ and $\vec{M}^{(3)}_{oln}(k\vec{r})$ are the vector spherical harmonics. $a_n$ and $b_n$ are some coefficients that depend on the particle and illumination properties. For a spherical and isotropic particle of radius $a$ and refractive index $n_\mathrm{p}$, which is illuminated by a linearly polarized plane wave, the $a_n$ and $b_n$ coefficients are expressed in terms of spherical Bessel $j_n$ and Hankel $h_n$ functions as \cite{f_bohren_absorption_1998}:

\begin{equation}
	a_n =
	\frac
	{
		\zeta^2 j_n (\zeta k a)k a j_n' (k a) - j_n(ka)[\zeta kaj_n(\zeta ka)]'
	}
	{
		\zeta^2 j_n (\zeta k a)k a h_n^{(1)'} (k a) - h_n^{(1)}(ka)\zeta kaj_n'(\zeta ka)
	} ~,
	\label{Eq:an}
\end{equation}

and

\begin{equation}
	b_n =
	\frac
	{
		j_n(\zeta k a) kaj_n'(ka) - j_n (ka) \zeta kaj_n'(mka)
	}
	{
		j_n(\zeta k a) kah_n^{(1)'}(ka) - h_n^{(1)} (ka) \zeta kaj_n '(mka)
	} ~,
	\label{Eq:bn}
\end{equation}

	where $\zeta = n_\mathrm{p} / n_m $ and the prime notation denotes differentiation with respect to the argument. As we can see, the holograms will depends on Eq.\ref{Eq.Lorenz-Mie-function} and will vary with a lot of parameters ($\lambda$, $n_m$, $n_\mathrm{p}$, $a$ and $\vec{r}_\mathrm{p}$) which can all be fitted. In general, the illumination wavelength $\lambda$ and medium index $n_\mathrm{m}$ are known and do not need to me fitted. From only one hologram it is could thus be possible to measure precisely the position of the particle $\vec{r}_\mathrm{p}$ and in the same time characterize the radius and optical index of the colloid. As a side note, it is even possible to characterize a particle without a priori knowledge of its characteristics using Bayesian approach \cite{gregory_bayesian_2005, dimiduk_bayesian_2016}.

Computing Eq.\ref{Eq.Lorenz-Mie-function} numerically brings another interesting question, as it is analytically written as a sum over $n$; one could ask after which number of terms $n_c$ the series will converge. It has actually been found that the series converge after a number of terms \cite{lentz_generating_1976}
\begin{equation}
	n_c = k a + 4.05 (k a)^{1/3} + 2 ~.
\end{equation}

Consequently, larger particles' holograms will need more terms to converge and, hence, are longer to fit. As an example, the largest particles used during my thesis have a radius $a = 2.5 ~ \mathrm{\mu m}$ leading to a number of terms $ n_c = 55$ in water and $\lambda = 532$ nm, for the smallest ones, where $a = 0.5 ~ \mathrm{\mu m}$ we find $n_c = 18$ which makes a huge difference in practice.

If a reader wants to evaluate an hologram given by the Lorenz-Mie theory for a peculiar particle and position, it can be done in a few lines with the \mintinline{python}{holopy} module using the following Python snippet which was used to make Fig.\ref{fig.holo_fix_n} and \ref{fig.holo_fix_z}:

\begin{minted}
	[
	frame=lines,
	framesep=2mm,
	baselinestretch=1.2,
	fontsize=\footnotesize,
	linenos
	]
	{python}
import holopy as hp
from holopy.scattering import calc_holo, Sphere

sphere = Sphere(n=1.59, r=1.5, center=(4/0.1, 4/0.1, 10))
# n is the optical index of the particle, r it's radius in microns
# center is its center position in microns.

medium_index = 1.33
illum_wavelen = 0.532
illum_polarization = (1, 0)
detector = hp.detector_grid(shape=100, spacing=0.1)
# shape is the size in pixel of the camera and the spacing is the pixel's size in microns.

holo = calc_holo(
	detector, sphere, medium_index, illum_wavelen, illum_polarization, theory="auto"
)
#the hologram can be directly be plotted using:
hp.show(holo)
\end{minted}

\begin{figure}[H]
	\centering
	\includegraphics[scale=1]{02_body/chapter2/images/lorenz_mie_fit_demo/plot_lorenz_mie.pdf}
	\caption{a) Raw hologram of a $2.5 ~ \mathrm{\mu m}$ polystyrene particle measured experimentally with the setup detailed in the chapter \ref{chap:exp-setup}. b) Background obtained by taking the median value of the time series of images of the diffusing particle. c) Normalized hologram given by dividing a) by b). d) Result of the fit of c) using Eq.{\ref{Eq.normalized_Mie}} the particle is found to be at a height $z = 14.77 ~ \mathrm{\mu m}$. e) Comparison of the normalized radial intensity, obtained experimentally form c) and theoretically from d).}
	\label{fig.Lorenz_mie_demo}
\end{figure}

\subsubsection{Hologram dependance on the particule characteristics}

As we can see with the Eq.\ref{Eq.Lorenz-Mie-function}, the in-line holograms vary with the that position, radius and optical index of the particle. For in-line holograms, since the incident and scattered field, the $x$ and $y$ position of the particle will simply be given by the center of the hologram. Thus, it is possible to track only movement of the particle only in 2 dimensions by using algorithm such as the hough transform to find the center. As a side note, in that case, it would be optimal to place the particle just above the focal plane to have an airy disk like hologram, as shown in Fig.\ref{fig.holo_fix_n} for $a = 2.5 ~ \mathrm{\mu m}$ and $z = 5 ~\mathrm{\mu m}$.

\begin{figure}[H]
	\centering
	\includegraphics{02_body/chapter2/images/holo_size_exemple/holos_only_z.pdf}
	\caption{Radial intensity profile stack as a function of the distance between the particle center and the focal place of the objective lens, generated for a particle of radius $a = 1.5 ~\mathrm{\mu m}$ and optical index $n = 1.59$.}
	\label{fig:holo_onlyz}
\end{figure}

In order to gain some insights on how the holograms vary with the different parameters, one can to compute holograms for particles of different size and height. We will start by looking at the a particle of a radius $a = 1.5 ~ \mathrm{\mu m} $ and $n = 1.59 $ as shown in the Fig.\ref{fig:holo_onlyz}. In this case, one can observe that as the distance between the particle and the focal plane $z$ increases, the hologram's rings gets larger. Unlike a Michelson interferometer or \gls{RICM} we do not observe the rings scrolling. Additionally, this thickening of the rings can also be observed on the Fig.\ref{fig:holo_z_fit}, where hologram's intensity profile are plot as a function of the height $z$ both theoretically and experimentally in the for a polystyrene colloidal particle of radius  $a = 1.5 ~ \mathrm{\mu m} $, and, for different couples of parameters on the Fig.\ref{fig.holo_fix_n}. Also, we can note that if $z$ is not large enough compared to the radius of the particle, the center of an hologram can be so bright that if the camera does not have a large enough dynamic range, the rings could not be seen.  Thus, for having an optimal condition for the fits, one should take care to defocus the enough the objective lens to have $z >> a$.

\begin{figure}[H]
	\centering
	\includegraphics{02_body/chapter2/images/holo_size_exemple/smallparticles.pdf}
	\caption{Radial intensity profile for of the particle radius $a << \lambda$ , generated for a particles of optical index $n = 1.59$ with a distance between the particle center and the focal place of the objective lens $z = 15 ~\mathrm{\mu m}$ and $\lambda = 532 ~ \mathrm{nm}$.}
	\label{fig:small_part_holo}
\end{figure}

\begin{figure}
	\centering
	\includegraphics{02_body/chapter2/images/holo_size_exemple/holos_only_r.pdf}
	\caption{Radial intensity profile stack as a function of the particle radius, generated for a particle of optical index $n = 1.59$ with a distance between the particle center and the focal place of the objective lens $z = 15 ~\mathrm{\mu m}$.}
	\label{fig:holo_onlyr}
\end{figure}

We can now take a look at the variation with respect to the radius of the particle as shown on the Fig.\ref{fig:holo_onlyr} for a particle of optical index $n = 1.59$ and at a distance $z = 15 ~\mathrm{\mu m }$. One can observe that for small particles compared to the wavelength $a << \lambda$ we do not observe the rings this is due to the fact that for the small particles, the scattering can be approximated using the Reyleigh theory which tells us that the scattering is isotropic. Thus, the variation of intensity around $I_0$ will be smaller for small particle. Also, in this small particle regime, the particle size will not affect the general shape of the hologram but just the intensity of the hologram as it can be seen on the Fig.\ref{fig:small_part_holo}, for particle of radius $a = 0.02 ~\mathrm{\mu m}$ to $a = 0.07 ~\mathrm{\mu m}$ for a wavelength $\lambda = 0.532 ~\mathrm{\mu m}$. Additionally, since the noise to signal ratio will be lower than for bigger particles, it will be less precise to characterize small colloids compared to the wavelength.

 As the particle gets bigger, the scattering become anisotropic and it mostly towards the incident plane wave direction. This effects leads to an increase of the amplitude of the rings $I/I_0$, as one can see on the Fig.\ref{fig:holo_onlyr}. Thus, the noise to signal ratio is high enough to easily discern the hologram on top of the noise as one can see on the experimental picture fig.\ref{fig.Lorenz_mie_demo}-a). One who wants to use this method should thus take care to use large enough particle for the holograms' intensity to be greater than the camera noise level.

\begin{figure}[H]
	\centering
	\includegraphics{02_body/chapter2/images/holo_size_exemple/holos_only_n.pdf}
	\caption{Radial intensity profile stack as a function of the particle optical index, generated for a particle of radius $a = 1.5 ~ \mathrm{\mu m}$ with a distance between the particle center and the focal place of the objective lens $z = 15 ~\mathrm{\mu m}$.}
	\label{fig:holo_onlyn}
\end{figure}

Finally, one can also check how the holograms are varying with the optical index a particle. In this case it is not the particle's optical index $n_\mathrm{p}$ which will matter the most but the ratio $\zeta = n_\mathrm{p} / n_\mathrm{m} $ which can be found in the $a_n$ and $b_n$ formulas, Eq.\ref{Eq:an} and \ref{Eq:bn}. Indeed, for the scatter to happen, the optical index $n_\mathrm{p}$ of the colloid needs to be different from the optical index of the surrounding medium $n_\mathrm{m}$. Additionally, the numerical solution of the Lorenz-Mie scattering is not stable for $n_\mathrm{p} \simeq n_\mathrm{m}$. In the Fig.\ref{fig:holo_onlyn}, we can observe holograms of a particle of radius $a = 1.5 ~ \mathrm{\mu m}$ at fix distance between the particle and the focal plane of the objective lens $z=15 ~ \mathrm{\mu m}$ with a varying colloid's optical index, in water $n_\mathrm{m} = 1.33$. On the Fig.\ref{fig:holo_onlyn} one can thus observe than for  $n_\mathrm{p} \simeq n_\mathrm{m}$ we do not oberve any holograms.  Additionally, one can observe that the noise to signal ratio gradually increases as $n_\mathrm{p}$ became different from $n_\mathrm{m}$. One who wants to use this method should thus take care to the particle material or the solvent to have $n_\mathrm{m}$ different enough to  $n_\mathrm{p}$ for the holograms' intensity to be greater than the camera noise level.

