Describe the vibration-rotation spectra of a polyatomic linear molecule. What are P, Q and R branches? Draw rotational energy levels of the vibrational states v=0 and v=1 and perpendicular vibrational-rotational transitions of a linear polyatomic molecule.
Consider the one-dimensional wavefunction \psi(x)=Axe^{-kx}, (0\leq x\leq\infty;\,k>0)
(i) Calculate A so that \psi(x) is normalized.
(ii) Using Schrödinger's equation find the potential V(x) and energy E for which \psi(x) is an eigenfunction. (Assume that as x \to \infty, V(x) \to 0).
(i) Solve the radial part of the time-independent Schrödinger equation for a hydrogen atom. Obtain an expression for the energy eigenvalues.
(ii) What is the degree of degeneracy of the energy eigenvalues? What happens if the spin of the electron is taken into account?
Given that \sigma_x,\sigma_y,\sigma_z are Pauli spin operators, prove the following relationships:
(i) \sin(\sigma_x\varphi)=\sigma_x\sin\varphi
(ii) \cos(\sigma_z\varphi)=\cos\varphi
Use the uncertainty principle to estimate the ground state energy of a linear harmonic oscillator.
The Einstein theory of specific heat of solids gives the expression C_V=\frac{3Nk_Bx^2e^x}{(e^x-1)^2} where x=\frac{\theta_E}{T} with \theta_E as the Einstein temperature.
(i) Mention Einstein's assumptions in deriving it. Also obtain low- and high-temperature limiting expressions for it.
(ii) Give schematic plot of \dfrac{C_V}{3N k_B} versus \dfrac{T}{\theta_E} and comment on the validity of expressions in (i) in comparison with experiments.
Consider non-equilibrium situation for a system in which the population inversion has been achieved. Explain that such a system can be treated as if it has negative absolute temperature.
Show that both Fermi-Dirac and Bose-Einstein distributions reduce under certain condition in a form which gives the total number of particles as N = A\int_0^\infty \sqrt{\varepsilon}e^{-\beta\varepsilon}\,d\varepsilon where A is a constant and \beta = 1/k_{\mathrm B}T. Further show that this expression is just the same as that obtained from the Maxwellian speed distribution.
Show that the Helmholtz free energy of a system never increases in any isothermal-isochoric transformation.
Establish the relation \left(\frac{\partial T}{\partial V}\right)_P=-\left(\frac{\partial P}{\partial S}\right)_T and then derive \left(\frac{\partial C_P}{\partial P}\right)_T=-T\left(\frac{\partial^2 V}{\partial T^2}\right)_P. Hence show that the heat capacity C_P of an ideal gas is independent of pressure P.
What is a displacement current? Prove that lines of conduction current plus displacement current are continuous.
Assume \vec{E}=0 inside a perfect conductor. Elaborate on any other four electrostatic properties that arise from this property.
A resistance R and a lossless capacitor C are connected through a switch. The capacitor is charged to potential V_0, and the switch is closed at t = 0. Prove that the energy stored in the capacitor is equal to the energy dissipated in the resistor.
The volume between two concentric conducting spherical surfaces of radii a and b (a<b) is filled with an inhomogeneous dielectric with \epsilon=\epsilon_0/(1+cr), where c is a constant and r is the radial coordinate. A charge +Q is placed on the inner surface, while the outer surface is grounded. Determine-
(i) \vec{D} in the region a<r<b;
(ii) capacitance of the device;
(iii) polarization charge density in the region a<r<b;
(iv) surface polarization charge densities at r=a and r=b.
A diffraction grating of width 5 cm with slits of width 10^{-4} cm separated by a distance of 2\times10^{-4} cm is illuminated by light of wavelength 550 nm. What will be the width of the principal maximum in the diffraction pattern? Would there be any missing orders?
Derive an expression for intermodal dispersion for a multimodal step-index fibre.
Distinguish between Fresnel and Fraunhofer classes of diffraction. Show that the area of each Fresnel half-period zone is same.
A pulse of \lambda_0=600\,\mathrm{nm} and \Delta\lambda=10\,\mathrm{nm} propagates through a fibre which has a material dispersion coefficient of 50\,\mathrm{ps} per km per nm at 600\,\mathrm{nm}. Calculate the pulse broadening in traversing a 10\,\mathrm{km} length of the fibre. If the pulse width at the input of the fibre is 12\,\mathrm{ns}, what will be the pulse width at the output of the fibre?
Show that a travelling wave on the string, clamped on both the ends, undergoes a phase change of \pi. Hence obtain the time-independent form of the wave equation representing a standing wave on the string.
Use matrix method to obtain an expression for the focal length of a coaxial combination of two thin lenses having focal lengths f_1 and f_2 separated by distance d.
