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lecture 16 notes converted to sort of coherent form.

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1 parent ca3daea commit b2bc966280b2ff37d1c34000fb370a0e19a1e590 @peeterjoot committed Mar 28, 2013
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@@ -7,8 +7,8 @@ confocal
ParametericPlot
Indistinguishability
Exponentiating
-Ficks
Fick's
+Ficks
cgs
phasor
linestyle
@@ -87,8 +87,8 @@ Eikonal
Goldstein's
GPS
colinear
-hoc
Hmm
+hoc
Peeter's
Benard
ijk
@@ -160,8 +160,8 @@ Hestenes's
parametrizations
imaginaries
reparametrize
-n'l'm
nlm
+n'l'm
PDE
Lut
quantized
@@ -188,8 +188,8 @@ arctan
entropic
invertible
pion
-outermorphism
OuterMorphism
+outermorphism
QFT
rescaling
spinors
@@ -310,8 +310,8 @@ df
LIGO
spacetime
dH
-BT
dj
+BT
Routhian
dk
dL
@@ -339,9 +339,9 @@ Prandtl
ia
isync
inferometer
-eV
ib
iB
+eV
orthonormalization
ic
ie
@@ -355,16 +355,16 @@ variates
im
Dekker's
ji
-jj
KE
+jj
elastostatics
amino
-ip
jk
+ip
indices
kj
-iu
kk
+iu
mc
mE
iz
@@ -377,8 +377,8 @@ Strang's
xyz
mk
resistive
-mn
kx
+mn
Eulerian
anticommutes
n'l
View
@@ -4732,6 +4732,11 @@ Generate figures for continuum mechanics problem set II figure 1. Using Show an
path => 'phy452/mathematica/lecture16DensityPlot.nb',
WHAT => qq(Plot the period boundary conditions density. Used Mathematica Map, pure functions, Placed PlotLegends, ToString, Text, text concatonation operator),
},
+{
+ DATE => 'March 27, 2013',
+ path => 'phy452/mathematica/largeTemperatureGaussianFermionDistributionIntegral.nb',
+ WHAT => qq(Lecture 16, Integral verification for thermal de Broglie lambda calculation.),
+},
# not all of these are committed to the repo. Some are, but are not described here.
#blogit/imageProcessingExperimentation.cdf
#blogit/streamSlowCurveFit.cdf
@@ -47,7 +47,7 @@ \section{Fermi gas}
gives
\begin{dmath}\label{eqn:basicStatMechLecture16:60}
-k_{\mathrm{F}} = (6 \pi \rho)^{1/3}
+k_{\mathrm{F}} = (6 \pi^2 \rho)^{1/3}
\end{dmath}
\begin{dmath}\label{eqn:basicStatMechLecture16:80}
@@ -58,7 +58,7 @@ \section{Fermi gas}
\Bk = \frac{2\pi}{L}(n_x, n_y, n_z)
\end{dmath}
-This is for periodic boundary conditions \footnote{I filled in details in the last lecture using a particle in a box, whereas this periodic condition was intented. We see that both achieve the same result}, where
+This is for periodic boundary conditions \footnote{I filled in details in the last lecture using a particle in a box, whereas this periodic condition was intended. We see that both achieve the same result}, where
\begin{dmath}\label{eqn:basicStatMechLecture16:120}
\Psi(x + L) = \Psi(x)
@@ -104,10 +104,10 @@ \section{Fermi gas}
Again
\begin{dmath}\label{eqn:basicStatMechLecture16:240}
-k_{\mathrm{F}} = (6 \pi \rho)^{1/3}
+k_{\mathrm{F}} = (6 \pi^2 \rho)^{1/3}
\end{dmath}
-\paragraph{Spin considerations}
+\makeexample{Spin considerations}{example:basicStatMechLecture16:1}{
%FIXME:
%\begin{itemize}
@@ -130,14 +130,15 @@ \section{Fermi gas}
\begin{dmath}\label{eqn:basicStatMechLecture16:300}
k_{\mathrm{F}} =
-\lr{\frac{ 6 \pi \rho }{2 S + 1}}^{1/3}
+\lr{\frac{ 6 \pi^2 \rho }{2 S + 1}}^{1/3}
\end{dmath}
and again
\begin{dmath}\label{eqn:basicStatMechLecture16:320}
\epsilon_{\mathrm{F}} = \frac{\hbar^2 \kF^2}{2m}
\end{dmath}
+}
\paragraph{High Temperatures}
@@ -165,22 +166,46 @@ \section{Fermi gas}
e^{\beta \mu}
\int \frac{d^3 \Bk}{(2 \pi)^3}
e^{-\beta \epsilon_k}
-\equiv
-e^{\beta \mu} \inv{\lambda^3}
+=
+e^{\beta \mu}
+\int dk \frac{4 \pi k^2}{(2 \pi)^3}
+e^{-\beta \hbar^2 k^2/2m}.
\end{dmath}
-where
+Mathematica (or integration by parts) tells us that
-\begin{dmath}\label{eqn:basicStatMechLecture16:400}
-\lambda = \frac{h}{\sqrt{2 \pi m \kB T}}.
+\begin{dmath}\label{eqn:basicStatMechLecture16:680}
+\inv{(2 \pi)^3}
+\int 4 \pi^2 k^2 dk
+e^{-a k^2} = \inv{(4 \pi a )^{3/2}},
\end{dmath}
-The density as a function of temperature is plotted in \cref{fig:lecture16:lecture16Fig6} for a few arbitrarily chosen values of the chemical potential $\mu$.
