Search, filter, and study UPSC Civil Services & IFoS Physics past year questions with rigorous derivations, formulas, and step-by-step solutions.
Set up the time-independent Schrodinger equation for an electron moving in Coulomb field, V(r) = \frac{Ze^2}{4 \pi \varepsilon_0 r}, in polar coordinates. Solve the radial equation to get the energy eigen values.
(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 ?
(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.
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.
(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.
(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}.
(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.
(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}).
(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.
(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.
(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.
(c) Explain how Einstein's A and B coefficients are related to the phenomena of spontaneous and stimulated emission of radiation, respectively. Derive the relation between A and B. Establish that at very high frequency around X-ray wavelength regime, lasers cannot be made as easily as at low frequencies, e.g. far infra-red regime.