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\coursename{Statistical and Thermal Physics}
\courseno{Phys 362}
\semester{Spring 2018}

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\hwnumber{18}

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\dueday{17}
\duemonth{April}
\dueyear{2018}
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%\problemsettitle{Quantum Theory: Problems}


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\textauthor{Gould and Tobochnik}
\texttitle{Statistical and Thermal Physics}

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\begin{document}

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\startproblemlist % starts the assignment list

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\startproblem{Thermodynamics of spin systems}
%   \label{xx}  % optional label

Consider a system of $N$ spin-1/2 particles, each with magnetic dipole moment $\mu$ and in a magnetic field of magnitude $B$. Let $n_+$ the number of particles with spin up. The entropy of the system is
%
\[ S = k \ln{\left[\Omega(n_+)\right]}\]
%
where $\Omega(n_+)$ is the multiplicity of the macrostate with $n_+$ particles with spin up. LONG but not HARD

\startproblemparts
  \item Determine an expression for the energy of the system given that there are $n_+$ particles with spin up.
  \item Show that the temperature of the system satisfies
	%
	\[ \frac{1}{T} = \frac{k}{2 \mu B}\, \ln{\left[ \frac{N - E/\mu B}{N + E/\mu B}\right]}\] 
	%
	and use  the result to determine an expression for the energy equation of state $E=E(T,N).$ 
	\item List the range of possible values of $E$ (in terms of $\mu B$ and N) and plot the temperature as a function of $E$ over the entire range. Describe when the temperature is positive and when it is negative. 
  \item Determine the probability with which a single particle is in the spin up state in terms of $T,N, \mu$ and $B$. Repeat this for spin down.
  \item Using these probabilities, determine the mean energy $\overline{E}$ for a single particle. How does this compare to the expression for the energy, $E$, of the entire system that you obtained earlier?
\stopproblemparts 


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\startproblem{Chemical potential for a system of spin-1/2 particles}
%   \label{xx}  % optional label

Consider a system of $N$ spin-1/2 particles, each with magnetic dipole moment $\mu$ and in a magnetic field of magnitude $B$. Let $n_+$ the number of particles with spin up. LONG MEDIUM

\startproblemparts

  \item Determine an expression for the chemical potential of the system. Express your answer in terms of E,N,B,T and $\mu$. Do not expand out T, or E just call it T or E.
  \item Determine conditions on $n_+$ that give a positive or a negative chemical potential. Your answer should be in terms of N and $n_+$. There are two regimes to answer. One for positive T and one for negative T. 
  \item Suppose that $n_+ = N/3.$ Determine an expression for the chemical potential. If particles are added to the system in such a way that this ratio is preserved, will the system gain or lose energy?
\stopproblemparts 

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\startproblem{Einstein solid: low temperature limit}
%   \label{xx}  % optional label

For any Einstein solid %
					\[ \begin{split} \Omega(N,q) & \approx \left(1 + \frac{q}{N-1}\right)^{N-1}
					                           \left(1 +\frac{N -1}{q}\right)^{q}
																		 \sqrt{\frac{N+q -1}{2 \pi (N-1)q}}\\
																		 & \approx \left(1 + \frac{q}{N}\right)^{N}
					                           \left(1 +\frac{N }{q}\right)^{q}
																		 \sqrt{\frac{N+q}{2 \pi N q}}\\
							\end{split}
					\] 
					%
					 whenever $ q\gg 1$ and $N \gg 1.$

In the low temperature limit $q \ll N.$ LONGISH MEDIUM
\startproblemparts

  \item Show that if $q \ll N$ then
	%
	\[ \Omega(N,q) \approx \left(\frac{eN}{q} \right)^q \frac{1}{\sqrt{2 \pi q}} \] 
	and 
	%
	\[ S = kq\left[ \ln{\left(\frac{N}{q}\right)} -1\right].\] You will discard one term in the entropy formula as q is large.
  \item Use this to determine an expression for the temperature of the solid in terms of energy. If possible, invert the expression to get energy in terms of temperature.
\stopproblemparts 


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\startproblem{Partition functions for artificial systems}
%   \label{xx}  % optional label

Consider three systems, each with a single particle at temperature $10^5\unit{K}$. System A has two states, one with energy $0\unit{eV}$ and the other with $10\unit{eV}.$ System B has three states, one with energy $0\unit{eV}$ and the other two each with $10\unit{eV}.$ System C has four states, two each with energy $0\unit{eV}$ and the other two each with $10\unit{eV}.$ Let $Z_A$ be the partition function for system A, etc,\ldots . EASY

\startproblemparts

  \item Which of the following is true? Explain your answer.


