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[QUE/SM-03005] Statistical Mechanics

Node id: 3233page

Consider an isolated system of $N$ non-interacting particles occupying two states of energies $-\epsilon$ and $+\epsilon$. The energy of the system is $E$. Let $x=\displaystyle{\frac{E}{N\epsilon}}.$

  1. Show that the entropy of the system is given by\footnote{HINT : Let $n_1$ and $n_2$ denote the number of particles in the two states of energy $-\epsilon$ and $+\epsilon$ respectively. We have $\widetilde{\Omega}=N!/(n_1!n_2!)$; $S=k_B\ln\widetilde{\Omega}$; Calculate $n_1$ and $n_2$ by solving : $n_1+n_2=N$ and $n_2\epsilon-n_1\epsilon=E$.} $$ S(E)=Nk_B\left[\left(\frac{1+x}{2}\right)\ln\left(\frac{2}{1+x} \right)+\left(\frac{1-x}{2}\right)\ln\left(\frac{2}{1-x}\right)\right] $$
  2. Show that ${\displaystyle \beta=\frac{1}{k_BT}=\frac{1}{2\epsilon}\ln\left(\frac{1-x}{1+x}\right)}$
 kapoor's picture 22-03-04 07:03:33 n

[QUE/SM-04005] Statistical Mechanics

Node id: 3238page

The canonical partition function of a system of $N$ hypothetical particles each of mass $m$, confined to a volume $V$ at temperature $T$ is given by, $$Q(T,V,N) = V^N\left(\frac{2\pi k_B T}{m}\right)^{5N/2}.$$ Determine the equation of state of the hypothetical system. Also find $C_V$ - heat capacity at constant volume. Identify the hypothetical system. How many degrees of freedom does each particle of the hypothetical system have?

 kapoor's picture 22-03-04 07:03:29 n

INFO --- Description and Up fields for Navigation

Node id: 5321page
 AK-47's picture 22-03-04 06:03:16 n

[NOTES/QM-18005]Born Approximation

Node id: 4833page

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qm-lec-18005

 AK-47's picture 22-03-03 22:03:07 y

[NOTES/QM-16001] Angular Momentum Algebra — Coordinate Representation

Node id: 4779page

$\newcommand{\DD}[2][]{\frac{d^2 #1}{d^2 #2}}$ 
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qm-lec-16001

 AK-47's picture 22-03-03 22:03:14 y

[Solved/QM-19002] Quantum Mechanics

Node id: 2257page
 kapoor's picture 22-03-03 13:03:40 n

[Short-Examples/QM-06003] ---- Probability and Average Values

Node id: 1963page
 kapoor's picture 22-03-03 12:03:10 n

Problem Sets --- Classical Mechanics

Node id: 2106curated_content
 kapoor's picture 22-03-03 12:03:45 n

[QUE/PDE-01002]

Node id: 1820page
 kapoor's picture 22-03-03 12:03:07 n

Solved/PDE-01003 Partial Differential Equations

Node id: 1821page
 kapoor's picture 22-03-03 12:03:25 n

Solved/PDE-01001

Node id: 1812page
 kapoor's picture 22-03-03 12:03:05 n

Six Lectures and a Problem Set Given on Lie Algebras --- SERC School (2014)

Node id: 4936curated_content

Lectures Given at SERC School (2014)

BITS PIlani Hyderabad

 

Page to View and Download All Resources

 kapoor's picture 22-03-03 12:03:18 n

Quantum Field Theory --- Notes for Lectures and Problems --- [QFT-MIXED-LOT]]

Node id: 4685collection

 

 AK-47's picture 22-03-01 20:03:46 n

Concepts in Thermodynamics

Node id: 5280multi_level_page
 ashok's picture 22-03-01 06:03:30 n

[Solved/PDE-01004] Partial Differentaial Equations

Node id: 1833page
 kapoor's picture 22-02-27 17:02:12 n

[Solved/EM-04001] A conducting grounded sphere and a point charge

Node id: 2205page
 kapoor's picture 22-02-27 17:02:19 n

[Solved/EM-04002] Two charged spherical shells separated by large distance

Node id: 2221page

Two spheres of conducting material have radii 1 cm and 10 cm and carry charge  \(100\) coul and \(1\) coul, respectively. The separation between the centers of the two spheres is 10 m.

  1.  What is the potential of  each sphere?
  2.  Find the charges on the two spheres, if the two spheres are connected by a fine wire.
  3.  Are the values obtained by you exact or approximate(give reasons)?

 

 kapoor's picture 22-02-27 17:02:37 n

[Solved/EM-05001] Electromagnetic Theory

Node id: 2363page

Question:
A dielectric sphere contains a dipole \(\vec{p}_0\). Find the net dipole moment of the system.

Solution:
The dielectric gets polarized and the dipole and acquires a polarization \(\vec{P}\). The dipole moment of the dielectric is
\begin{eqnarray}
\vec{p} &=& \int_V \vec{P} d^3r   = \int_V \vec{D}-\epsilon_0 \vec{E} d^3 r\\      &=& \int_V\epsilon_0(\kappa-1) \vec{E} d^3r  = \epsilon_0(\kappa-1) \int_V\vec{E} d^3r \end{eqnarray}
For arbitrary charge distribution the average of electric field, \( \int_V\vec{E} d^3r\), over sphere is \(-\dfrac{\vec{p}_\text{tot}}{3\epsilon_0}\).  Hence we get

\[ \vec{p}= - \frac{(\kappa-1)}{3} \vec{p}_\text{tot}\]

using \(\vec{p}_\text{tot}= \vec{p}_0 + \vec{p}\), we get

\begin{eqnarray}\vec{p}_\text{tot} &=& \vec{p}_0 -\frac{(\kappa-1)}{3} \vec{p}_\text{tot}\\\text{or } \vec{p}_\text{tot} &=&\frac{3}{\kappa+2}  \vec{p}_0\end{eqnarray}

Zangwill

 kapoor's picture 22-02-27 17:02:33 n

[QUE/CM-01005#Solu] Periodic Motion $V(x) = V_0 \left( e^{-2x/\alpha} - 2 e^{-x/\alpha}\right) $

Node id: 1912page

Question Plot the Morse potential
$$V(x) = V_0 \left( \exp(-2x/\alpha) - 2 \exp(-x/\alpha)\right) $$
and find the period of small oscillations for a particle having energy
$E$ and moving in fore field described by the Morse potential.}

 kapoor's picture 22-02-27 17:02:21 n

[Solved/QM-20002] Quantum Mechanics --- Solved Problem

Node id: 2264page
 kapoor's picture 22-02-27 17:02:01 n

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