\subsubsection{Lorenz-Mie conclusion}

The combination of the height, optical index and radius of colloid thus gives unique holograms. This uniqueness of the holograms permits extracting precisely the position, optical index and radius an a colloid. In order to see how holograms are for different couples for parameter on the Figs.\ref{fig:holo_fix_z} and \ref{fig:holo_fix_n}, one can see possible holograms for different size and height. Additionally, one can use the the Jupyter Notebook on my github repository in order to plot any hologram \href{https://github.com/eXpensia/Ma-these/blob/main/02_body/chapter2/images/holo_size_exemple/holosize_variation.ipynb}{\faGithub}.  Finally, Lorenz-Mie is the most versatile in-line holographic method, indeed, it permits tracking and characterize unique particles even without a priori knowledge. Besides, it is possible to write the Lorenz-Mie function $\vec{f}_s$ for particular cases such as anisotropic \cite{fung_holographic_2013}, non-spherical particles \cite{wang_using_2014} or particle clusters \cite{fung_holographic_2013, perry_real-space_2013} to name a few; such possibilities open the door to a lot of experimental studies. Additionally, it can reach really high precision as the tenth of nanometer on the position and radius as well as $10^{-3}$ on the optical index \cite{lee_characterizing_2007}. Unfortunately, the Lorenz-Mie fitting suffer from a major drawback which is the time needed to fit one image. For example, a 200 by 200 pixels image, of a $2.5 ~ \mathrm{\mu m}$ particle's hologram, can take up to two minutes to be fitted using a pure and straightforward python algorithm. A lot of work as been done to have faster tracking, such as random-subset fitting \cite{dimiduk_random-subset_2014}, GPU (graphical processing unit) acceleration, machine-learning \cite{yevick_machine-learning_2014, hannel_machine-learning_2018} and deep neural networks \cite{altman_catch_2020}.

\begin{figure}[H]
	\centering
	\includegraphics{02_body/chapter2/images/test_tableau2.pdf}
	\caption{On the left, experimentally measured  holograms' radial intensity profile stack, generated from a polystyrene bead of nominal radius $a = 1.5 \pm 0.035 ~ \mathrm{\mu m} $ using the experimental setup explained in chapter \ref{chap:exp-setup}. The calibration of this particle radius and optical index is shown in Fig.\ref{fig:KDErn}. On the right, the corresponding theoretical stack using the result of each individual hologram's fit.}
	\label{fig:holo_z_fit}
\end{figure}

\begin{figure}
	\centering
	\includegraphics{02_body/chapter2/images/holo_size_exemple/holos_fix_z.pdf}
	\caption{On the left, experimentally measured  holograms' radial intensity profile stack, generated from a polystyrene bead of nominal radius $a = 1.5 \pm 0.035 ~ \mathrm{\mu m} $ using the experimental setup explained in chapter \ref{chap:exp-setup}. The calibration of this particle radius and optical index is shown in Fig.\ref{fig:KDErn}. On the right, the corresponding theoretical stack using the result of each individual hologram's fit.}
	\label{fig:holo_fix_z}
\end{figure}

\begin{figure}
	\centering
	\includegraphics{02_body/chapter2/images/holo_size_exemple/holos_fix_n.pdf}
	\caption{On the left, experimentally measured  holograms' radial intensity profile stack, generated from a polystyrene bead of nominal radius $a = 1.5 \pm 0.035 ~ \mathrm{\mu m} $ using the experimental setup explained in chapter \ref{chap:exp-setup}. The calibration of this particle radius and optical index is shown in Fig.\ref{fig:KDErn}. On the right, the corresponding theoretical stack using the result of each individual hologram's fit.}
	\label{fig:holo_fix_n}
\end{figure}

\clearpage
\newpage

\subsection{Rayleigh-Sommerfeld back-propagation}

Rayleigh-Sommerfeld back-propagation \cite{wilson_3d_2012} works on the same principle as the Lorenz-Mie fitting but assumes that we have small scatterers., such as :

\begin{equation}
	|\zeta - 1| << 1 \text{ and } ka|\zeta - 1| << 1 ~.
\end{equation}

In this case, at the focal plane, the intensity of the scattered field is smaller than the incident field, hence, the term $|\vec{E}_s|^2$ can be ignored. Thus, the normalized intensity, Eq.\ref{Eq.normalized_Mie} can be rewritten as:

\begin{equation}
\frac{I(\vec{r})}{I_0(\vec{r})}= 1 + 2\operatorname{Re}\left( \frac{E_s(\vec{r},0)}{E_0(\vec{r})} \right) ~.
\end{equation}

If one can retrieve completely the scattered field from an image, it is possible to reconstruct it above the focal plane by convolution using the Rayleigh-Sommerfeld propagator \cite{goodman_introduction_2005}:

\begin{equation}
	h_{-z}(\vec{r}) = \frac{1}{2 \pi} \frac{\partial}{\partial z} \frac{\mathrm{e}^{ikR}}{R} ~,
	\label{Eq:propagator}
\end{equation}

where $ R^2 = r^2 + z^2 $ and the sign convention on the propagator indicates if the particle is below or above the focal plane. Using this propagator we have:

\begin{equation}
	E_s(\vec{r}, z) = E_z(\vec{r}, 0) \otimes h_{-z}(\vec{r})
\end{equation}

By using the convolution theorem \cite{cheong_strategies_2010, goodman_introduction_2005, sherman_application_1967,schnars_digital_1994} and supposing a uniform illumination, one can write the reconstructed scattered field at a height $z$ as:

\begin{equation}
	E_s(\vec{r}, z) \approx \frac{\mathrm{e}^{ikz}}{4\pi ^2}
	\int ^\infty _{- \infty}
	B(\vec{q}) H(\vec{q}, -z) \mathrm{e}^{i \vec{q} \cdot \vec{r}} d^2 q
	\label{Eq.RS} ~,
\end{equation}

where $B(\vec{q})$ is the Fourier transform of $I/I_0$ and $H(\vec{q}, -z)$ is given by

\begin{equation}
	H(\vec{q}, -z) = \mathrm{e}^{iz \sqrt{k^2 - q^2}} ~.
\end{equation}

Finally, using Eq.\ref{Eq.RS} one can reconstruct the scattered field and intensity since $I(\vec{r}) = |E_s(\vec{r})|^2$ as shown in Fig.\ref{fig.sommerfeld}. Moreover, by finding the position where the field is the brightest as shown in red the Fig.\ref{fig.sommerfeld}, we measure the position of the particle.
Those equations are way less computational intensive than the Lorenz-Mie function Eq.\ref{Eq.Lorenz-Mie-function}. Thus tracking can be way faster, moreover, Fourier transforms can be largely accelerated using GPU. Additionally, as the propagator Eq.\ref{Eq:propagator} take only into account the intensity of the image, this method does not require any information on the particle and number of particles. As a matter of fact,   to write Eq.\ref{Eq.RS} one just need to assume that we have spherical colloids. Thus, this method is great to reconstruct the 3D position of a lot of particles or clusters formations. However, the major drawback is that it is the less precise of the presented measurements and that we can't use it to characterize the particles generating the holograms.

\subsubsection{Numerical Rayleigh-Sommerfeld back-propagation}

The \mintinline{python}{holopy} Python module also provide a set of method that permits to user the Rayleigh-Sommerfed back-propagation. Given the \mintinline{python}{hologram} variable containing all the needed metadata about the hologram such as the pixel size, medium index $n_\mathrm{n}$ and wavelength $\lambda$ and the actual image, one can then use the \mintinline{python}{propagate} method to back-propagate an hologram over a set of height \mintinline{python}{zstack} using the following Python snippet.

\begin{minted}
	[
	frame=lines,
	framesep=2mm,
	baselinestretch=1.2,
	fontsize=\footnotesize,
	linenos
	]
	{python}
	import holopy as hp
	import numpy as np

	zstack = np.linspace(0, 20, 11)
	rec_vol = hp.propagate(holo, zstack)
\end{minted}

Please note that using the \mintinline{python}{propagate} this each propagation will be done by performing a convolution of the reference hologram over the distance to be propagated. However, better reconstruction can be obtained iteratively propagating holograms over short distance. The latter method is called Cascaded Free Space Propagation, and is particularly useful when the reconstructions have fine features or when propagating over large distances \cite{kreis_frequency_2002}. It can be done by specifying the argument \mintinline{python}{cfsp} to the  \mintinline{python}{propagate} method. For example, to propagate three steps over each distance, we can use \mintinline{python}{hp.propagate(holo, zstack, cfsp=3)}.

\begin{figure}[!ht]
	\centering
	\includegraphics[scale=2]{02_body/chapter2/images/sommerfel_demo.jpg}
	\caption{Figure from \cite{cheong_strategies_2010} a) Volumetric reconstruction using Eq.\ref{Eq.RS} of the scattered intensity of single colloidal sphere, colored by intensity. b) Volumetric reconstructions of $22$ individual $1.58 ~ \mathrm{\mu m}$ diameter silica spheres organized in bcc lattice using holographic optical tweezers in distilled water. Colored regions indicate the isosurface of the brightest 1 percent of reconstructed voxels.}
	\label{fig.sommerfeld}
\end{figure}

\subsubsection{Conclusion}

Finally, the method we choose is the Lorenz-Mie fitting method, since this it permits the characterization of single particles. Indeed, since we are interested to fine effects near the surface, we need to know perfectly the radius of the particle we have recorded. This feature also make our all process calibration free, as we don't need to assume any physical properties. In the following, the experimental setup is going to be detailed.

\subsection{Experimental setup}
\label{chap:exp-setup}
In order to observe the holograms we use an homemade inverted microscope as shown on the Fig.\ref{fig:picture} and shematized in Fig.\ref{fig:schema}. A sample consists of a parallelepipedic chamber ($1.5 ~ \text{cm} ~ \times ~ 1.5 ~ \text{cm} ~ \times ~ 150 ~ \mathrm{\mu m} $), made from two glass covers, a parafilm spacer, and sealed with vacuum grease, containing a dilute suspension of spherical polystyrene beads. We used 3 different sizes, of nominal radii $0.56 ~ \mathrm{\mu m}, ~ 1.5 ~ \mathrm{\mu m} \text{ and } 2.5 ~ \mathrm{\mu m} $, at room temperature $T$, in distilled water (type 1, MilliQ device) of viscosity $\eta = 1 ~ \mathrm{mPa.s}$. The sample is illuminated by a collimated laser beam with a $521 ~ \mathrm{\mu m}$ wavelength. As depicted in the chapter \ref{chap:LM_fit}, the light scattered by one colloidal particle at a given time $t$ interferes with the incident beam. An oil-immersion objective lens (x60 magnification, $1.30$ numerical aperture) collects the resulting instantaneous interference pattern, and relays it to a camera (Basler acA1920-155um) with a $51.6$ nm/pixel resolution (see Fig.\ref{fig.Lorenz_mie_demo}a)). The exposure time of the camera is set to $\tau_{\mathrm{expo}} = 3$ ms to avoid motion-induced blurring of the image, as a general rule, the particle should not diffuse more than the pixel size during that time such that here $2D\tau_{\mathrm{expo}} < 51.6$ nm.

\begin{figure}[!ht]
	\centering
	\includegraphics{02_body/chapter2/images/figures_setup/photo_setup.pdf}
	\caption{Photo of the custom build microscope used along my thesis. It is mainly composed of Thorlabs cage system. The camera used is a Basler acA1920-155um, we use a x60 magnification and $1.30$ numerical aperture oil-immersion objective lens. The light source is a colimated  $521 ~ \mathrm{\mu m}$ wavelength laser.}
	\label{fig:picture}
\end{figure}

\begin{figure}[!ht]
	\centering
	\includegraphics[scale=0.9]{02_body/chapter2/images/figures_setup/schema_setup.pdf}
	\caption{Schematic of the experimental setup. A laser plane wave of intensity $I_0$ illuminates the chamber containing a dilute suspension of micro-spheres in water. The light scattered by a particle interferes with the incident beam onto the focal plane of an objective lens, that magnifies the interference patten and relays it to a camera.}
	\label{fig:schema}
\end{figure}

\subsection{Hologram fitting strategy}

\subsubsection{How to fasten the process ?}

As presented in the section \ref{chap:LM_fit} about the Lorenz-Mie fitting, the main drawback is the time to fit an image, from $30$ seconds for the images of $100 \times  100$ pixels to a few minutes for the $500\times 500$ pixels. We can directly see a bottleneck, indeed, if we want to track one trajectory made of $100~000$ images we would need to wait a minimum of $\approx 70$ days; for a series of images that need only a few minutes to be shot experimentaly. When I started my PhD, two groups, the Grier's lab and the Manoharan's lab, had already introduced python packages, respectively, Pylorenzmie and Holopy in order to inverse holograms. They had introduced ways to only fit a set of randomly chosen pixels, and demonstrated that taking only $1\%$ of the image pixels, could lead to similar precision and improve considerably the fit's execution time \cite{dimiduk_random-subset_2014}. Unfortunately, even if this is faster, it leads to a few images per second and still is too long for the amount of data we wanted to have. Ironically, this part of my project is certainly the one where I spent the most my time, and I actually learned a lot of things on code optimization and computer cluster usage. It's around the half of my thesis, that Pylorenzmie got a new commit on their github repository which was telling that they succeeded on using GPU acceleration using CUDA. This was not an easy task since they needed to reconstruct the Bessel functions in an understandable way for the GPU, fortunately it is possible to do so by using continued fractions \cite{lentz_generating_1976}. This humongous update permits fitting whole images at a whooping speed improvement of 20 fps. At this speed, we fit the tridimension position of the particle, the radius and optical index. To have a more reliable and fast tracking, what we do is that we fit with all free parameters the first $10~000$ images of a movie. We then determine the physical properties of the colloidal particle and then fit the whole movie with only the position as a free parameter.

