Browse & Filter

 Consider a photon gas in two dimensions at temperature T  in area A. Find the energy density $u(\omega)$  as a function of temperature and various physical constants. Show that the total energy is proportional to $T^3$. ( you can assume that the internal degree of freedom is 1. )

In this lecture the method of canonical ensemble is applied to paramagnetism and ideal gases. For a paramagnetic substance the variation of paramagnetic susceptibility with temperature is derived. For an ideal gas Maxwell’s distribution of velocities is obtained using canonical

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For conservative systems, we show how solution of time dependent Schrodinger equation can be found by separation of variables. Explicit expression for the wave function at arbitrary time \(t\) is obtained in terms of energy eigenfunctions and eigenvalues. 

Using the identity $\Gamma(N\,+\,1)\,=\,N!=\,\int_0^\infty x^Ne^{-x} dx $
and write the integral in the form
$$ \int_0^\infty e^{Ng(x)}dx $$
Find the maximum of $g(x)$ ( which occurs say at $x_0$ ). Assume the integral
is dominated by the contribution from the neighbourhood of $x_0$ for large N. Expanding $g(x)$ up to second order in $(x-x_0)$ derive the Stirling's approximation ( for $n\rightarrow\,\infty$)
$$ \rm{ln}N!\,=\,N\rm{ln}N\,-\,N\,+\,\frac{1}{2}\rm{ln}(2\pi N)$$

( More exact formula , just for information, is
$$ N!\,=\,\left(\sqrt{2\pi}\right)e^{-N}N^{N+1/2}\left[1\,+\,\frac{1}{12N}\,+\,\frac{1}{288N^2}\,+\,\ldots\right]$$

By means of a few examples we show how dimensional analysis can be used to quickly estimate numerical values of expressions occurring in problems.