  \startmulti[-0.5ex]
		 \item $Z_A = Z_B = Z_C$.
		 \item $Z_A = Z_B \neq Z_C$.
		 \item $Z_A = Z_C \neq Z_B$.
		 \item $Z_B = Z_C \neq Z_A$.
		 \item None of the partition functions are the same.
	\stopmulti
	
  \interrupt{Now suppose that system A has a single particle and two states, one with energy $0\unit{eV}$ and the other with $10\unit{eV}.$ System B has two distinguishable particles and each could be in one of the two states of system A.} % Space for comments

  \item Is $Z_A = Z_B$ in this case? Explain your answer.
\stopproblemparts 

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%%% Problem XX, page yy, label

\textproblem{4.24}{211}{ch4pr24}EASY



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\startproblem{Interstellar heat bath}
%   \label{xx}  % optional label

The molecule  \textsf{CN} is often found in interstellar molecular clouds. This molecule has many states associated with rotational motion. Observations indicate that about $10\%$ of all such molecules are in any one of the first three excited states, each of which has the same energy, $\sci{4.7}{-4}\unit{eV}$, above the ground state. Assuming that the molecules are in thermal equilibrium with a heat bath, determine the temperature of the heat bath. 

This addresses a famous issue in cosmology and is discussed in detail in P.~Thaddeus,  \textit{Annual Review of Astronomy and Astrophysics,} vol.~10, p.~305 (1972).



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\startproblem{Uncertainty in energy for the canonical ensemble.}
%   \label{xx}  % optional label

Show that, for the canonical ensemble, the uncertainty in the energy satisfies
%
\[ \sigma_E^2 = - \frac{\partial \overline{E}}{\partial \beta}.\] 

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\startproblem{Five state system}
%   \label{xx}  % optional label

Consider a particle which could be in one of five possible states; the energies of these states are $0,\epsilon,2\epsilon,3\epsilon$ and $4\epsilon$. 

\startproblemparts
  \item Suppose that $\epsilon=0.050\unit{eV}$ and $T=100\unit{K}$. Determine the probabilities with which the particle could be in each state and plot a histogram of these.
  \item Suppose that $\epsilon=0.050\unit{eV}$ and $T=500\unit{K}$. Determine the probabilities with which the particle could be in each state and plot a histogram of these.
  \item Suppose that $\epsilon=0.20\unit{eV}$ and $T=500\unit{K}$. Determine the probabilities with which the particle could be in each state and plot a histogram of these.
	\item Use your results to describe qualitatively the appearance of the entire histogram as the temperature increases. 
	\item Use your results to describe qualitatively the appearance of the entire histogram as the energy gap, $\epsilon$, increases. 
\stopproblemparts 


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\startproblem{Systems with degenerate energy levels}
%   \label{xx}  % optional label

Some systems have degenerate energy levels, meaning that there are different states that have the same energy. Consider a system that has five states with energies $0, \epsilon, \epsilon, 2\epsilon, 2\epsilon$ (electrons in a hydrogen atom have this type of degeneracy). If $kT = 2\epsilon,$ determine the probabilities with which the system will have energy $0,$ energy $\epsilon$ and energy $2\epsilon.$



\stopproblemlist %ends the assignment list

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% Graph Sheet

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    \psplot{0.325}{9.2}{1 x div 3 mul} 

    % axes
    \psaxes[Dx=2,Dy=2]{->}(10.5,10.5)

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    \uput[180](0,5){$P$ in $\unit{N/m}^2$}
    \uput[270](5,-0.5){$V$ in $\unit{m}^3$}
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  \end{center}
  \caption{Question 1}
  \label{fig:graph1}

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