\subsubsection{Radius and optical index characterization}

Once the data of the radius and optical index retrieved, the quantity we can look at is the the distribution of measurements. Using $10 ~ 000$ measurements we can plot the histograms of the measured $a$ and $n_\mathrm{p}$.

This simple histogram could suffice to measure the physical properties of the colloidal particle. However, we can go a bit further and look at the 2D histogram of the $a$ and $n_\mathrm{p}$ as presented in the fig.\ref{fig:KDErn} here smoothed using a Gaussian kernel density estimator. As we can see it is not isotropic, and it seems that the measurement of $n_\mathrm{p}$ and $a$ are correlated.

\begin{figure}[!ht]
	\centering
	\includegraphics{02_body/chapter2/images/KDErn.pdf}
	\caption{2D Probability density function of the measurements of the optical index $n_\mathrm{p}$ and radius $a$. Black lines indicate iso-probability. Taking the $10\% $ top probability, we measure $n_\mathrm{p} = 1.585 \pm 0.002$ and $a=1.514 \pm 0.003 ~ \mathrm{\mu m}$. }
	\label{fig:KDErn}
\end{figure}


\chapter{Risk}

\begin{multicols}{2}[\subsubsection*{Contents of this chapter}]
   \printcontents{}{1}{\setcounter{tocdepth}{2}}
\end{multicols}

\section{Checklist}

Ten sequential steps to model, assess and improve a portfolio.

\begin{itemize}
\item Data Processing: Steps 1-5
\item Risk Management: Steps 6-8
\item Portfolio Management: Steps 9-10
\end{itemize}

The model comes first, and the data comes after. There are two core components in modern quantitative finance. The first is the checklist, and the second is linear factor models. Linear factor models are the core for anything you need to know in econometrics, machine learning, estimation risk, etc..

The temptation might be to take the data and train some deep neural network on it, but you will not get anything out of that. You have to extract a repetitive behavior from the past that will repeat in the future. For equities, this turns out to be true for the log values. The changes in the log values are more or less i.i.d..

\subsection{Step 1: Risk Drivers Identification}

The identification of risk drivers is more of an art than a science.

Risk drivers are random processes that (1) behave homogenously as time evolves, similar to a random walk, and (2) completely determine the P\&L of the financial instruments at hand. Log-values for equity (i.e. stock prices), yield curve in fixed income, implied volatility surfaces for derivatives, number of defaults for credit, activity time for high frequency and cumulative P\&L for strategies.

Translate from a product specific to a product agnostic time series.

The risk driver will be multivariate stochastic process that should behave homogeneously as time as time evolves. It should also be solely responsible for the P\&L of the financial instrument. I.e., knowing the risk driver at all times would mean knowing the value of the financial instrument at all times.

\subparagraph{Example: Equities} The data available for two stocks is simply their stocks value over several business days. The values themselves do not evolve homogenously, but the log of the stock values do. Hence the log-value is a risk driver. This example is probably too simple to really be helpful.

\subsubsection{Equities}
log Values

\subsubsection{Fixed Income}
Zero coupon bond is a (imaginary) bond that does not give coupons. The value of a bond is the discounted values of the coupons until maturity plus the zero coupon bond.

Log rolling price of a zero coupon bond, i.e. at each time look at the price of a product that expires (for example) one year ahead.

You can normalize by dividing by time to maturity. I.e. 2 year forwards less volatile than 10 year forwards.

Yield to maturity: rolling value Vzcb(t+tau) ==> log Vzcb ==> rescale by diving by tau ==> opposite sign

Standardization process, previously and art, now standard.

Annualized compounded return of the zero-coupon bond from generic time t to $t_{end} = t+\tau$.

Yt = 1/tau ln(vzcb/Vzcb)

Swap / yield curve -- times to maturitiy and rate. Infinite dimensional parameters. Lower dimensional representation possible by fitting the curve.

Fixed-income parsimonious representations: Nelson-Siegel parametrization. Fit a parametric shape through the yield curve. Level, slope (steepness) curvature, decay. Four parameters. Therefore, 4 risk drivers. Parametrization extremely close to empirical yield.

Meucci: feeling that Nelson-Siegel perfectly good, though not necessarily arbitrage-free. Provides alternative (Vasicek).

Covers fixed income modeling through zero coupon bond.

\subsubsection{Derivatives}

Focus on European style derivatives, call options. Payoff is hockeystick. Different profiles (as long as european) vcan be expressed through call options.

Moneyness:

\begin{equation}
m = \frac{1}{\tau}ln (S_t / k^{strk})
\end{equation}

If price is above strike price then options are in the money, if price is below strike price then it is out of the money.

Rolling value is decent risk driver, avoiding implied volatility surface.

Implied volatility via Black-Scholes-Merton function, gives the price of a call option. Five inputs, four of which are observed. Remaining one is sigma, volatility, the implied volatility.

Long horizons you assume changes proportional of the level. Short horizons you assume changes identical to the level (i.e. geometric vs arithmetic brownian motion).

Heston arbitrage-free implied volatility surface is derivative counterpart to Vasicek in fixed income.

There is so much uncertainty in forecasting the future that it's better to just stick to the simple stuff, because the uncertainty will swamp out any benifit from being finniky.

\subsubsection{Credit}
Credit risk has to do with not being able to trust the counterparty.

Important concepts:

Time of default

Default indicator

Exposure at default

Loss given default

Credit ratings give categorical distribution, rather than binary default/not default. Loss given default often in terms of Beta distribution.

Risk drivers, when they are too many then you perform dimensionality reduction.

Key metric is cumulative number of transitions between credit ratings.

\subsection{Step 2: Quest for Invariance}
The invariants are shocks that, together with information available at the previous time step, fully determine the current risk drivers. They are i.i.d. across time ("white noise"). The unpredictable element. Once identified, they can be extracted from the past time series. The invariants introduce the stochastic element, which allows prediction of the future via estimation.

What is the white noise behind the risk drivers? The quest for invariance is the quest for i.i.d. components in the time series data.

\subparagraph{Example: Equities} Assuming the daily compounded returns are i.i.d., the risk driver can be written as:

\begin{equation}
X_{n,t} = X_{n,t-1} + \ln\frac{V_{n,t}}{V_{n,t-1}}
\end{equation}

So the invariant is $\epsilon_{n,t} = \ln\frac{V_{n,t}}{V_{n,t-1}}$.

Risk drivers display empirical features that deviate from the random walk. Efficiency, mean reversion, long memory, and volatility clustering.

\subsubsection{Efficiency}

Random walk as model for efficiency. Kolmogorov Smirnoff test can test for i.i.d.ness. Also, ellipsoid invariance test.

\subsubsection{Mean-reversion}

AR(1) processes, for example. Exponentially decaying autocorrelation. Markov Chains

\subsubsection{Volatility Clustering}

Modeled with GARCH. Stochastic mean and stochastic volatility.

GARCH models as random walk, except innovations have variance that undergoes an autocorrelated random walk.

GARCH has been used for stocks, yield, etc., model the increment and it applies across all asset classes.

Estimate via maximum likelihood.

Extensions of GARCH: generalize dispersion, or change market to which it's applied.

GARCH, EGARCH, ACD,

\subsection{Step 3: Estimation}

Estimating the joint distribution of the invariants from their past realizations. This can be done because of the i.i.d. property of the invariants. Often this involves assigning different weights to different observations. For example, emphasizing more recent observations or emphasizing observations is more similar market environments.

Estimating the i.i.d. components in the risk drivers.

\subsection{Step 4: Projection}

Based on the distribution of the invariants, project the distribution of the process of the risk drivers for a set of future times. For example, write down the distribution of the risk driver at the horizon, conditioned on information available now, based on square-root rule (assuming normally distributed innovations).

\subsection{Step 5: Pricing at the horizon}

The previous step had to do with projecting the risk driver. This step projects the actual value. If in step 4, the probability distribution of the risk driver at some point in the future was calculated, in this step there is a transformation of the random variable to obtain the probability distribution of the value at some point in the future.

\subsection{Step 6: Aggregation}
Steps 1-5 treated markets. In this step, the portfolio and the allocation policy are introduced. The goal is the ex-ante market performance of a portfolio stemming from an allocation policy.

This includes portfolio value, ex-ante performance and stress test. Stress test is a projection conditioned on extreme events, so for example: what would the returns be if the market went down 10\%.

\subsection{Step 7: Ex-ante Evaluation}
A risk or satisfaction measure is assigned to the ex-ante performance of the allocation policy, as calculated by step 6. Satisfaction measures are mainly of three types:

\begin{itemize}
\item expected utility / certainty-equivalent, including mean-variance trade-off
\item spectral / distortion measures, which include VaR, cVaR and actuarial measures
\item Sharpe and other non-dimensional ratios
\end{itemize}

One very simple example for a satisfaction measure is negative standard deviation. Analogous to the loss function in machine learning, there is a measure assigned to the quality of the distribution.

\subsection{Step 8: Ex-ante Attribution}

Attribution is attribution of the risk to different risk factors.

Express the ex-ante performance $\mathbf{Y}$ as a linear combination of relevant risk factors $\mathbf{Z} = (Z_1,...,Z_{\overline{k}}$ with exposures $\mathbf{\beta}$ plus a shift term $\alpha$ and a zero-center residual $U$.

\begin{equation}
\mathbf{Y} = \mathbf{\alpha} + \mathbf{\beta Z} + \mathbf{U}
\end{equation}

Attribution models can be constructed bottom-up, by starting from exposures of each financial instrument to pre-defined factors, or top-down, extracting surprise factors from the attribution model.

An example for a risk factor may be the return over the same future interval of some stock market index (i.e. SP500). The ex-ante performance metric may simply be the normally distributed ex-ante return $\mathbf{Y}_{h(\cdot)} = \mathbf{R}_{\mathbf{w},t_{now}\rightarrow t_{hor}}$. The attribution model reads:

\begin{equation}
\mathbf{R} = \mathbf{\alpha} + \mathbf{\beta Z} + \mathbf{U}
\end{equation}

\subsection{Step 9: Construction}

Optimal portfolio, optimal policy.

\subsection{Step 10: Execution}

\section{Distributions}

\subsection{Univariate distributions}
CDF, pdf, and characteristic function. Characteristic function is the expected value of $e^{i\omega X}}$, which is the fourier transform of the pdf. Similarly, there is the moment generating function, but the moment generating function may not be defined (for example for the student-t distribution). The Quantile function is the inverse of the CDF, basically, giving the value of $X$ that is bigger than $c$ of the data. There's also the sub-quantile function, which is the integral of the quantile function, which is the cvar, the conditional value at risk. There's also the Lorenz curve. Are all representations of the univariate variable.

