The wavelengths (\lambda) in the visible spectrum of H atom can be expressed by the empirical formula \lambda = \left( \frac{n_1^2}{n_1^2 - 4} \right) \cdot G where n_1 is an integer and n_1 = 3, 4 etc. and G is an empirical constant. Prove from the above, the wave number \bar{\nu} = R_H \left( \frac{1}{2^2} - \frac{1}{n_1^2} \right), \text{ where } R_H = \frac{4}{G}.
Obtain the term symbols for two singlet states and two triplet states for two electron atoms.
What are Pauli spin matrices ? Show that : (\vec{\sigma}\cdot\vec{A})(\vec{\sigma}\cdot\vec{B}) = \vec{A}\cdot\vec{B} + i\vec{\sigma}\cdot(\vec{A}\times\vec{B}) where \vec{\sigma} are the Pauli spin matrices and \vec{A} and \vec{B} are vector operators which commute with \vec{\sigma}, but do not necessarily commute with each other.
For a quantum mechanical system prove that all energy eigen-values E_n are real and if E_n \neq E_k, then the corresponding eigen functions are orthogonal.
On the basis of uncertainty principle calculate the size of Hydrogen atom.
(i) What angles do the \vec{L} \text{(vector)} make with the z-axis when l = 2 for an electron ? (ii) Determine the values of the total angular momentum for a 3d electron.
Solve the Schrodinger equation for a potential step function given by \begin{aligned} v(x) &= 0 \text{ for } x < 0 \\ &= v_0 \text{ for } x > 0 \end{aligned} and calculate the reflection and transmission coefficients. Show that for E < v_0 there is a finite probability of finding the particle in a classically forbidden region.
Solve the eigen value equation L^2 Y(\theta, \phi) = \lambda \hbar^2 Y(\theta, \phi) and obtain the eigen values and eigen functions of L^2.
Show that the elemental quantity of heat \delta Q is not a total differential.
Consider a gas of photons in equilibrium and contained in a volume V at a temperature T. Using the Bose-Einstein statistics, calculate the total energy of the photon gas.
Write down the expression for the Bose-Einstein distribution function and explain the meaning of the symbols used.
Prove that the Maxwell's equations in a medium contain the conservation of charge in differential form.
The electric field in a medium is given by \vec{E} = \vec{E}_0 e^{-\alpha z} \sin(kz - \omega t), where \vec{E}_0 is a constant vector with dimensions of the electric field. Prove that \vec{E} cannot have a component along the unit vector, \hat{z}, parallel to the z-axis. Here \alpha is a positive constant.
Consider an infinite line charge with charge density \rho\text{ coulomb/meter} located at a distance d\text{ meters} from a grounded conducting plane z = 0. Determine : (i) the magnitude of the potential V for z > 0 and z \le 0. (ii) the surface charge density induced on the conducting plane.
The current density in spherical co-ordinates is given by \vec{J} = \frac{1}{r^3} \left[ 2 \cos \theta \hat{r} + \sin \theta \hat{\theta} \right] \text{A/m}^2 where \hat{r} and \hat{\theta} are unit vectors. Calculate the amount of current passing through a hemisphere of radius 20\text{ cm}.
Consider an infinite current sheet with a uniform current density \vec{K} \text{ (Amp/m)}. Show that the magnetic field \vec{H} at a point away from the sheet is \vec{H} = \frac{1}{2} \vec{K} \times \hat{n} where \hat{n} is a unit normal vector directed from the current sheet to the point.
The plane y = 5 carries a current of density 10\\ \hat{z} \text{ (Amp/m)}. Calculate the value of the magnetic field \vec{H} at the point (0, 1, -5).
If the magnetic field \vec{B}, at a point with position vector \vec{r} is uniform, show that the corresponding vector potential \vec{A}(\vec{r}) is given by \vec{A}(\vec{r}) = -\frac{1}{2} \left[ \vec{r} \times \vec{B} \right].
Consider the Earth as a black body. Radiations from the Sun arrive at the surface of the Earth with an average intensity of S\text{ watts/m}^2. If the reflection coefficient of the Earth's surface is \alpha, determine the temperature of the Earth under equilibrium conditions.
An inductor of inductance 5\text{ H} is suddenly connected to a 10\text{ V} d.c. power supply through a resistor of 10\\ \Omega. After what time will the current in the circuit be 1/10\text{th} of its steady state value ?
