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.
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.
Normalize the wave function \psi(x)=e^{-\lvert x\rvert}\sin\alpha x
Calculate the wavelength of de Broglie waves associated with electrons accelerated through a potential difference of 200 Volts.
Estimate the size of the hydrogen atom and the ground state energy from the uncertainty principle.
(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.
Calculate (\Delta x)^2, where \Delta x = x - \langle x \rangle.
What led van der Waals to modify the ideal gas equation? Using the concepts of critical temperature T_c, pressure P_c and volume V_c, show that the critical constant for a real gas is 8/3.
Find out the expressions for van der Waals constants a and b.
Calculate the values of van der Waals constants a and b for oxygen with T_c=154.2\,\mathrm{K}, P_c=49.7 atmosphere and R=80\,\mathrm{cm^3\,atmosphere/K}.
Write down the expressions for Bose-Einstein and Fermi-Dirac distribution functions, and show how Fermi-Dirac distribution leads to the explanation of Pauli's exclusion principle.
Write down the expression for the Bose-Einstein distribution function and explain the meaning of the symbols used.
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.
Show that the elemental quantity of heat \delta Q is not a total differential.
Justify which of the four Maxwell's equations imply that there are no magnetic monopoles. How these equations would have been written if they were?
A resistor R(= 6.2\,\mathrm{M}\Omega) and a capacitor C(= 2.4\,\mu\mathrm{F}) are connected in series and a 12 V battery of negligible internal resistance is connected across their combination.
(i) What is the capacitive time constant of this circuit?
(ii) At what time, after the battery is connected, does the potential difference across the capacitor become 5.6 V?
An electrical circuit consists of a resistance R, inductance L and capacitance C in series. If a charge is put on the capacitor at some instant, determine the condition that V_C, the voltage across the capacitor, is subsequently oscillatory. Derive an expression for the quality factor Q of the circuit by considering the decay of the oscillation, using the result that the amplitude falls by a factor of e in \left(\frac{Q}{\pi}\right) period.
A long solenoid has 220 turns /cm; its diameter is 3.2 cm. Inside the solenoid at its centre, we place a 130-turn closepacked coil of diameter 2.1 cm along its axis. The current in the solenoid is increased from zero to 1.5 amperes at a steady rate over a period of 0.16 second. What is the magnitude of the induced e.m.f. that appears in the central coil when the current in the solenoid is being changed?
The four arms of a Wheatstone bridge have the following resistances: AB = 100\,\Omega, BC = 10\,\Omega, CD = 5\,\Omega, DA = 60\,\Omega A galvanometer of 15\,\Omega resistance is connected across BD. Calculate the current through the galvanometer when a potential difference of 10 volts is maintained across AC.
Find whether the discharge of a condenser through the inductive circuit is oscillatory when C = 0.1\,\mu\mathrm{F}, L:=10\,\mathrm{mH} and R = 200\,\Omega. If it is oscillatory, calculate its frequency.
