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= \hbar L_+$ (iii) [L_+, L_-] = 2\hbar L_z (iv) L_+ L_y = L^2 - L_z^2 + \hbar L_z Prove that : (i) [L^2, L_z] = 0 (ii) [L_z, L_+] = \hbar L_+ (iii) [L_+, L_-] = 2\hbar L_z (iv) L_+ L_y = L^2 - L_z^2 + \hbar L_z where \hbar = \frac{h}{2\pi} (h is Planck's constant)
Show that the square of the orbital angular momentum operator (L^2) commutes with any of the components of angular momentum operator L. Is it possible to measure L^2, L_x, L_y and L_z simultaneously ? Give reasons for your answer.
$, [J_z^2, J_y] and [J^2, J_y], and then show that \langle J, m | J_x^2 | J, m \rangle = \langle J, m | J_y^2 | J, m \rangle
Consider a particle of mass m moving in the potential V(x) = \begin{cases} +\infty & ; x \le 0 \\ \frac{1}{2}m\omega^2 x^2 & ; x > 0 \end{cases} Estimate the ground state energy of this particle using the WKB method.
Consider a particle of mass m moving freely between x = 0 and x = a inside an infinite square well potential. Calculate the expectation values \langle x \rangle_n, \langle p \rangle_n, \langle x^2 \rangle_n and \langle p^2 \rangle_n, and compare them with their classical counterparts.
The ground state wave function of a harmonic oscillator is \psi_0(x) = \left(\frac{m\omega}{\hbar \pi}\right)^{1/4} \exp\left(-\frac{m\omega x^2}{2\hbar}\right). (i) At which point is the probability density maximum ? (ii) What is the value of the maximum probability density ?
(i) Assuming the potential seen by a neutron in a nucleus to be schematically represented by a one-dimensional, infinite rigid wall potential of length 10^{-15}\text{ m}, estimate the minimum kinetic energy of the electron. (ii) Estimate the minimum kinetic energy of neutron bound within the nucleus as described above. Can an electron be confined in a nucleus ? Explain.
Consider a particle of mass m and charge q moving under the influence of a one-dimensional harmonic oscillator potential. Assume that it is placed in a constant electric field E. The Hamiltonian of this particle is therefore given by H = \frac{p^2}{2m} + \frac{1}{2}m\omega^2 x^2 - qEx Derive the energy expression and wave function of the n\text{th} excited state.
Derive the degeneracy of a harmonic oscillator g_n = \frac{1}{2}(n+1)(n+2).
Find out the boiling temperature of water at the top of Mount Everest. Given : Pressure at the top of Everest is 0.36\text{ atm}. The density of water vapour at 100^\circ\text{C} is 0.598\text{ kg/m}^3. The latent heat is 2.257 \times 10^3\text{ J/g}.
What do you understand by macrostates and microstates? Briefly explain.
Deduce the thermodynamic relation : \left(\frac{\partial S}{\partial V}\right)_T = \left(\frac{\partial P}{\partial T}\right)_V Using the expression establish Clausius-Clapeyron latent heat equation \frac{dP}{dT} = \frac{L}{T(V_2 - V_1)}
(i) Prove that for perfect gas the specific heat at constant pressure C_p is always greater than the specific heat at constant volume by a constant value R. (ii) Consider for hydrogen, the density at NTP is 0.0899\text{ gm/lit}, molecular weight 2.016\text{ gm} and specific heat at constant pressure 6.85\text{ cal/gm}. Calculate the specific heat of hydrogen gas at constant volume. Given : J = 4.18 \times 10^7\text{ erg}.
Calculate the Fermi energy in electron-volt for sodium assuming that it has one free electron per atom. The density of sodium = 0.97\text{ gm/cc} and the atomic weight of sodium is 23.
Prove that the work done by a perfect gas during a quasi-static adiabatic expansion is given by W = \frac{P_i V_i}{\gamma - 1} \left[ 1 - \left( \frac{P_f}{P_i} \right)^{\frac{\gamma - 1}{\gamma}} \right] where \gamma is the ratio of specific heats.
Explain the T-s diagram for the reversible Carnot cycle and hence obtain the expression for the efficiency of the Carnot engine.
The specific heat of a solid at low temperatures is given by the relation C_V = AT^3, where A is a constant and T is the absolute temperature. How much heat will be required to raise the temperature of m\text{ gm} of the solid from 300\text{ K} to 500\text{ K}?
Define internal energy U, Helmholtz's function F, enthalpy H, Gibbs' potential G and hence obtain the four Maxwell's thermodynamic relations.
The volume of a mole of liquid \text{He}^4 is 27 \times 10^{-6}\text{ m}^3 and the mass of a \text{He}^4 atom is 6.65 \times 10^{-27}\text{ kg}. Assuming that liquid \text{He}^4 is an ideal Bose gas, calculate (i) the concentration of boson in this volume. (ii) Bose temperature.
The magnitude of the average electric field normally present in the Earth's atmosphere just above the surface of the Earth is about 150\text{ N/C}, directed radially inward, toward the centre of the Earth. What is the total net surface charge carried by the Earth? Assume the Earth to be a conductor. (The radius of the Earth is 6.37 \times 10^6\text{ m})