\subsection{Multivariate Distributions}
CDF, pdf, characteristic function and moment generating function genrealize as expected to multivariate random variables. Quantile function doesn't really exist.

\subsection{Elliptical Distributions}

A market that is jointly elliptical has portfolios that are also jointly elliptical. Any elliptical distribution can be represented in terms of $X = \mu + R \sigma Y$.

The important property is affine equivariance. Elliptical distributions are very significant for analytical approaches. Elliptical distributions can be factored,via transformation, to a uniform distribution on the unit sphere and a radial distrubtion.

\subsection{Scenario Probability Distributions}

\subsection{Exponential Family Distributions}

\section{Expectation and Variance}

\subsection{Affine Equivariance}

\subsection{Variational Principle}


\section{Our method}

Our method uses a \emph{break table} just like the method by Haddon
and Waite \cite{Haddon:1967}, but instead of building, moving, and
sorting the table while objects are moved, we first move the objects
and then construct the table.  For that, additional space in the form
of a bitmap is required.  The bitmap has one bit per word of
memory in the heap, which amounts to less than $2$\% additional memory
on a $64$-bit machine.  The bits in the bitmap are set by the
\emph{mark} phase of the garbage collector.

\refFig{fig-example-a} shows a heap in which shaded areas indicate
live objects and white areas indicate dead objects.  The heap contains
16 word as shown by the addresses.  At the bottom of the figure is
shown the bitmap after the mark phase (phase 1) is complete.

\begin{figure}
\begin{center}
\inputfig{fig-example-a.pdf_t}
\end{center}
\caption{\label{fig-example-a}
Example of initial heap.}
\end{figure}

In phase 2, the heap is compacted by sliding the live objects to the
beginning of the heap.  In this phase, two pointers are used, a
\emph{source} pointer pointing to words containing live objects, and a
\emph{destination} pointer pointing to words containing dead objects,
as illustrated by \refFig{fig-example-b}.  Words are copied from the
source location to the destination location.  In each iteration, the
destination location is incremented by one unit, whereas the source
location is incremented until either it reaches the end of the heap,
or a word containing a live object as indicated by the bitmap
containing a $1$.

\begin{figure}
\begin{center}
\inputfig{fig-example-b.pdf_t}
\end{center}
\caption{\label{fig-example-b}
Pointers to source and destination locations.}
\end{figure}

\refFig{fig-example-c} shows the situation when phase 2 is complete.

\begin{figure}
\begin{center}
\inputfig{fig-example-c.pdf_t}
\end{center}
\caption{\label{fig-example-c}
Heap after compaction.}
\end{figure}

In phase 3, a \emph{break table} is built at the position of the
destination pointer.  The break table consists of a sequence $a_0,
d_0, a_1, d_1, \ldots, a_n, d_n, a_{n+1}$ of alternating
\emph{addresses} and \emph{deltas}.  The value $a_0$ is always $0$.
The table has an odd number of elements, because it both starts and
ends with an address.  Each $a_i$ (except possibly $a_0$ and
$a_{n+1}$) is the index of the beginning of a zone of dead objects.
Each $d_i$ is the sum of the sizes of the dead zones preceding
$a_{i+1}$.

\refFig{fig-example-d} shows the break table of the example heap in
\refFig{fig-example-a}.

\begin{figure}
\begin{center}
\inputfig{fig-example-d.pdf_t}
\end{center}
\caption{\label{fig-example-d}
Break table.}
\end{figure}

Notice that if both the bottom zone and the top zone of the heap
contain live objects, then our break table may require three additional
words of storage compared to the total number of free words available.
The reason for this additional requirement is that our table contains
\emph{sentinels} (the first two words and the last word) that are not
strictly required, but by including them, we avoid special cases in
our procedure for searching the table.  We can easily make sure that
three additional words are available by triggering a collection when
granting a request for memory would leave fewer than three free words
at the end of the heap.  Haddon and Waite \cite{Haddon:1967} did not
have this luxury, because they assumed an existing mark-and-sweep
collector.  For that reason, their paper contains an extensive
argument that their break table can fit in the available space.

The break table is built by scanning the bitmap from start to end.  In
practice, since the heap is likely to contain fairly large contiguous
zones, the bitmap will contain long runs of $0$s and long runs of
$1$s.  It is therefore advantageous to scan the bitmap a word at a
time, making this phase quite efficient.

In phase 4, the lower part of the heap is traversed word by word in
order to adjust the pointers according to the contents of the break
table.  For each pointer value $p$, the table is searched for values
$a_i, d_i, a_{i+1}$ such that $a_i \le p < a_{i+1}$.  The
pointer value $p$ is adjusted by subtracting $d_i$ from it.  To find
the entry, the break table is search using \emph{binary search}.

While the overhead of the binary search may seem unacceptably high,
two properties contribute to keeping this overhead low:

\begin{enumerate}
\item While the break table could contain as many as $N/4$ entries,
  where $N$ is the number of words in the heap, in practice, it
  contains far fewer than that, again because the heap is likely to
  contain relatively few relatively large zones.
\item It is very likely that the entry in the break table required to
  adjust a particular pointer is the same as the entry required to
  adjust the pointer immediately preceding it.  By testing this case
  first, the vast majority of full binary searches can be avoided.
\end{enumerate}


\documentclass{memoir}
\usepackage{notestemplate}

%\logo{~/School-Work/Auxiliary-Files/resources/png/logo.png}
%\institute{Rice University}
%\faculty{Faculty of Whatever Sciences}
%\department{Department of Mathematics}
%\title{Class Notes}
%\subtitle{Based on MATH xxx}
%\author{\textit{Author}\\Gabriel \textsc{Gress}}
%\supervisor{Linus \textsc{Torvalds}}
%\context{Well, I was bored...}
%\date{\today}

%\makeindex

\begin{document}

% \maketitle

% Notes taken on

\section{Properties of Ideals}
\label{sec:properties_of_ideals}

\begin{defn}
	Let \(A\leq R\) be a subset of a ring \(R\).
	\begin{itemize}
		\item  The \textbf{ideal generated by \(A\)} is the  smallest ideal of \(R\) containing \(A\), written by \((A)\).
		\item
			\begin{align*}
				RA = \left\{r_1a_1 + r_2a_2 + \ldots + r_na_n \mid r_i \in R,\, a_i \in A,\, n \in \Z_+ \right\} \\
				AR = \left\{a_1r_1 + a_2r_2 + \ldots + a_nr_n \mid a_i \in A,\,r_i \in R,\,  n \in \Z_+ \right\} \\
				RAR = \left\{r_1a_1r'_1 + r_2a_2r_2' + \ldots + r_na_nr_n' \mid r_i,r'_i \in R,\, a_i \in A,\, n \in \Z_+ \right\}
			\end{align*}
			are the \textbf{left, right, and two-sided ideal generated by \(A\)}.
		\item If \(\left| A \right| = n < \infty\), then \((A)\) is a \textbf{finitely generated ideal}.
		\item If \(\left| A \right| =1\) then \((A)\) is a \textbf{principal ideal}.
	\end{itemize}
\end{defn}
If \(A = \left\{ a_1,a_2,\ldots \right\} \) then we write \((A) = (a_1,a_2,\ldots)\) for simplicity. Notice that \(b\in R\) is in \((a)\) if and only if \(b = ra\) for some \(r \in R\), which is equivalent to \((b) \subset (a)\).

\begin{prop}
	Let \(I \triangleleft R\). Then \(I = R\) if and only if there exists a unit \(u \in I\).\\

	If \(R\) is commutative, then \(R\) is a field if and only if the only ideals of \(R\) are \(0\) and \(R\).
\end{prop}

\begin{cor}
	If \(R\) is a field and \(R'\) an arbitrary ring, then any nonzero ring homomorphism \(\varphi :R\to R'\) is injective.
\end{cor}

\begin{defn}[Maximal Ideal]
	A proper ideal \(M\) of a ring \(R\) is a \textbf{maximal ideal} of \(R\) if there does not exist another ideal of \(R\) that contains \(M\) besides \(R\) itself.
\end{defn}
Every proper ideal is containd in a maximal ideal.

\begin{anki}
TARGET DECK
Current Math::Abstract Algebra II

% Up to 5 consequences
START
Definition
Name: Maximal Ideal
Premise 1: \(M\) is a proper ideal of ring \(R\)
Consequence 1: \(M\) is maximal if there does not exist another ideal of \(R\) that contains \(M\) besides \(R\) itself
Tags: ring_ideals
<!--ID: 1611701297894-->
END
\end{anki}

\begin{thm}[Classification of Maximal Ideals]
	Let \(R\) be a commutative ring with identity and \(M\) an ideal in \(R\). Then \(M\) is a maximal ideal of \(R\) if and only if \(R / M\) is a field.
\end{thm}

\begin{anki}
% Up to 4 premises
% Up to 4 equivalences
START
Theorem
Name: Classification of Maximal Ideals
Premise 1: \(R\) commutative ring with identity
Premise 2: \(M\) ideal in \(R\)
Consequence 1: \(M\) maximal ideal \(\iff\) \(R/M\) is a field
Tags: ring_ideals
<!--ID: 1611701297914-->
END
\end{anki}

\begin{defn}[Prime Ideal]
	A proper ideal \(P\) in a commutative ring \(R\) is a \textbf{prime ideal} if whenever \(ab \in P\), then either \(a \in P\) or \(b \in P\).
\end{defn}
Note that it is possible to define prime ideals in a noncommutative setting.

\begin{anki}
% Up to 5 consequences
START
Definition
Name: Prime Ideal
Premise 1: Proper ideal \(P\) in commutative ring \(R\)
Consequence 1: \(P\) prime ideal iff \(ab \in P \implies a \in P\) or \(b \in P\)
Tags: rings_ideals
<!--ID: 1611701297931-->
END
\end{anki}

\begin{prop}
	Let \(R\) be a commutative ring with identity \(1_R\neq 0\). Then \(P\) is a prime ideal in \(R\) if and only if \(R / P\) is an integral domain.
\end{prop}

\begin{anki}
% Up to 4 premises
% Up to 4 equivalences
START
Theorem
Premise 1: \(R\) commutative ring w identity
Consequence 1: \(P\) prime ideal in \(R\) iff \(R / P\) integral domain
Tags: rings_ideals
<!--ID: 1611701297949-->
END
\end{anki}

As an example, note that every ideal in \(\Z\) is of the form \(n\Z\), and that \(\Z_n\) is an integral domain only when \(n\) is prime. This is why ideals of the form \(\Z_p\) are viewed as prime ideals.
\begin{cor}
	Every maximal ideal in a commutative ring with identity is also a prime ideal.
\end{cor}

\begin{anki}
START
MathJaxCloze
Text: Every maximal ideal in a commutative ring with identity is also a {{c1::prime ideal}}.
Tags: rings_ideals
<!--ID: 1611701297965-->
END
\end{anki}
% \printindex
\end{document}


\documentclass[12pt]{article}  %%%,twocolumn is too scrunched. decrease margin, increase column seperation

%%%
%%% Composition of formulas for MST Qualifying Exam
%%% All mistakes are mine, all work is cited by Author
%%% Creative Commons:
%%% Attribution-Noncommercial-Share Alike 3.0 Unported
%%% http://creativecommons.org/licenses/by-nc-sa/3.0/
%%%

\usepackage{verbatim} %%% multi-line comments
\usepackage[pdftitle={equations for the qualifying exam}, pdfauthor={Ben Payne}, pdfsubject={equations}, pdfkeywords={maxwell, lagrange}]{hyperref}
\usepackage{times}

%%% AMS packages and font files
\usepackage{amsmath} %%% necessary for gathered equations
\usepackage{amsfonts}
%%% For figures
%%%\usepackage[dvipdfm,colorlinks=true]{hyperref}
\usepackage[dvips]{graphicx}
\usepackage[usenames,dvipsnames]{color}
\usepackage{setspace}

