(i) What is the origin of paramagnetism ? Discuss the temperature dependence of paramagnetic susceptibility. (ii) Calculate the diamagnetic susceptibility of atomic Hydrogen in the ground state at STP. Assume the mean square distance of electronic charge distribution of atomic Hydrogen from the nucleus r^2 = 3 a_0^2, a_0 being the radius of the first Bohr orbit of Hydrogen.
Does the nucleus possess magnetic moment ? Justify your answer. Define nuclear magneton (\mu_{\text{N}}) and Bohr magneton (\mu_{\text{B}}). Calculate their values.
(i) Write semi-empirical mass formula. Calculate the atomic number (Z) of most stable nucleus for given mass number (A) using the above formula. (Use the value of fitted coefficients for Coulomb energy a_3 = 0\cdot 711\text{ MeV} and that for asymmetry energy a_4 = 23\cdot 702\text{ MeV}). (ii) Calculate the Q-value of the following nuclear reaction : {}_4\text{Be}^9 + {}_2\text{He}^4 = {}_6\text{C}^{12} + {}_0\text{n}^1 Given : the mass of neutral atoms of \text{Be}, \text{He} and \text{C} are 9\cdot 015060, 4\cdot 003874 and 12\cdot 003815\text{ amu}, respectively. The mass of neutron is 1\cdot 008986\text{ amu}.
What is meant by strength of the interactions of elementary particles ? Classify the different forces on the basis of this strength of interaction.
For the ground state of deuteron, prove that the radius of nucleon is of the order of \sim 2\cdot 15 \times 10^{-13}\text{ cm}.
Compare nuclear density of hydrogen ({}_1\text{H}^1) with its atomic density. (Assume the atom to have the radius of its first Bohr orbit). What inference can one get from the above comparison ?
What is nuclear resonance absorption of gamma rays ? Describe the working and applications of Mossbauer spectrometer.
(ii) \bar{\nu}_e + p^+ \rightarrow n^0 + e^- What are the various conservation laws for elementary particles ? Apply these conservation laws to confirm whether the following reactions are possible or not : (i) \pi^+ + n^0 \rightarrow K^0 + K^+ (ii) \bar{\nu}_e + p^+ \rightarrow n^0 + e^-
Show that for a diatomic molecule with two nuclei of mass 'M' separated by a distance 'a', the rotational energy of nuclear motion is lower than electronic energy by a factor of \frac{m_e}{M}.
Explain how the hydrogen spectrum is used for imaging the universe.
What is Lamb shift in the fine structure of hydrogen spectrum ? Discuss its theory based upon second quantization.
How do Stokes lines appear in Raman spectrum as per classical and quantum theory of Raman effect ?
Differentiate between L-S coupling and J-J coupling. What are the possible orientations of \vec{J} for the J = \frac{3}{2} and J = \frac{1}{2} states that correspond to l = 1 ?
(i) Using free electron theory of metals, calculate the Fermi energy level of sodium atom at absolute zero. Assume that sodium has one free electron per atom and its density is 0\cdot 97\text{ gm/cm}^3. (ii) Draw the energy level diagram and mathematical expressions for the following : I. E_n of an electron confined in a one-dimensional box II. Linear harmonic oscillator Make a qualitative comparison of the above two cases.
Describe Electron Paramagnetic Resonance. Highlight its differences with NMR and discuss its applications.
How is Rydberg constant related to emission wavelength of hydrogen spectrum ?
(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.
(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 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)
A particle limited to the x-axis has the wave function \phi(x) = bx^2 between x = 0 and x = 2; the wave function \phi(x) = 0 elsewhere. (i) Find the probability that the particle can be found between x = 1.0 and x = 1.5. (ii) Find the expectation value <x> of the particle position.
Find the energy of the particle of mass m moving in a potential field V(x) = \frac{2\hbar^2 b^2 x^2}{m} for which the time independent wave function is \psi(x) = \exp(-bx^2). Here b is a constant.
