(i) Explain why a neutral particle such as neutron possesses a finite value of magnetic moment, and how it is determined. (ii) Write down the electronic configuration of {}^7\text{Li}, \, {}^{13}\text{C} \quad \text{and} \quad {}^{25}\text{Mg} in the ground state of the nuclear shell model.
Which conservation laws are obeyed in nuclear reactions? Explain the significance of Q value of a nuclear reaction.
Derive an expression for the spin-orbit interaction energy in one electron system. Calculate the energy separation between the levels {}^2P_{1/2} and {}^2P_{3/2}.
Calculate the change in rotational constant (B_v) when deuterium is substituted for hydrogen in a hydrogen molecule.
Explain the nature, origin and significance of 21 cm hydrogen line.
A diatomic gas is found to have a number of absorption bands in the ultraviolet region. Each band is also observed to have a fine structure. How can the observed spectrum be explained and analyzed?
(i) Treating CH molecule as an united atom, derive its resultant molecular electronic states. (ii) Calculate the moment of inertia of HCl molecule from the expression u_{\text{HCl}} = 20.68 (J+1), where J = 0, 1, 2, \dots
(i) A sample is irradiated by a 5000 Å\ radiation to give a Raman line at 5050.5 Å. Calculate the Raman frequency. (ii) Explain, both red and violet degraded bands have been observed in electronic band systems, but rotation-vibration spectra show bands degraded to red only.
Write the electronic configuration of mercury (Z = 80). Obtain the spectral term for the normal and the first excited configuration of the atom.
Solve the one dimensional Schrodinger wave equation with potential: V(x) = \begin{cases} 0 & \text{for } x < -a \\ V_0 & \text{for } -a < x < a \\ 0 & \text{for } x > 0 \end{cases}
Derive an expression for the electrical conductivity of metals on the basis of free electron theory of metals.
Distinguish between a classical and a quantum mechanical harmonic oscillator. Explain the existence of zero point energy.
(i) The wave function of a particle is . \psi(x) = A \exp \left( -\frac{x^2}{a^2} + i k_0 x \right) Find the expectation values of position (x) and momentum (p) for the particle. (ii) A 200 eV increase in the energy of an electron changes its De Broglie wavelength by a factor of two. Calculate the initial De Broglie wavelength of the electron.
Describe Carnot cycle and show that efficiency is given by \eta = \frac{Q_1 - Q_2}{Q_1} = \frac{T_1 - T_2}{T_1} where the symbols have their usual meaning.
Derive the Bose-Einstein distribution for an ideal gas.
Discuss the phenomenon of Bose-Einstein condensation. Obtain the expression for the condensation temperature. Briefly comment on observation of Bose-Einstein condensate.
Using Kirchhoff's laws find currents in each branch of the circuit shown in the following diagram.
A bulb filament is constructed from a tungsten wire of length 2\text{ cm} and diameter 50\text{ }\mu\text{m}. It is enclosed in a vacuum bulb. What temperature does it reach when it is operated at a power of 1 watt? Given:
(i) Emissivity of tungsten \varepsilon = 0.4
(ii) Stefan's constant \sigma = 5.67 \times 10^{-8}\text{ watt/m}^2\text{ K}^4.
What are vector and scalar potentials for the electromagnetic field? Are they unique? Explain what are Coulomb's and Lorentz gauges. Derive the electromagnetic wave equation in Lorentz gauge and show that it is equivalent to Maxwell's equation.
Why did Maxwell have to introduce the idea of displacement current? Derive the wave equation from Maxwell's laws. Obtain Fresnel's formula for reflection and transmission coefficients of the electric vector when it is perpendicular to the plane of incidence.
