Solve the Schrödinger equation for a particle of mass m confined in one dimensional potential well of the form: V(x)= \begin{cases} 0; & 0 \leq x \leq L,\\ \infty; & x<0,\ x>L. \end{cases} Obtain the discrete energy values and the normalized eigen functions.
An electron is moving in a one-dimensional box of infinite height and width 1\,\mathring{\mathrm{A}}. Find the minimum energy of electron.
Find the de Broglie wave length of
(i) a neutron
(ii) an electron moving with kinetic energy of 500\,eV (1\,eV = 1.602 \times 10^{-19}\,J)
One kg of water at 20^\circ\mathrm{C} is converted into ice at -10^\circ\mathrm{C} at constant pressure. Heat capacity of water is 4{,}200\ \mathrm{J\,kg^{-1}\,K^{-1}} and that of ice is 2{,}100\ \mathrm{J\,kg^{-1}\,K^{-1}}. Heat of fusion of ice at 0^\circ\mathrm{C} is 335\times10^3\ \mathrm{J\,kg^{-1}}. Calculate the total change in entropy of the system.
Show that both Fermi-Dirac and Bose-Einstein distribution functions at an energy E are given by: f(E) \simeq \exp\left[\frac{\mu-E}{k_{\mathrm B}T}\right], where f(E) is much smaller than unity, \mu and k_{\mathrm B}T are the chemical potential and thermal energy of the atom.
Define Enthalpy and show that it remains constant in a throttling process.
Consider a system of free gas particles having f degrees of freedom. Use equipartition theorem to establish the relation f=\frac{2}{\left(\dfrac{C_p}{C_V}-1\right)}, where C_p and C_V are molar specific heats at constant pressure and constant volume respectively. Obtain the values of \dfrac{C_p}{C_V} for diatomic and triatomic gases.
Explain the four thermodynamic relations of Maxwell. Using the same, obtain the Clausius-Clapeyron equation \frac{\mathrm{d}P}{\mathrm{d}T}=\frac{L}{T(V_2-V_1)}.
Starting from Maxwell's equation, obtain the wave equation for the electric field \vec{E} in free space and appropriate wave equation for the electric field \vec{E}=E_z(x,y,z)\hat{z}.
Show that the energy flow due to a plane electromagnetic wave propagating along z-direction in a dielectric medium is given by \hat{z}\frac{k}{\omega\mu}E_0^2\cos^2(kz-\omega t), where \mathbf{k} and \omega are the propagation vector and angular frequency, E_0 is electric field amplitude, \mu is the relative permeability of the medium.
State and explain Stefan-Boltzmann Law. Show that \log P = \log K + 4\log R, where P is the power emitted by black body and R is the resistance of the black body, K is a constant.
In deriving radiation laws, we consider a cubical container of volume V containing a photon gas in equilibrium. Calculate the differential number of allowed normal modes of frequency \omega.
For initial current conditions I=I_0 and \frac{dI}{dt}=0 at t=0, show that the time dependent current in the critical damping case for an LCR circuit is given by I=I_0\left(1+\frac{\gamma t}{2}\right)e^{-\gamma t/2} where \gamma=\frac{R}{L}, \omega_0^2=\frac{1}{LC}, \omega=\sqrt{\omega_0^2-\frac{\gamma^2}{4}} and \tan\delta=\frac{-\gamma}{2\omega}.
Using Ampere's Law and continuity equation, show that the divergence of the total current density is zero.
When connected in series, L_1, C_1 have the same resonant frequency as L_2, C_2 also connected in series. Prove that if all these circuit elements are connected in series, the new circuit will have the same resonant frequency as either of the circuits first mentioned.
How does holography differ from conventional photography? What are the requirements for the formation and reading of a hologram?
For a multimode step index optical fibre, the core refractive index is 1.5 and fractional index difference is 0.001. Calculate the pulse broadening for 1\,\mathrm{km} length of the fibre. Over a length of 2\,\mathrm{km} of the fibre, calculate the minimum pulse separation that can be transmitted without overlap.
Explain the working principle of a 3-level laser with a specific example. Comment on why the third level is needed.
In a certain engine, a piston undergoes vertical SHM with an amplitude of 10\,\mathrm{cm}. A washer rests on the top of the piston. As the motor is slowly speeded up, at what frequency will the washer no longer stay in contact with the piston?
Show that the group velocity is equal to particle velocity. Also prove that the group velocity of the photons is equal to c, the velocity of light.
