What is the main inference drawn from the Stern Gerlach experiment ? What would be the change in the result in the following cases :
(i) electron spin is \frac{3}{2}\hbar instead of \frac{\hbar}{2}.
(ii) a beam of neutral atoms is passed through a homogeneous magnetic field.
Describe the differences in the behaviour of a quantum harmonic oscillator with the corresponding classical oscillator.
An electron is described by the following wave function : \begin{aligned} \psi(x) &= 0 && \text{for } x < 0 \\ &= C e^{-x}(1 - e^{-x}) && \text{for } x \ge 0 \end{aligned} where C is a constant. Find out the average position of the electron.
Use the uncertainty principle to estimate the binding energy of the hydrogen atom in the ground state in terms of m_\text{e}, e, \hbar and c. Estimate the answer in eV.
A particle of mass m is confined in a one dimensional box on the x-axis between x = 0 and x = L. Obtain the allowed energy states of the particle and the corresponding eigenfunctions. What will be the energy pattern in the limit of L \to \infty ?
Show that
(i) \hat{L} \times \hat{L} = i \hbar \hat{L}
(ii) [\hat{L}_+, \hat{L}_-] = 2\hbar \hat{L}_z
(iii) \hat{\sigma}_x \hat{\sigma}_y \hat{\sigma}_z = -i
A beam of particles of mass m and energy E is incident on a step potential of height V_0 from the left as shown in the figure. Discuss the behaviour of the particle for (i) E < V_0 and (ii) E > V_0. Obtain expressions for the reflection and the transmission coefficients and sketch them as a function of the incident energy of the particle.
Estimate the probability of finding the particle between x = 0 and x = \frac{L}{3} in the lowest energy state.
Generalize the results obtained in part (a) to obtain the allowed energy states and the corresponding eigenfunctions of a particle in a three dimensional isotropic box of dimension L. What is the degeneracy of the first two excited states of the particle ?
State Maxwell's distribution law of molecular speeds. Draw and explain a curve between n(c) and c in a gas at a given temperature T, where n(c) dc is the number of molecules having speed between c and c + dc. Discuss the effect of T and mass m of the molecule on the nature of the curve.
Describe neutron star on the basis of Fermi-Dirac statistics and obtain the condition of critical mass for a neutron star.
Derive the mathematical expression for the total energy of a degenerate Fermi gas at a temperature T and calculate the specific heat of the Fermi gas at this temperature.
Prove the thermodynamic relation : \left( \frac{\partial S}{\partial V} \right)_T = \left( \frac{\partial P}{\partial T} \right)_V and hence show that \frac{\mathrm{d}P}{\mathrm{d}T} = \frac{L}{T (V_2 - V_1)} ; all the terms have their usual meanings.
Use the method of electric images to find the electric field on the surface of a grounded conducting sphere.
How does one explain the observed spectrum of black-body radiation using Planck's quantum hypothesis ? State and obtain Wien's displacement law. Also explain the important features of this law.
(i) Define and explain the significance of the quality factor of an electrical machine. (ii) Discuss in brief, the working principle of a transformer.
Discuss the reflection and refraction of plane electromagnetic waves at plane dielectric boundaries for normal incidence and also find the reflection and transmission coefficients.
(i) State Faraday's law of electromagnetic induction and prove that it can be expressed in the following vector form : \mathrm{Curl}~\vec{E} = -\frac{\partial \vec{B}}{\partial t} with \vec{E} and \vec{B} being the electric and magnetic fields. (ii) A coil of 10 turns has dimension 9~\text{cm} \times 7~\text{cm}. It rotates at the rate of 15\pi~\text{rad/sec} in a uniform field whose flux density is 0\cdot 6~\text{weber/m}^2. What is the maximum e.m.f. induced in the coil ?
(i) Using Maxwell's equations, obtain the relation \frac{1}{c} \frac{\partial}{\partial t} \left( \frac{E^2 + B^2}{2} \right) + \vec{\nabla} \cdot (\vec{E} \times \vec{B}) = 0 (ii) What is Poynting vector ? Deduce Poynting theorem for the flow of energy in an electromagnetic field.
Write down the four Maxwell's equations and explain the contribution of Maxwell in the development of these equations.
