Obtain the allowed eigenenergies of the half 1-D simple harmonic oscillator defined by the Hamiltonian H = \frac{p^2}{2m} + V(x), \quad \text{where} V(x) = \begin{cases} \dfrac{1}{2}m\omega^2 x^2 & \text{for } x \ge 0 \\ \infty & \text{for } x < 0 \end{cases} Express its eigenstates in terms of eigenstates of the full 1-D oscillator.
A particle is in the ground state of a 1-D simple harmonic oscillator potential, V(x) = \dfrac{1}{2}m\omega^2 x^2. Calculate the position uncertainty of the particle.
Two solids A and B have Debye temperatures 200\text{ K} and 300\text{ K} respectively. At T = 20\text{ K}, compare their specific heats.
A gas has only two particles a and b. Show with the help of diagrams how these two particles can be arranged in three energy states 1, 2, 3 using (i) Maxwell–Boltzmann, (ii) Fermi–Dirac and (iii) Bose–Einstein statistics.
The average kinetic energy of hydrogen atoms in a certain stellar atmosphere, assumed to be in thermal equilibrium, is 1\cdot 2\text{ eV}. Calculate the ratio of the number of atoms in the second excited state (n = 3) to the number in the ground state.
What are the limitations of the first law of thermodynamics? One mole of a gas, assumed to be perfect, at 0\text{ }^\circ\text{C} is heated at constant pressure till its volume is twice its initial value. Calculate the amount of heat absorbed. Given, C_v = 20\cdot 9\text{ J}\cdot\text{mol}^{-1}\cdot\text{K}^{-1} and R = 8\cdot 3\text{ J}\cdot\text{mol}^{-1}\cdot\text{K}^{-1}.
Calculate the pressure at which water will boil at 150\text{ }^\circ\text{C}, given that the change in specific volume when 1\text{ gram} of water is converted into steam is 1676\text{ cm}^3. Given, latent heat of vaporization for steam = 540\text{ cal per gram}, J = 4\cdot 2 \times 10^7\text{ ergs/cal} and one atmospheric pressure = 10^6\text{ dynes/cm}^2.
Why does the specific heat of solids depend on material at low temperatures but become independent of material at high temperatures?
Discuss the origin of hysteresis in ferromagnetic materials on the basis of domain theory. Explain how domain wall motion and pinning lead to energy loss during cyclic magnetization.
A charge of -3\cdot 30\text{ }\mu\text{C} is fixed at a point. From a horizontal distance of 0\cdot 0455\text{ m}, a charged particle of mass 7\cdot 35 \times 10^{-3}\text{ kg} and charge -7\cdot 45\text{ }\mu\text{C} is fired with an initial velocity 62\cdot 5\text{ m/s} directly towards the fixed charge. How far does this charge travel before its speed becomes zero?
Blackbody radiation in a cavity at 2000\text{ K} is subject to isothermal reversible expansion through 10^3\text{ cm}^3. Calculate (i) the heat transferred and (ii) the work done. If the initial volume was 10\text{ cm}^3 and expansion had been adiabatic, calculate the change in temperature of the radiation. (Given, Stefan–Boltzmann constant, \sigma = 5\cdot 672 \times 10^{-8}\text{ J}\cdot\text{m}^{-2}\cdot\text{K}^{-4}\cdot\text{s}^{-1})
In a homogeneous non-conducting region where \mu_r = 1, find \epsilon_r and \omega, if
\begin{align*} \vec{E} &= 30\pi e^{j[\omega t - (4/3)y]} \vec{a}_z \quad (\text{V/m}) \\ \vec{H} &= 1\cdot 0 e^{j[\omega t - (4/3)y]} \vec{a}_x \quad (\text{A/m}) \end{align*}
Calculate the effective resistance of the following combination of resistances as shown in the figure and determine the voltage drop across each resistance when a potential difference of 120\text{ volts} is applied between points A and B :
A wire of length 2\text{ m} is perpendicular to the X\text{--}Y plane. It is moved with velocity \vec{V} = 2\hat{i} + 3\hat{j} + \hat{k}\text{ m/s} through a region of uniform magnetic induction \vec{B} = \hat{i} + 2\hat{j}\text{ Wb/m}^2. Compute the potential difference induced between the ends of the wire.
A very long solenoid of radius a, with n turns per unit length, carries a current I_s. Coaxial with the solenoid, at radius b \gg a, is a circular ring of wire with resistance R. When the current in the solenoid is gradually reduced, a current I_r is induced in the ring. Calculate I_r in terms of \frac{dI_s}{dt}. Also, calculate the power dissipated through Joule effect and the electric field \vec{E} near the solenoid.
The space between the plates of a parallel-plate capacitor is filled with a dielectric material whose susceptibility varies linearly from 0 at the bottom plate (x = 0) to 1 at the top plate (x = d). The capacitor is connected to a battery of voltage V. Calculate all the bound charges and check that the total charge is zero. Assume that the battery is connected in such a way that the electric field points along the x-direction while the free charge density, \sigma_f, is positive at the bottom plate and negative at the top plate.
The magnetic field H of an electromagnetic wave travels in the -a_z direction in free space with a phase shift constant of 30\text{ rad/m} and an amplitude of \left(\frac{1}{3\pi}\right)\text{ A/m}. If the field has the direction -a_y when t = 0 and z = 0, write the suitable expressions for E and H. Determine the frequency and wavelength of the wave.
Find the required thickness of the calcite plate to convert plane polarized light (\lambda = 6000\text{ \AA}) into circularly polarized light. (For calcite, \mu_O = 1\cdot 658 and \mu_E = 1\cdot 486)
A particle executes simple harmonic motion of amplitude A and angular frequency \omega. A damping force proportional to velocity acts on the particle. Derive the expression for the displacement of the particle as a function of time and discuss the effect of damping on amplitude and frequency. Also, calculate the time at which the amplitude reduces to half its initial value, if the damping coefficient is b and mass is m.
What is meant by achromatic combination of lenses? Derive the condition for achromatization of a pair of lenses separated by a distance x. The two lenses have different dispersive powers.
