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An ideal Bose-Einstein gas is made up of particles, each of it is a two level system ( Internal energies are \(0\) and \(\Delta\), besides the kinetic energy.). Assuming \(N/V\) is a constant, obtain the transcendental equation for the condensation temperature. Will the temperature higher or lower than for the system with no internal degrees of freedom.

HSMani

Obtain \(C_v\) for the ideal Bose gas for \(T < T_c\) and \(T > T_c\). 

HSMani

Consider a three particle system at temperature T in three levels having energies $0,\,,\,\epsilon\,,\,2\epsilon $. Write the canonical partition function for

(a) Classical particles implementing Gibb's ad hoc rule
(b) Bose Einstein statistics and
(c) Fermi Dirac statistics.

Calculate the average energy for all the three cases and comment on the high temperature and low temperature limit.

KPN

Consider N classical particles of mass M confined to a volume V. Besides the kinetic energy ($\frac{p^2}{2M}$ for each particle) , there is a two particle interaction between the particles given by $$ V(r_{ij})\,=\,\frac{K}{r_{ij}^l} $$ Show that the canonical partition function scales as $$ Z_c(\xi T, \xi^{-3/l} V)\,=\,\xi^{3N(\frac{1}{2}-\frac{1}{l})}Z_c(T,V) $$ where $\xi$ is the scaling parameter. Using the above result show that the internal energy $U$ is given by $$ U\,=\,c_1pV\,+\,c_2 NkT $$ where $$c_1\,=\,\frac{3}{l}\qquad c_2\,=\, \frac{3(l\,-\,2)}{2 l}$$

HSMani

1. {Show that for ideal bosons in an open system, the total number of particles is given by $$N =\frac{\lambda}{1-\lambda}+\frac{V}{\Lambda^3}\ g_{3/2}(\lambda).$$ In the above $\lambda=\exp\left(\frac{\mu}{k_BT}\right)$ is fugacity. $\mu$ is chemical potential. $\Lambda$ is thermal/quantum wavelength given by $\Lambda = {\displaystyle \frac{h}{\sqrt{2\pi mk_BT}}.}$ Also, $g_{3/2}(\lambda)={\displaystyle \sum_{k=1}^\infty\ \frac{\lambda^k}{k^{3/2}}.}$
2. Postulate that $0\ \le\ \lambda\ \le 1$ can be as large as $1-{\displaystyle\frac{a}{N}}$ where $a$ is constant. 

1. Show that $$a=\frac{\rho\Lambda^3}{\rho\Lambda^3-g_{3/3}(\lambda)}.$$ The number of bosons in the ground state is denoted by $N_0=\lambda/(1-\lambda)$.
2.  Derive an expression for $N_0/N$ in terms of $a$ and show that \begin{eqnarray*} \frac{N_0}{N}=\left\{\begin{array}{lll} 1-\left(\frac{T}{T_C}\right)^{3/2} & \ {\rm for}\ T\ \le \ T_C\ ,\\[10mm] \approx 0 & \ {\rm for}\ T\ > T_C\ , \end{array}\right. \end{eqnarray*} where the Bose condensation temperature, $T_C$ is given by the relation, $$\frac{N}{V}\left(\frac{h}{\sqrt{2\pi mk_BT_C}}\right)^3=g_{3/2}(\lambda=1)=\zeta(3/2)\approx 2.612.$$

KPN

Show that in the canonical ensemble formalism, the entropy $S$ of the system is related to the partition function $Q$ as given below. $$S=k_B\left[\ln Q+T\left(\frac{\partial\ln Q}{\partial T}\right)_V\right].$$}

KPN

 Consider a container of volume V with N non interacting particles. Consider a small volume $v$ inside it. Making the assumption that the particles have equal probability to be anywhere inside the container find the probability $P(n)$ of the volume $v$ containing $n$ particles. Maximize the expression and show that the most probable value is $\frac{N}{V}v$. Find the width of the distribution. If $N\,=\,10^{22}$ , what is the width of the distribution. If the deviation of the density should be less that $1$ percent what should be the volume $v$?

HSMani

{A particular system obeys the fundamental equation, $$U=A\ \frac{N^3}{V^2}\exp\left( \frac{S}{Nk_B}\right), $$ where $A$ (joule metre$^2$) is a constant. Initially the system is at $T=317.48$ kelvin, and $P=2\times 10^5$ pascals. The system expands reversibly until the pressure drops to a value of $10^5$ pascals, by a process in which the entropy does not change. What is the final temperature\footnote{HINT: Take partial derivatives of $U$ with respect to $S$ and $V$ and get $T$ and $P$ respectively. Find a relation between $P$ and $T$ when entropy does not change.

KPN

Find the density matrix of an ensemble of spin half prepared with 60 percent have $\sigma_z\,=\,+1$ and 40 percent have $\sigma_x\,=\,1$. Use the eigenstates of $\sigma_z$ as the basis. If the system is subjected to a magnetic field ( from time $t\,=\,0$) find the density matrix as a function of time. Assume the Hamiltonian is given by $H\,=-\,\mu B_0\sigma_z $ where $\mu\,B_0$ are a constants.

HSMani