%%% screws up the paragraph formatting
%%%\raggedright

\begin{comment}
\setlength{\topmargin}{-.5in}
\setlength{\textheight}{9.5in}
\setlength{\oddsidemargin}{0in}
\setlength{\textwidth}{6.5in}
\end{comment}
\setlength{\topmargin}{-1in}
\setlength{\textheight}{10in}
\setlength{\oddsidemargin}{-.5in}
\setlength{\textwidth}{7.5in}

%%% Commonly used macros
%%% from https://facetsproject.org/facets/browser/trunk/docs/gsdocs/nondimtf2hall.tex
\newcommand{\pfrac}[2]{\frac{\partial #1}{\partial #2}}
\newcommand{\pfraca}[1]{\frac{\partial}{\partial #1}}
\newcommand{\pfracb}[2]{\partial #1/\partial #2}

\def\includeSchaums{T}
\def\qualifyingyear{F}

\setlength{\columnsep}{.5in}
\begin{document}
%%%\twocolumn
%%% or you can put it in \documentclass[11pt,twocolumn]{article}

%%% don't hyphenate so much - default = 200, max (never hyphenate) = 10,000
\hyphenpenalty=800

\title{Physics equations}
\author{Ben~Payne\footnote{Electronic address: ben.is.located@gmail.com}}
%%%\affiliation{Department~of~Physics, Missouri~University~of~Science~\&~Technology, Rolla,~MO~65409}
\date{\today}

\thispagestyle{empty}  %%% hides page number of the title page
\pagestyle{empty} %%% no page numbers on the following pages
%%%\begin{abstract}
%%%equations for the qualifying exam
%%%\end{abstract}

version 0.64, 20090308

%%% http://clarku.edu/~djoyce/trig/identities.html
\if\includeSchaums T
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{coordinate systems}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
(from cover of \cite{GriffithED}.) Most can be found in Schaum's.
%%%\begin{comment}
 the following is in Schaum's, pg 120
\subsection{cartesian}
%%%\Huge
\begin{equation}
dV = 	dx\cdot dy\cdot dz
\label{eq:cartesian_dV}
\end{equation}
%%%\normalsize
gradient:
%%%\Huge
\begin{equation}
\vec{\nabla} t = \widehat{x} \pfrac{t}{x}  + \widehat{y} \pfrac{t}{y} + \widehat{z} \pfrac{t}{z}  %%% can't seem to get \hat to work
\label{eq:cartesian_gradient}
\end{equation}
%%%\normalsize
divergence (equ 1.40, page 17 \cite{GriffithED})
%%%\Huge
\begin{equation}
 \vec{\nabla} \cdot v = \pfrac{v_x}{x} + \pfrac{v_y}{y} + \pfrac{v_z}{z}
\label{eq:cartesian_divergence}
\end{equation}
%%%\normalsize
laplacian
\begin{equation}
\nabla^2 f = \pfrac{^2 f}{x^2} + \pfrac{^2 f}{y^2} + \pfrac{^2 f}{z^2}
\label{eq:cartesian_laplacian}
\end{equation}
%%%\end{comment}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{spherical}

\begin{equation}
dV = 	r^2 \ \sin \theta \ dr \ d\theta \ d\phi
\label{eq:spherical_dV}
\end{equation}

page 40, \cite{GriffithED}
\begin{equation}
  d \vec{a} = \hat{r} \ r^2 \sin \theta \ d\theta \ d\phi
\end{equation}

\begin{equation}
\vec{\nabla} = \hat{r}\pfraca{r}+ \hat{\theta}\frac{1}{r}\pfraca{\theta}+\hat{\phi}\frac{1}{r\sin\theta}\pfraca{\phi}
\label{eq:spherical_gradient}
\end{equation}

%%%\begin{comment}
 Schaum's, pg 126
laplacian. See \htmladdnormallink{planetMath laplacian rectangualr to cartesian}{http://planetmath.org/encyclopedia/DerivationOfTheLaplacianFromRectangularToSphericalCoordinates.html}
and \htmladdnormallink{planetMath laplacian}{http://planetmath.org/?method=l2h&from=collab&id=76&op=getobj}
(equ 3.53, page 137 \cite{GriffithED})
\begin{equation}
\nabla^2 f = {1 \over r^2} {\partial \over \partial r}  \left( r^2 {\partial f \over \partial r} \right)
+ {1 \over r^2 \sin \theta} {\partial \over \partial \theta}  \left( \sin \theta {\partial f \over \partial \theta} \right)
+ {1 \over r^2 \sin^2 \theta} {\partial^2 f \over \partial \phi^2}
\label{eq:spherical_laplacian}
\end{equation}
%%%\end{comment}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{cylindrical}

\begin{equation}
dV = 	s \ ds \ d\phi \ d\theta
\label{eq:cylindrical_dV}
\end{equation}

\begin{equation}
\vec{\nabla} = \hat{s}\pfraca{s} + \hat{\phi}\frac{1}{s} \pfraca{\phi} + \hat{z}\pfraca{a}
\label{eq:cylindrical_gradient}
\end{equation}

%%%\begin{comment}
\begin{equation}
\nabla^2 f = {1 \over \rho} {\partial \over \partial \rho}  \left( \rho {\partial f \over \partial \rho} \right)
+ {1 \over \rho^2} {\partial^2 f \over \partial \theta^2} + {\partial^2 f \over \partial z^2 }
\label{eq:cylindrical_laplacian}
\end{equation}
%%%\end{comment}
\fi
\section{formulas}

material in this section can NOT be found in Schaum's \htmladdnormallink{mathematical handbook of formulas}{http://site.ebrary.com/lib/umr/Doc?id=10015341}, which is allowed on the day of the exam.
%%%\begin{comment}
 Schaum's, 123 \\
Stoke's thm (equ 1.57, page 34 \cite{GriffithED})
\begin{equation}
\int_{surface} (\vec{\nabla}\cdot\vec{v})\cdot d\vec{a} = \oint_{volume}\vec{v}\cdot d\vec{l}
\label{eq:stokesThm}
\end{equation}
%%%\end{comment}

Greene's thm (equ 1.56, page 31 \cite{GriffithED})
\begin{equation}
\int_{volume}(\vec{\nabla}\cdot\vec{v}) d\tau = \oint_{surface} \vec{v}\cdot d\vec{a}
\label{eq:greenesThm}
\end{equation}

Taylor series expansion is in Schaum's, but the easy approximation is not. (assuming small x)
\begin{equation}
(1+x)^n \approx 1 + n x
\label{eq:TaylorSeriesApprox}
\end{equation}

%%%\begin{comment}
 in Schuam's \\
Taylor series expansion
%%% \begin{widetext} %%% not recognized
%%%\begin{multicols}{2}%%% not recognized
\begin{equation}
f(x) = f(a) + f'(x)|_a (x-a) + \frac{1}{2!} f''(x)|_a (x-a)^2 + \cdots
\label{eq:TaylorSeries}
\end{equation}
%%%\end{multicols}%%% not recognized
%%%\end{widetext} %%% not recognized
%%%\end{comment}

\section{Math}
\begin{equation}
\left[ \begin{array}{cc}
  \cos \theta & \sin \theta \\
-\sin \theta  & \cos \theta \\
 \end{array} \right]
\left[ \begin{array}{c}
A_x\\
A_y
\end{array}\right]
=
\left[ \begin{array}{c}
A_{x'}\\
A_{y'}
\end{array}\right]
\end{equation}
\begin{center}
 \includegraphics[scale=0.3]{pictures/coordinate_rotation}
\end{center}

Variance
\begin{equation}
(\Delta A)^2 = \langle A^2 \rangle - \langle A \rangle^2
\end{equation}

Quadratic equation
\begin{equation}
 x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\end{equation}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Electromagnetics} %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Maxwell equations} %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% see http://hyperphysics.phy-astr.gsu.edu/HBASE/electric/maxeq.html
See also page 22, \cite{ReifThermo}; page 2, \cite{JacksonED} \\
%%%\begin{align} %%% utilizes the "&"
\underline{Faraday}'s law (equation 7.16, page 302, \cite{GriffithED})
\begin{equation}
  \vec{\nabla}\times \vec{E} = -\pfrac{\vec{B}}{t}
  \label{eq:maxwellEcurl}
\end{equation}

\underline{Ampere}'s law (equation 5.54, page 225 and equation 5.44, page 222 \cite{GriffithED})
\begin{equation}
  \vec{\nabla}\times \vec{B} =  \mu_0\vec{J}+\frac{1}{c^2}\pfrac{\vec{E}}{t}
  \label{eq:maxwellBcurl}
\end{equation}
where
\begin{equation}
 c = \frac{1}{\sqrt{\epsilon_0 \mu_0}}
\end{equation}

\underline{Gauss}'s law, electricity (equation 2.14, page 69 \cite{GriffithED})
\begin{equation}
  \vec{\nabla}\cdot\vec{E} = \frac{\rho}{\varepsilon_0}
  \label{eq:maxwellEdiv}
\end{equation}

\underline{Guass}'s law, magnetism (equation 5.48, page 223 \cite{GriffithED})
\begin{equation}
  \vec{\nabla}\cdot\vec{B} = 0
  \label{eq:maxwellBdiv}
\end{equation}
%%%\end{align}
Integral forms:

flux of electric field (equation 2.11, page 67 \cite{GriffithED})
\begin{equation}
\Phi_E = \int_{any\ surface} \vec{E} \cdot d\vec{a}
\end{equation}

\underline{Gauss}'s law, electricity. (equation 2.13, page 68 \cite{GriffithED})
\begin{equation}
  \oint_{closed\ surface} \vec{E}\cdot d\vec{a} = \frac{Q_{enclosed}}{\epsilon_0}
	\label{eq:GaussLaw}
\end{equation}

\underline{Faraday}'s Law (equation 7.18, page 306 \cite{GriffithED})
\if\qualifyingyear T
(2008 A3)
\fi
\begin{equation}
  \oint \vec{E} \cdot d\vec{l} = - \frac{d\Phi _B}{dt}
	\label{eq:Faraday_law}
\end{equation}

\underline{Ampere}'s law (equation 5.55, page 225 and 5.42, page 222 \cite{GriffithED})
\if\qualifyingyear T
(2008 A3)
\fi
\begin{equation}
  \oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enclosed}
	\label{eq:Ampere}
\end{equation}

\underline{Ampere}'s law (page 306, \cite{GriffithED})
\begin{equation}
  \oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enclosed} + \mu_0 \epsilon_0 \int \left( \frac{d}{dt}\vec{E}\right) \cdot d\vec{a}
	\label{eq:Ampere_large}
\end{equation}

\underline{Lorentz} force law (equation 5.2, page 204 \cite{GriffithED}). Also equation 1-61, page 22 \cite{GoldsteinCM}
\begin{equation}
\vec{F} = q (\vec{E} + \vec{v}\times \vec{B})
\end{equation}

%%%%%%%%%%%%%%%%%%%%%% end maxwell %%%%%%%%%%
\subsection{general EM}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

equation 1-62, page 22 \cite{GoldsteinCM}
\begin{equation}
\vec{B} = \vec{\nabla} \times \vec{A}
\end{equation}

Coulomb's Law
\begin{equation}
\vec{F} =\frac{1}{4\pi \epsilon _{0} } \frac{q_{1} q_{2} }{r^{2} } \hat{r}
\end{equation}
 where $q_{2} $ is the test charge

\begin{equation}
\vec{E}=\frac{1}{4\pi \epsilon_0} \int_v \frac{\rho}{r^2} d \tau \hat{r}
\end{equation}

electric displacement (see also eq \ref{eq:electricDisplacementwithPolarization})
\begin{equation}
\vec{D} = \epsilon_0 \vec{E}
	\label{eq:electricDisplacement}
\end{equation}

