(i) Write the commutation relations for the position variable x and the momentum components p_x, p_y and p_z. Explain the physical significance of these relations. (ii) Calculate the de Broglie wavelength of an electron moving with a kinetic energy of 1 MeV.
Discuss WKB approximation and apply the same to determine the transition probability for leakage through a potential barrier.
(i) The ground state wavefunction of a linear simple harmonic oscillator is \psi = A \exp \left( - \frac{\alpha^2 x^2}{2} \right) Calculate the constant A and the average values of x^2 and x. Given that \int_0^\infty e^{-x^2} = \frac{\pi^{1/2}}{2} (ii) How can the pure rotation spectrum of \text{H}_2 molecule be observed ? If the bond length of \text{H}_2 molecule is 0 \cdot 07417 \text{ nm}, what would be the spacing of lines in its spectrum ?
Consider a particle of mass m in an infinite one dimensional potential well of width a. The particle is found in the state given by \psi (x) = c \left[ \sin \frac{\pi x}{a} + \frac{1}{2} \sin \frac{2 \pi x}{a} \right] (i) Calculate c. (ii) If a measurement of energy is made, what are the possible results and what are the probabilities for each one of them ?
(c) Discuss the differences in the assumptions underlying Einstein and Debye theories of specific heat C_v. Give schematic plots of C_v versus reduced temperature for these theories and elucidate the differences therein. Elaborate the meaning of the “law of corresponding states” for these plots.
(c) Show that the chemical potential of a system is an intensive quantity and is a function of temperature and pressure only.
(b) Utilize the above expression to elaborate the concept of the Bose-Einstein condensation and discuss that the phenomenon explains qualitatively the properties in the low-temperature phase of liquid {}^4\text{He}.
(d) Consider the expression C_P - C_V = - T \left( \frac{\partial V}{\partial T} \right)_P^2 \left( \frac{\partial P}{\partial V} \right)_T and give reasoning regarding the values of T when C_P = C_V for water. Also, evaluate C_P - C_V for a vander Waals gas to elaborate that its value is larger for any real gas as compared to an ideal gas.
(b) Show that the electric and magnetic field vectors, \vec{E} and \vec{B}, in plane electromagnetic waves are mutually perpendicular in a plane normal to the direction of propagation. How are phases of \vec{E} and \vec{B} related to each other ?
(b) Describe physical significance of the displacement current considering the example of current flow through a capacitor.
(a) Prove the relation \nabla^2 \left( \frac{1}{|\vec{r} - \vec{r}'|} \right) = - 4 \pi \delta \left(|\vec{r} - \vec{r}'|\right) and hence show that \phi(\vec{r}) = \int \frac{\rho(\vec{r}')}{|\vec{r} - \vec{r}'|} \, d\vec{r}' is a solution of the Poisson equation \nabla^2 \phi(\vec{r}) = - 4 \pi \rho(\vec{r}).
(c) Use the Planck formula for the blackbody radiation u(\omega, T) = \frac{\hbar^2}{\pi^2 c^3} \frac{\omega^3}{\exp(\beta \hbar \omega) - 1} with \beta = \frac{1}{k_B T} to derive Wien's law, Rayleigh-Jeans law and Stefan-Boltzmann law.
(b) Derive approximate expressions for the potential and the radial as well as the azimuthal components of the field due to an electric dipole at points far away from it. Also derive expression and hence describe the effect of a unifrom electric field on a dipole which can rotate freely.
Differentiates between Rayleigh and Raman scatterings. Why is Raman scattering considered to be a breakthrough in molecular spectroscopy ? What are the advantages of using laser light in Raman spectroscopy ?
(c) Why does one see two image points for a single object point while viewed through a calcite crystal ? What is this property of the crystal known as ? What is an optic axis of a crystal ? Explain the meanings of positive and negative crystals with one example for each kind.
(b) For a transverse sinusoidal wave of wavelength \lambda propagating along negative x direction through a string fixed at a point, show that the nodes are located at x = 0, \lambda/2, \lambda, 3\lambda/2, \dots while the kinetic energy / unit length at the antinodes is given by E = 2 \rho A^2 \omega^2 \cos^2 \omega t where \rho, A and \omega, are the mass density / unit length, amplitude of transverse displacement and angular frequency of the wave, respectively.
(c) What do you understand by paraxial rays ? Show that the effect of translation of a paraxial ray while travelling along a homogeneous medium is represented by a 2 \times 2 matrix if the ray is initially defined by a 2 \times 1 matrix.