+so we have
-\imageFigure{lecture16Fig6}{Density as a function of temperature}{fig:lecture16:lecture16Fig6}{0.3}
+\begin{dmath}\label{eqn:basicStatMechLecture16:700}
+\rho
+=
+e^{\beta \mu} \lr{ \frac{2m}{ 4 \pi \beta \hbar^2} }^{3/2}
+=
+e^{\beta \mu} \lr{ \frac{2 m \kB T 4 \pi^2 }{ 4 \pi h^2} }^{3/2}
+=
+e^{\beta \mu} \lr{ \frac{2 m \kB T \pi }{ h^2} }^{3/2}
+\end{dmath}
-Here $\lambda$ is the \underlineAndIndex{thermal de Broglie wavelength}, $\lambda^3 \sim T^{-3/2}$.
+Introducing $\lambda$ for the \underlineAndIndex{thermal de Broglie wavelength}, $\lambda^3 \sim T^{-3/2}$
+\begin{dmath}\label{eqn:basicStatMechLecture16:400}
+\lambda \equiv \frac{h}{\sqrt{2 \pi m \kB T}},
+\end{dmath}
+
+we have
+\begin{dmath}\label{eqn:basicStatMechLecture16:720}
+\rho = e^{\beta \mu} \inv{\lambda^3}.
+\end{dmath}
+
+Does it make any sense to have density as a function of temperature? An inappropriately extended to low temperatures plot of the density is found in \cref{fig:lecture16:lecture16Fig6} for a few arbitrarily chosen numerical values of the chemical potential $\mu$, where we see that it drops to zero with temperature. I suppose that makes sense if we are not holding volume constant.
+
+\imageFigure{lecture16Fig6}{Density as a function of temperature}{fig:lecture16:lecture16Fig6}{0.3}
We can write
@@ -194,13 +219,13 @@ \section{Fermi gas}
\frac{\mu}{\kB T} = \ln \lr{ \rho \lambda^3 } \sim -\frac{3}{2} \ln T
\end{dmath}
-or
+or (taking $\rho$ (and/or volume?) as a constant) we have for large temperatures
\begin{dmath}\label{eqn:basicStatMechLecture16:460}
\mu \propto -T \ln T
\end{dmath}
-The chemical potential is plotted in \cref{fig:lecture16:lecture16Fig3}, whereas this $- \kB T \ln \kB T$ function is plotted in \cref{fig:lecture16TLogTPlot:lecture16TLogTPlotFig7}. The contributions to $\mu$ from the $\kB T \ln (\rho h^3 (2 \pi m)^{-3/2})$ term are dropped for the high temperature approximation. It's not entirely clear to me how we justify dropping the $\ln \rho$, since we see that $\rho = \rho(\mu, T)$ gets very small in the high temperature limit, but if all we are about is that $\mu$ is large and negative, then this isn't inconsisent.
+The chemical potential is plotted in \cref{fig:lecture16:lecture16Fig3}, whereas this $- \kB T \ln \kB T$ function is plotted in \cref{fig:lecture16TLogTPlot:lecture16TLogTPlotFig7}. The contributions to $\mu$ from the $\kB T \ln (\rho h^3 (2 \pi m)^{-3/2})$ term are dropped for the high temperature approximation. %It's not entirely clear to me how we justify dropping the $\ln \rho$, since we see that $\rho = \rho(\mu, T)$ gets very small in the high temperature limit, but if all we are about is that $\mu$ is large and negative, then this isn't inconsisent.
\imageFigure{lecture16Fig3}{Chemical potential over degenerate to classical range}{fig:lecture16:lecture16Fig3}{0.3}
\imageFigure{lecture16TLogTPlotFig7}{High temp approximation of chemical potential, extended back to $T = 0$}{fig:lecture16TLogTPlot:lecture16TLogTPlotFig7}{0.3}
@@ -220,10 +245,13 @@ \section{Fermi gas}
\end{dmath}
For a Fermi gas at $T = 0$ we have
+%FIXME: \eqnref{eqn:basicStatMechLecture15:400}
\begin{dmath}\label{eqn:basicStatMechLecture16:560}
E
= \sum_\Bk \epsilon_k n_k
+= \sum_\Bk \epsilon_k \Theta(\mu_0 - \epsilon_k)
+= \frac{V}{(2\pi)^3} \int_{\epsilon_k < \mu_0} \frac{\hbar^2 \Bk^2}{2 m} d^3 \Bk
= \frac{V}{(2\pi)^3} \int_0^{\kF} \frac{\hbar^2 \Bk^2}{2 m} d^3 \Bk
= \frac{V}{(2\pi)^3}
\frac{\hbar^2}{2 m}
View
@@ -10,8 +10,8 @@
#~/bin/lgrep landauSection11Problem2b.tex | tee o ; . ./o
#~/bin/lgrep pipeFlowConstPressureGradient.tex | tee o ; . ./o
#~/bin/lgrep t.tex | tee o ; . ./o
-#~/bin/lgrep basicStatMechLecture16.tex | tee o ; . ./o
-#perl -p -i ./p basicStatMechLecture16.tex
+~/bin/lgrep basicStatMechLecture16.tex | tee o ; . ./o
+perl -p -i ./p basicStatMechLecture16.tex
#perl -p -i ./p energyProbabilityPathriaQuestion.tex
#~/bin/lgrep basicStatMechProblemSet6Problem1.tex | tee o ; . ./o
#~/bin/lgrep basicStatMechLecture19.tex | tee o ; . ./o
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