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Write down Schr"{o}dinger equations for a particle of energy E < V_0, incident on a step potential height V_0. Solve them to find out the transmission and reflection coefficients in terms of k and k', where k = \sqrt{\frac{2mE}{\hbar^2}} and k' = \sqrt{\frac{2m(V_0 - E)}{\hbar^2}}. Show that in this case, there is a finite probability of finding the particle in a classically forbidden region.
Show that the spherical harmonics Y_{lm}(\theta, \phi) are simultaneous eigenfunctions of L^2, L_z and L_z^2. What are their corresponding eigenvalues?
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(i) Enumerate the possible values of quantum numbers j and m_j for state in which l = 2 and s = 1/2.
(ii) Draw the corresponding vector model diagram.
The coefficient of viscosity of helium at 27^\circ\mathrm{C} is 2 \times 10^{-5}\ \mathrm{kg\,m^{-1}\,s^{-1}}. Calculate \begin{qroman}\item the average speed and \item the diameter of a helium molecule,\end{qroman} if it is assumed that the gas obeys Maxwell-Boltzmann distribution. Given Boltzmann constant k_{\mathrm B} = 1.38 \times 10^{-23}\ \mathrm{J\,K^{-1}} and mass of helium atom = 6.67 \times 10^{-27}\ \mathrm{kg}.
N particles obeying Classical Statistics are distributed among three states having energies \varepsilon_1 = 0, \varepsilon_2 = k_{\mathrm B}T and \varepsilon_3 = 2k_{\mathrm B}T, where k_{\mathrm B} is Boltzmann constant. If the total equilibrium energy of the system is 1000k_{\mathrm B}T, calculate the value of N.
Discuss the significance of Saha's ionisation formula in classification of stars.
For a completely degenerate BE gas, discuss the condition for onset of BE condensation.
A thermally insulated ideal gas is compressed quasi-statically from an initial state with volume V_0 and pressure P_0 to a final state of volume V_f and pressure P_f. Show that the work done on the gas in the process is given by W=\frac{C_V}{R}\left(P_fV_f-P_0V_0\right) where C_V and R having standard meanings.
What is transport phenomenon ? Obtain the expression for the coefficient of viscosity. Discuss its temperature dependence.
Establish Van der Waals equation of state for a real gas. Deduce expressions for critical constants and show that critical coefficient is independent of the nature of gas.
Calculate the net change in entropy when 10\text{ g} water at 60^\circ\text{C} is mixed with 30\text{ g} water at 20^\circ\text{C}.
State Dulong-Petit's law and discuss its limitations in explaining heat capacities of solids. How were these efficiencies overcome by Debye ?
In Leh, temperature of ice on a cold winter night is measured as -20^\circ\mathrm{C}. Calculate the change in entropy when 1\ \mathrm{kg} of ice is converted into steam at 100^\circ\mathrm{C}. Given specific heat capacity of ice is 500\ \mathrm{cal\,kg^{-1}\,K^{-1}}, latent heat of ice is 3.36\times10^5\ \mathrm{J\,kg^{-1}}, latent heat of steam is 2.26\times10^6\ \mathrm{J\,kg^{-1}} and J=4.2\ \mathrm{J\,cal^{-1}}.
The vapour pressure, in mm of Hg, of a substance in solid state is given by the relation \ln p=23.03-\dfrac{3754}{T}, where T is in Kelvin. The vapour pressure, in mm of Hg, of the substance in liquid state is given by the relation \ln p=19.49-\dfrac{3063}{T}. Calculate \begin{qroman}\item the coordinates of the triple point, and \item the latent heat of vaporisation at the triple point.\end{qroman} Take Gas constant R=8.314\ \mathrm{J\,mol^{-1}\,K^{-1}}.
Define a black body. How can we realise a black body in practice ? Derive expression for Planck's radiation law.
Show that Wien's law and Stefan-Boltzmann law are limiting cases of Planck's radiation law.
(i) Considering an isotropic, linear, non-conducting, non-magnetic and inhomogeneous dielectric medium with \vec{D}=\epsilon\vec{E}=\epsilon_0n^2(x,y,z)\vec{E}, show that the electromagnetic wave equation for the field \vec{E} is given by \nabla^2\vec{E}+\vec{\nabla}\left(\frac{1}{n^2}\vec{\nabla}n^2\cdot\vec{E}\right)-\mu_0\varepsilon_0n^2\frac{\partial^2\vec{E}}{\partial t^2}=0. \setcounter{enumi}{1}
(ii) Write down the scalar equation for E_x from the above equation.
(iii) Interpret physically the situation if we move from homogeneous to an inhomogeneous medium.
(iv) Obtain the similar vector equation for the magnetic field \vec{H} in inhomogeneous medium.
A plane electromagnetic wave is given by E_z = a \cos \omega x \cos \omega t and H_y = - a \sin \omega x \sin \omega t. Evaluate the instantaneous value of the Poynting vector \vec{S} and show that <\vec{S}> = 0.
Sea water has resistivity 0\cdot 3\ \Omega\text{m} and its dielectric constant is 81. Calculate the ratio of the amplitudes of the conduction and polarisation current intensities when the applied field is oscillating at 100\text{ MHz}.