Electric field due to point charge (equation 2.4, page 60 \cite{GriffithED})
\begin{equation}
\vec{E} = \frac{1}{4 \pi \epsilon_0} \sum_i \frac{q_i}{r_i^2} \hat{r}_i
%%%\label{eq:another}
\end{equation}

electrostatic potential (equ 2.21, page 78 and equ 7.10, page 293\cite{GriffithED})
\begin{equation}
V = - \int^r_{\infty} \vec{E}\cdot d\vec{l}
%%%\label{eq:another}
\end{equation}

electric field and electrostatic potential (equation 2.23, page 78,  \cite{GriffithED})
\begin{equation}
  \vec{E} = -\vec{\nabla}V
	\label{eq:electric_field_potential}
\end{equation}

electrostatic potential (equation 2.27, page 84 \cite{GriffithED})
\begin{equation}
 V = \frac{1}{4 \pi \epsilon_0} \sum_i \frac{q_i}{r_i}
\end{equation}

electrostatic potential (equation 2.29, page 84 \cite{GriffithED})
\begin{equation}
 V = \frac{1}{4 \pi \epsilon_0} \int \frac{\rho}{r} d \tau
\end{equation}

work (equation 2.42, page 92  \cite{GriffithED})
\begin{equation}
  W = \frac{1}{2} \sum q_i V(r_i)
%%%\label{eq:}
\end{equation}

work and electrostatic potential (equ 2.45, page 94 \cite{GriffithED})
\begin{equation}
  W = \frac{\epsilon_0}{2} \int_{all space} E^2 d \tau
\label{eq:work_in_electric_field}
\end{equation}

surface charge density and electrostatic potential (where $n$ is the normal direction \\ (equation 2.49, page 102  \cite{GriffithED})
\begin{equation}
\sigma = -\epsilon_0 \frac{\partial V}{\partial n}
\label{eq:surfacechargedensity}
\end{equation}

equation 2.48, page 102 \cite{GriffithED}
\begin{equation}
\vec{E} = \frac{\sigma}{\epsilon} \hat{n}
\end{equation}

electric displacement, including polarization (equ 4.21, page 175 \cite{GriffithED})
(see also eq \ref{eq:electricDisplacement})
\begin{equation}
\vec{D} = \epsilon_0 \vec{E} + \vec{P}
\label{eq:electricDisplacementwithPolarization}
\end{equation}

work and electrostatic potential (equ 2.55, page 106, \cite{GriffithED})
\begin{equation}
  W = \frac{1}{2} C V^2
	\label{eq:workCapacitor}
\end{equation}

electrostatic potential due to a monopole (equation 3.97, page 149  \cite{GriffithED})
\begin{equation}
V_{monopole} = \frac{1}{4 \pi \epsilon_0} \frac{Q}{r}
\end{equation}

electrostatic potential due to a dipole (equation 3.99, page 149  \cite{GriffithED})
\begin{equation}
V_{dipole} = \frac{1}{4 \pi \epsilon_0} \frac{\vec{p}\cdot \hat{r}}{r^2}
\end{equation}

where dipole moment (which points from the (-) charge to the (+) is \\
(equation 3.98, page 149  \cite{GriffithED}) (equation 3.100, page 150  \cite{GriffithED})
\begin{equation}
\vec{p} = \int \vec{r} \; ' \rho ( \vec{r}\;' ) d \tau \;' = \sum_i q_i r_i
\end{equation}

Note: $E_{dipole}$, equation 3.104, page 155; is given on 2002 B10,
but as a derivation (see also problem 3.33).

Torque ($N$) is on a dipole ($\vec{p}=q\vec{d}$) due to E field (equation 4.4, page 164, \cite{GriffithED})
\begin{equation}
\vec{N} = \vec{p} \times \vec{E}
\end{equation}

potential energy of a dipole (problem 4.7) (equation 4.6, page 165, \cite{GriffithED})
\begin{equation}
U = -\vec{p} \cdot \vec{E}
\end{equation}

Similarly for magnetic moment ($\vec{\mu} = (\vec{I} \cdot \vec{A})\hat{n}$) \\
(equation 6.1, page 257, \cite{GriffithED})
\begin{equation}
\vec{N} = \vec{\mu} \times \vec{B} \\
\end{equation}

potential energy (equation 6.34, page 281, \cite{GriffithED})
\begin{equation}
U = -\vec{\mu} \cdot \vec{B}
\end{equation}

force due to magnetic field (equ 5.16, page 205 \cite{GriffithED})
\if\qualifyingyear T
(1997 A4)
\fi
\begin{equation}
 F_{magnetic} = I \int d\vec{l} \times \vec{B}
\end{equation}

(equ 6.31, page 275 \cite{GriffithED})
\begin{equation}
 \vec{B} = \mu \vec{H}
\end{equation}

(equ 6.32, page 275 \cite{GriffithED})
\if\qualifyingyear T
(2009 A)
\fi
\begin{equation}
 \mu = \mu_0 (1+\chi_m)
\end{equation}

(equ 7.3, page 285 \cite{GriffithED})
\begin{equation}
 \vec{J} = \sigma \vec{E}
\end{equation}

flux (equ 7.12, page 295 \cite{GriffithED})
\if\qualifyingyear T
(1997 A4, 2009 A)
\fi
\begin{equation}
 \Phi = \int \vec{B} \cdot d\vec{a}
\end{equation}

work (equ 7.29, page 317 \cite{GriffithED}). Used in problems 7.26, 7.27.
\if\qualifyingyear T
(2008 A3)
\fi
\begin{equation}
  W = \frac{1}{2} L I^2
	\label{eq:work_self_inductance}
\end{equation}

work (equ 7.34, page 317 \cite{GriffithED})
\if\qualifyingyear T
(2009 A)
\fi
\begin{equation}
  W = \frac{1}{2 \mu_0} \int_{all space} B^2 d \tau
	\label{eq:work_in_magnetic_field}
\end{equation}

flux, inductance (equ 7.25, page 313 \cite{GriffithED})
\if\qualifyingyear T
(2009 A)
\fi
\begin{equation}
 \Phi = L I
\end{equation}

Power (equ 8.11, page 347 \cite{GriffithED}), positive on page 349
\begin{equation}
  Power = \int \vec{S} \cdot d\vec{a}
	\label{eq:power_fields}
\end{equation}

total potential energy in a field (equation 8.5, page 346 \cite{GriffithED}) (combines \ref{eq:work_in_magnetic_field} and \ref{eq:work_in_electric_field})
\begin{equation}
U_{em} = \frac{1}{2} \int ( \epsilon_0 E^2 + \frac{1}{\mu_0} B^2 ) d \tau
\end{equation}

Poynting vector (equation 8.10, page 347 \cite{GriffithED})
\begin{equation}
  \vec{S} = \frac{1}{\mu} \vec{E}\times\vec{B}
	\label{eq:Poynting_vector}
\end{equation}

Power and Poynting vector (equation 8.11, page 347 \cite{GriffithED})
\begin{equation}
P = \frac{dW}{dt} = - \frac{dU_{em}}{dt}- \oint \vec{S} \cdot d\vec{a}
\end{equation}

\subsection{circuits}

capacitance (equ 2.53, page 104, \cite{GriffithED})
\begin{equation}
  C \equiv \frac{Q}{V}
\end{equation}

\begin{equation}
 P=I^2 R
\end{equation}

\begin{equation}
 V=IR
\end{equation}

Power and electrostatic potential ($P=I^2 R$ can be derived from $V=IR$) \\
(equation 7.7, page 290, \cite{GriffithED})
\begin{equation}
  Power = I V
	\label{eq:power_current}
\end{equation}

circuit sum (equation 7.4, page 290, \cite{GriffithED}) [resistor, capacitor, inductor]
\begin{equation}
V = IR + \frac{Q}{C} + L \frac{dI}{dt}
\end{equation}
 \begin{center}
  \includegraphics[scale=0.3]{pictures/V_R_C_L}
 \end{center}

\begin{equation}
I = \frac{dQ}{dt}
\end{equation}

capacitance, A is the area, d is the seperation (equation 2.54, page 105 \cite{GriffithED})
\begin{equation}
C = \frac{A \epsilon_0}{d}
\end{equation}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{relativity}

Lorentz factor, [$\gamma > 1$](equation 12.6, page 486 \cite{GriffithED})
\begin{equation}
\gamma = \frac{1}{\sqrt{1-(\frac{v}{c})^2}}
\label{eq:gamma}
\end{equation}
\if\qualifyingyear T
GIVEN on 2008 A1, not given on 2001 A6\\
\fi
 total relativistic energy for free particle (equation 12.55, page 551 \cite{GriffithED})
\begin{equation}
  E = \sqrt{m^2 c^4 + p^2 c^2}
	\label{eq:relativistic_energy}
\end{equation}

\begin{equation}
  E = \gamma m c^2
	\label{eq:Einsteins_energy}
\end{equation}

relativistic momentum (equation 2-7, page 71 \cite{TiplerMP})
\if\qualifyingyear T
\\
2001 A5 d
\fi
\begin{equation}
  p = \gamma m v
  \label{eq:relativistic_momentum}
\end{equation}

\subsection{EM concepts}
\begin{itemize}
	\item hollow charged conductors have no interior electric field (page 97, \cite{GriffithED})
	\item solid evenly-charged spheres have linearly increasing electric field inside;
	outside E field falls of as $\frac{1}{r^2}$
	\item potential of a charge falls off as $\frac{1}{r}$
	\item length contraction, time dilations (chapter 12, \cite{GriffithED})
	\item images above an $\infty$ grounded conducting plane $\rightarrow$ use method of images
	(section 3.2, see also problem 4.6 in \cite{GriffithED})
	\item changing electric field induces a magnetic field (page 323, \cite{GriffithED})
	\item magnetic forces do no work (page 207, \cite{GriffithED})
	\item $E_{tangential}$ is continuous, $E_{normal}$ is discontinuous
	\item to solve Laplace's equation ($\nabla^2 V = 0$) with V specified on the
	boundaries, use seperation of variables (Chapter 3, \cite{GriffithED})
	\item magnetic forces do no work (page 207, \cite{GriffithED})
\end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Thermodynamics}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
most generally, specific heat is
\begin{equation}
C_x = \frac{dQ}{dT}|_x
\end{equation}
where x is the variable that is held constant.
\begin{equation}
 \frac{1}{T} = \pfrac{S}{E} |_v
\end{equation}
Then from equation  $5 \cdot 6 \cdot 1$ page 164; page 153; \cite{ReifThermo}
\begin{equation}
T \cdot dS = dE + p \cdot dV
\label{eq:TdS}
\end{equation}
solve for $dE$, which is $dQ$, and hold V constant to get \\
specific heat, constant volume;
\if\qualifyingyear T
given on 2002 A6
\fi
\begin{equation}
C_V = \pfrac{E}{T}|_v
\end{equation}
using the definition of enthalpy, add $d(pV)$ to Eq.~\ref{eq:TdS} to get
\begin{equation}
dH = T \cdot dS + V \cdot dp
\end{equation}
and hold pressure p constant to get \\
specific heat, constant pressure
\begin{equation}
C_p = \pfrac{H}{T}|_p
\end{equation}
\begin{equation}
 C_P = C_V + N K_{Boltzmann} = C_V + \nu R
\end{equation}
$C_P > C_V$ always
\begin{equation}
 pV = \nu R T
\end{equation}
where $\nu$ is the number of moles, $\frac{N}{N_{Av}}$ and $R=N_{Av} K_{Boltzmann}$.

equations $3 \cdot 3 \cdot 12$; $3 \cdot 11 \cdot 6$, page 123; \cite{ReifThermo}
\if\qualifyingyear T
 \\
given on 2002 B6, 2006 A6
\fi
\begin{equation}
S = k \ln \Omega
\end{equation}

ideal gas; relating average pressure, Volume, Temperature (equations  $3 \cdot 12 \cdot 8$ page 125; $7 \cdot 2 \cdot 8$ page 241; \cite{ReifThermo})
\begin{equation}
\overline{p} V = N k_{Boltzmann} T
\end{equation}
$pV = nRT$ \\
ideal gas; relating to average Kinetic energy
\begin{equation}
\overline{p} V = \frac{2}{3} N \overline{K}
\end{equation}

where number density $ n = \frac{N}{V}$ is number of molecules per volume.

mean kinetic energy per particle; applies to fluids in closed systems
\begin{equation}
\overline{K} = \frac{3}{2} k_{boltzmann} T
\end{equation}

partition function for a single particle (equation $6 \cdot 5 \cdot 3$ page 213; \cite{ReifThermo})
\if\qualifyingyear T
\\
given 2002 A6
\fi
\begin{equation}
z \equiv \sum_r^\infty e^{-\beta E_r}
\end{equation}
and total partition function $Z = z^{3N}$ \\
partition function and energy (equation $6 \cdot 5 \cdot 4$ page 213; \cite{ReifThermo})
\if\qualifyingyear T
\\
given 2002 A6
\fi
\begin{equation}
E = - \frac{\partial (\ln Z)}{\partial \beta} |_V
\end{equation}

Fermi-Dirac distribution  (equation $9 \cdot 3 \cdot 14$ page 341; \cite{ReifThermo}) \\
mean occupation number; number of particles with energy $\epsilon_s$
\begin{equation}
n = \frac{1}{e^{\alpha + \beta \epsilon_s} +1}
\end{equation}

Bose-Einstein distribution  (equation $9 \cdot 3 \cdot 22$ page 342; \cite{ReifThermo})
\begin{equation}
n = \frac{1}{e^{\alpha + \beta \epsilon_s}-1}
\end{equation}
since when the exponent is 0, then $n \rightarrow \infty$, so all the particles form a BEC.\\
Maxwell-Boltzmann distribution  (equation $9 \cdot 4 \cdot 7$ page 345; \cite{ReifThermo})
(see also page 133, \cite{TiplerMP})
\begin{equation}
n = N \frac{ e^{-\beta \epsilon_s }}{ \sum_r e^{-\beta \epsilon_r} }
\end{equation}

\subsection{Thermodynamics concepts}
\begin{itemize}
	\item good review of basics on page 122 of \cite{ReifThermo}, including the laws
	\item second law: $\Delta S \geq 0$
	\item classical equipartition theorm: each squared term of the Hamiltonian
	has $E=\frac{1}{2}k_b T$ \\ (equation $7 \cdot 5 \cdot 7$ page 249; \cite{ReifThermo})
	\item entropy is maximum for a closed, isolated system at equilibrium
	\item all states are equally probable. Equilibrium is the most probable configuration -- the largest number of states sharing the same configuration
\end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Classical Mechanics}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
the ``basics:'' \\
Force and potential energy, (page 20, \cite{GoldsteinCM})
\begin{equation}
\vec{F} = - \vec{\nabla} V
\end{equation}
\begin{equation}
 T = \frac{1}{2} I \omega^2
\end{equation}

for circular motion, tangential velocity is
\begin{equation}
v = \omega r
\end{equation}
and centripital acceleration (also circular motion)
\if\qualifyingyear T
(2009 A)
\fi
\begin{equation}
\frac{v^2}{R}
\end{equation}

Angular momentum $\vec{L}$
\begin{equation}
\vec{L} = \vec{r} \times \vec{p}
\end{equation}

Torque $\vec{N}$
\begin{equation}
 \vec{N} = \vec{r}\times \vec{F} = \pfrac{\vec{L}}{t}
\end{equation}

Force $F$, potential $V$, momentum $p$
\begin{equation}
 \vec{F} = -\vec{\nabla} V = \pfrac{p}{t}
\end{equation}

\begin{equation}
\begin{gathered}
F_{spring} = -k \ x \\
V_{spring} = \frac{1}{2}k \ x^2
\end{gathered}
\end{equation}

\subsection{Lagrange's Equations of Motion}

kinetic energy (cartesian, cylindrical)
\begin{equation}
T = \frac{1}{2} m v^2 = \frac{1}{2} m ( \dot{x}^2 + \dot{y}^2 + \dot{z}^2) =
\frac{1}{2} m (\dot{r}^2 + r^2 \dot{\theta}^2 + \dot{z}^2)
\end{equation}

Lagrangian
$L=L\left( q_{j} ,\dot{q} _{j} ,t\right) =T-U$ (equation 1-56, page 20 \cite{GoldsteinCM}). \\
(equation 1-57, page 21 \cite{GoldsteinCM})
\if\qualifyingyear T
(2009 A)
\fi
\begin{equation}
\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q} _{j} } \right) -\frac{\partial L}{\partial q_{j} } = 0
\end{equation}

\subsection{Hamilton's Equations of Motion}
$H=H\left( q_{j} ,p_{j} ,t\right) $ \\
usually $ H = T+V$ \\
(equation 8-8, page 341 \cite{GoldsteinCM})
\begin{equation}
H=\sum\limits_{j}p_{j}  \dot{q} _{j} -L
\end{equation}

Once Lagrangian is found, get canonical momentum
\begin{equation}
p_j \equiv \frac{\partial L}{\partial \dot{q}_j }
\label{eq:canonical_momentum}
\end{equation}

and solve for $\dot{q}_j$ for later

\begin{equation}
h = \left( \sum \dot{q_j} \frac{\partial L}{\partial \dot{q} _{j} } \right) - L
\end{equation}

then substitue in $\dot{q}$ to get H. \\
Finally, get Hamilton Equations of motion by \\
(equation 8-12, page 342 \cite{GoldsteinCM})
\begin{equation}
\begin{gathered}
\dot{q} _{j} =\frac{\partial H}{\partial p_{j} } \\
\dot{p} _{j} = - \left( \frac{\partial H}{\partial q_{j} } \right)
\end{gathered}
\end{equation}

then integrate to get $q$ and $p$

\subsection{2-body central force}
standard orbit is Counter Clock-Wise (CCW). \\
Given the Lagrangian in polar coordinates, one derives the canonical angular momentum \\
(equation 3-8, page 73 \cite{GoldsteinCM}) (see eq \ref{eq:canonical_momentum})
\begin{equation}
l = \mu r^2 \dot{\theta}
\end{equation}
for perturbed orbits, $ \tau > \tau_c$ implies CCW (positive $\Delta\theta$) motion (forward orbital motion)
\begin{equation}
\Delta \theta = \frac{l}{m r^2}(\tau - \tau_c)
\end{equation}

circular orbit condition (equation 3-12, page 74  \cite{GoldsteinCM})
\begin{equation}
f^{effective}(r_c) = 0 = f(r_c) + \frac{l^2}{\mu r_c^3}
\end{equation}
where for the kepler problem $f(r) = -\frac{k}{r^2}$

small oscillations of orbit
\if\qualifyingyear T
(2009 A)
\fi
\begin{equation}
r = r_c + \epsilon
\end{equation}

not included: small oscillation (chapter 6 of  \cite{GoldsteinCM}).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Quantum}

expectation value, average, mean
\begin{equation}
\langle A \rangle = \langle \psi | A | \psi \rangle = \int_{-\infty}^{\infty} \psi^* A \psi \ dx
\end{equation}

probability amplitude ( \cite{LiboffQM} page 118 calls this the
add-mixture coefficient $|C_n|^2$)
\begin{equation}
| \psi | ^2 = \langle \psi | \psi \rangle = \int \langle \psi | x \rangle \langle x | \psi \rangle dx
\end{equation}

Ehrenfest Theorem
\begin{equation}
 \frac{d}{dt} \langle a\rangle = \left\langle \pfrac{a}{t} \right\rangle + i \hbar \left\langle [a,H] \right\rangle
\end{equation}

and for sudden change, from the ground state (1) to the first excited state (2), probability of transition is
\begin{equation}
\left| \langle \phi_2 | \psi_1 \rangle \right| ^2
\end{equation}

uncertianty
\begin{equation}
\Delta x = \sqrt{ \langle x^2 \rangle - \langle x \rangle ^2 }
\end{equation}

commutator (equation 7.24, page 194 \cite{LiboffQM})
\begin{equation}
\left[ x,p\right] =i\hbar
\end{equation}

\if\qualifyingyear T
GIVEN on 2005, A5:
\fi
frequency
\begin{equation}
  \omega = \sqrt{\frac{k}{m}}
	\label{eq:frequency_wavenum}
\end{equation}

time evolution (equ 2.490, page 98, \cite{ParrisQM})
\begin{equation}
  |\Psi (t)\rangle = e^{-i \frac{H}{\hbar} t} |\Psi_0\rangle
%%%\label{eq:frequency_wavenum}
\end{equation}

\begin{equation}
  |\Psi (t)\rangle = \sum_n e^{-i \frac{E_n}{\hbar} t} \Psi_n(0) |n\rangle
%%%\label{eq:frequency_wavenum}
\end{equation}

%\begin{comment}
%%% second semester graduate quantum
total spin number
\begin{equation}
m_{total} = m_1 + m_2
\end{equation}
where $m = -j,\ldots,j$

total momentum
\begin{equation}
s_{total} = (s_1 + s_2),(s_1 + s_2 -1),\ldots,| s_1 - s_2 |
\end{equation}
where $s_{total} \geq 0$
%\end{comment}

\subsection{energy eigenvalue equation}

In position representation, derive the energy eigenvalue equation from
\begin{equation}
  \vec{K} = -i \vec{\nabla}
	\label{eq:wavenumberinpositionspace}
\end{equation}
in position representation, (equation 130)
\begin{equation}
  \vec{P} = \hbar \vec{K}
	\label{eq:momentumwavenumber}
\end{equation}

thus momentum (page 14, \cite{ParrisQM}) (equation 3.2, page 69 \cite{LiboffQM}
\begin{equation}
  \vec{P} = - i \hbar \vec{\nabla}
	\label{eq:momentumtopositionspace}
\end{equation}
From eq \ref{eq:momentumtopositionspace}, knowing the Hamiltonian is $H = T+V$
\begin{equation}
  \hat{H} = \frac{p^2}{2 m} + V
	\label{eq:hamiltonian_energy}
\end{equation}
Then plug \ref{eq:momentumtopositionspace} into \ref{eq:hamiltonian_energy} and use
\begin{equation}
H \psi = E \psi
\end{equation}
 to get the energy eigenvalue equation,
\begin{equation}
  \hat{H} = \frac{-\hbar ^2 \nabla ^2}{2 m}\Psi + V\Psi = E \Psi
	\label{eq:energy_eigenvalue}
\end{equation}

Schrodinger Equation
\begin{equation}
  i \hbar \frac{d}{dt}|\Psi \rangle = H |\Psi \rangle
	\label{eq:schrodinger_equ}
\end{equation}

equation 9.19, page 358 \cite{LiboffQM}
\begin{equation}
\begin{gathered}
J_+ = J_x + i J_y \\
J_- = J_x - i J_y
\end{gathered}
\end{equation}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{harmonic oscillator}

\if\qualifyingyear T
GIVEN on 2005, A5:
\fi
SHO energy levels
\begin{equation}
  E_n = (n+\frac{1}{2})\hbar \omega
	\label{eq:SHO_energy}
\end{equation}

harmonic oscillator Hamiltonian (equation 3.2, page 103  \cite{ParrisQM})
\begin{equation}
H = \frac{p^2}{2 m}+ \frac{1}{2} m \omega^2 x^2
\label{eq:H_p_x}
\end{equation}
factor out $\frac{\hbar \omega}{2}$ to derive harmonic oscillator Hamiltonian in dimensionless operators \\
Note: $\hat{P}$ and $\hat{Q}$ are dimensionless, whereas $\hat{p}$ and $\hat{q}$ are dimensional
(equation 3.18, page 105 \cite{ParrisQM})
\begin{equation}
H = \frac{\hbar \omega}{2} (\hat{P}^2 + \hat{Q}^2)
\label{eq:H_p_q}
\end{equation}
by comparing \ref{eq:H_p_q} and \ref{eq:H_p_x} the relationship between
dimensionless q,p and dimensional x,p can be found.

in terms of energy operators, harmonic oscillator Hamiltonian (equation 3.34, page 107  \cite{ParrisQM})
\begin{equation}
H = \hbar \omega (N+ \frac{1}{2})
\end{equation}
where $N = a^+ a$ (equation 3.33)

\if\qualifyingyear T
GIVEN on 2006 A4, 1998 A3, 2009 A1.
\fi
Lowering, raising operators
\begin{equation}
\begin{gathered}
 a|n\rangle =\sqrt{n}|n-1 \rangle \\
 a^+|n\rangle =\sqrt{n+1}|n+1 \rangle
  \label{eq:raising_lowering}
\end{gathered}
\end{equation}

dimensionless position, momentum (equation 3.29, page 106 \cite{ParrisQM})
\begin{equation}
\begin{gathered}
\hat{q} = \frac{1}{\sqrt{2}} (\hat{a}^+ + \hat{a}) \\
\hat{p} = \frac{i}{\sqrt{2}} (\hat{a}^+ - \hat{a})
\label{eq:q_p_dimensionless}
\end{gathered}
\end{equation}

\subsection{time independent non-degenerate perturbation theory}
perturbation theory says the exact solution can be approximated as
\begin{equation}
E \cong E^{(0)} + E^{(1)} + E^{(2)} + \ldots
\end{equation}

(eq. \ref{eq:first_order_correction_to_energy} through \ref{eq:second_order_correction_to_energy}
\if\qualifyingyear T
given on 1998 A3, not given on 2002 A2) \\
\fi
first order correction to energy (equ 5.41, page 138, \cite{ParrisQM}), (page 685 \cite{LiboffQM})
\begin{equation}
E^{(1)}_n = \langle n^{(0)} |V|n^{(0)}\rangle
\label{eq:first_order_correction_to_energy}
\end{equation}

first order correction to state (equ 5.42, page 138, \cite{ParrisQM}), (equation 13.8, page 685 \cite{LiboffQM})
\begin{equation}
|n^{(1)}\rangle = \sum_{m \neq n} | m^{(0)}\rangle \langle m^{(0)} |n^{(1)}\rangle
\end{equation}
where (equation 5.43, page 138, \cite{ParrisQM})
\begin{equation}
\langle m^{(0)} |n^{(1)}\rangle = -\frac{\langle m^{(0)}|V|n^{(0)}\rangle}{E_m^{(0)} - E_n^{(0)}}
\end{equation}
where $m\neq n$

second order correction to energy
\begin{equation}
E_n^{(2)} = \sum_{m\neq n} \frac{ | \langle m^{(0)}|V|n^{(0)}\rangle |^2 }{E_m^{(0)} - E_n^{(0)}}
\label{eq:second_order_correction_to_energy}
\end{equation}

\subsection{time-independent degenerate perturbation theory}
characteristic equation
\begin{equation}
det(H - \lambda I) = 0
\label{eq:characteristic_equation}
\end{equation}
where $\lambda$ are eigenvalues

\subsection{particle in a box}
infinite square well of size L, $0 \leq x \leq L$ (equation 4.15, page 93 \cite{LiboffQM})
\begin{equation}
\psi_n(x) = \sqrt{\frac{2}{L}} sin \left( \frac{n \pi x}{L} \right)
\label{eq:particle_in_box_wavefuction}
\end{equation}
where $n = 1,2,3, \ldots$ (equation 4.14, page 93 \cite{LiboffQM})
\begin{equation}
E_n = \frac{n^2 \pi^2 \hbar^2}{2 m L^2}
\label{eq:particle_in_box_energy}
\end{equation}
Whereas if $\frac{-L}{2} \leq x \leq \frac{L}{2}$
\begin{equation}
\begin{gathered}
\psi_m(x) = \sqrt{\frac{2}{L}} cos \left( \frac{m \pi x}{L} \right) \\
\psi_n(x) = \sqrt{\frac{2}{L}} sin \left( \frac{n \pi x}{L} \right)
\end{gathered}
\end{equation}
where $m = 1,3, \ldots$ and $n = 2,4, \ldots$\\
this can also be derived by shifting eq. \ref{eq:particle_in_box_wavefuction} by $\frac{L}{2}$.

\subsection{free particle}
for a free particle, assume $V=0$, then $H = \frac{p^2}{2m}$. The wave function is
\begin{equation}
\phi_k(r) = \frac{1}{(2 \pi)^{3/2}} e^{i \vec{k} \cdot \vec{r}}
\label{eq:free_particle_wave}
\end{equation}
and the energy is (equation 3.19, page 71 \cite{LiboffQM})
\begin{equation}
E_k = \frac{\hbar^2 k^2}{2m}
\label{eq:energy_free_particle_wave}
\end{equation}
and since $E = \hbar \omega$, then
\begin{equation}
\omega_k = \frac{\hbar k^2}{2m}
\end{equation}
time dependent wave function
\begin{equation}
\psi(\vec{r},t) = \langle \vec{r} | \psi(t) \rangle = \langle \vec{r} | \int d^3 k \ e^{-i \omega_k t} \psi(\vec{k},0) | \vec{k} \rangle
\label{eq:time_dependent_free_particle_wave_function}
\end{equation}

\subsection{quantum concepts}
\begin{itemize}
	\item the same spectrum (eigenvalues) occur in any basis
	\item Q: when is perturbation valid? A: when the perturbation term is
	much smaller than the unperturbed term
	\item a $\delta(x)$ perturbation in the middle of an $\infty$ square well
	has no effect on even states
\end{itemize}

\begin{tabular}{l|l}
Bosons                &     Fermions \\\hline
zero or integer spin  &  half-integer spin\\
symmetric wave functions & anti-symmetric wave functions\\
Example: photons         & Example: electrons, proton, neutron\\
more than one particle may occupy the same quantum state & Pauli exclusion principle: one particle per state\\
Bose-Einstien statistics & Fermi-Dirac statistics\\
\end{tabular}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{modern}

\begin{equation}
\frac{h}{2 \pi} = \hbar
\end{equation}

period and frequency
\begin{equation}
T = \frac{1}{f}
\end{equation}

Bragg condition (equation 3-38, page 144 \cite{TiplerMP}; equ 1 page 25 {KittelSS})
\begin{equation}
2 d \ sin \theta  = m \lambda
\end{equation}
where m is an integer.

angular frequency (equation 5-13a, page 211 \cite{TiplerMP})
\begin{equation}
\omega = 2 \pi f
\end{equation}

de~Broglie relations (equation 5-1, page 198 \cite{TiplerMP})
\begin{equation}
E = h f
\end{equation}

de Broglie wavelength (equation 5-2, page 198  \cite{TiplerMP}) (equation 3.10, page 70  \cite{LiboffQM})
\begin{equation}
\lambda = \frac{h}{p}
\end{equation}

frequency
\begin{equation}
f = \frac{c}{\lambda}
\end{equation}

(page 199,  \cite{TiplerMP})
\begin{equation}
E = pc = hf = \frac{hc}{\lambda}
\end{equation}

wavenumber (equation 5-13b, page 211  \cite{TiplerMP}) (equation 3.9, page 70  \cite{LiboffQM})
\begin{equation}
k = \frac{2 \pi}{\lambda}
\end{equation}

phase velocity (equation 5-14, page 211 \cite{TiplerMP})
\begin{equation}
v_{phase} = f \lambda
\end{equation}

group velocity (equation 5-22, page 211 \cite{TiplerMP}) (equation 3.62, page 83 \cite{LiboffQM})
\begin{equation}
v_{group} = \frac{d\omega}{dk}
\end{equation}

energy and frequency (equation 5-24, page 218  \cite{TiplerMP})
\begin{equation}
E = \hbar \omega
\end{equation}

momentum (equation 5-25, page 218  \cite{TiplerMP})
\begin{equation}
p = \hbar k
\end{equation}

uncertainty (equation 5-29, page 223  \cite{TiplerMP})
(problem 5.28, page 143 \cite{LiboffQM})
\begin{equation}
\begin{gathered}
\Delta x \Delta p \geq \frac{\hbar}{2} \\
\Delta E \Delta t \geq \frac{\hbar}{2}
\end{gathered}
\end{equation}

work function: energy need to remove an electron
(equation 3-37, page 139 \cite{TiplerMP})
\if\qualifyingyear T
(1999 A4b)
\fi
\begin{equation}
\phi = h f_{threshold} = \frac{h c}{\lambda_{threshold}}
\end{equation}

\subsection{Modern concepts}
\begin{itemize}
\item The mass of a bound system is greater than that of the individual
constituent particles due to contribution of binding energy. For
instance, $E_{hydrogen} = 13.6 eV$ is the binding energy for
hydrogen. (see page 86, \cite{TiplerMP})
\item 4 forces in order of decreasing strength and [force carrier particle]
\if\qualifyingyear T
(2007 B6)
\fi
%%% http://en.wikipedia.org/wiki/Fundamental_interaction#Overview
%%% http://hyperphysics.phy-astr.gsu.edu/hbase/forces/funfor.html
\begin{enumerate}
	\item strong [gluons]
	\item electromagnetic [photon]
	\item weak [W and Z bosons]
	\item gravity [graviton]
\end{enumerate}
\end{itemize}

\subsection{constants}
\begin{itemize}
\item radius of a simple atom is $\approx .5 \dot{A}$ (half an angstrom).
(Bohr radius, page 173 \cite{TiplerMP})
\item $hc = 1240 \ eV \ nm$
\end{itemize}

%\begin{comment}
\section{optics}

snell's law
\begin{equation}
n_1 sin(\theta_1)=n_2 sin(\theta_2)
\end{equation}
%\end{comment}

\newpage

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Symbol notations}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

See page 631-642 of \cite{ReifThermo}

$C_v \equiv$ specific heat for constant volume

$k_b \equiv$ Boltzmann constant

$\vec{E} \equiv$ electric field

$\vec{B} \equiv$ magnetic field

$\vec{D} \equiv$ electric displacement, equ 4.21, page 175 \cite{GriffithED}

$\vec{N} \equiv$ torque

$W \equiv$ work

$\vec{v} \equiv$ velocity

$V \equiv$ electrostatic potential [Volts]

$V \equiv$ volume

$U \equiv$ potential energy [Volts]

$\vec{P} \equiv$ polarization, page 166 \cite{GriffithED}

$I \equiv$ current [Amps]

$\vec{F} \equiv$ force [Newtons, $\frac{kg \cdot m}{s^2}$]

$E \equiv$ energy

$T \equiv$ kinetic energy

$T \equiv$ temperature

$L \equiv$ lagrangian

$L \equiv$ capacitance

$\vec{L} \equiv$ classical (orbital) angular momentum

$l \equiv$ quantum (orbital) angular momentum

$\vec{p} \equiv$ linear momentum

$\omega \equiv$ angular frequency

$z \equiv$ single particle partition function

$Z \equiv$ total partition function

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{thebibliography}{99}

\bibitem{GriffithED}
Griffith \textit{Intro to Electrodynamics}, (1999)

\bibitem{JacksonED}
Jackson \textit{Classical Electrodynamics}, (1999)

\bibitem{GoldsteinCM}
Goldstein \textit{Classical Mechanics}, (1980) (second edition)\\
Available from Aaron

\bibitem{ReifThermo}
Reif \textit{Fundamentals of statistical and thermal physics}, (1965)

\bibitem{KittelSS}
Kittel \textit{Solit State, Eigth edition}, (19)

\bibitem{ParrisQM}
Parris's book on \textit{Quantum mechanics}, (2008) \\
\htmladdnormallink{463 course page}{http://physics.mst.edu/classes/class_463notes.html}

\bibitem{LiboffQM}
Liboff \textit{Introduction to Quantum mechanics}, (2003) Fourth edition

\bibitem{MarionCM}
Marion \textit{Classical Dynamics of Particles and systems}, (1970) Second edition. \\
Available from Prof Waddill, Elizabeth

\bibitem{TiplerMP}
Tipler, Llewellyn \textit{Modern Physics}, (1999)

\end{thebibliography}

\end{document}