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Pune University - Sem - II : Statistical Mechanics in Phy, April 2010 (Old Course)

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Total No. of Questions : 7] P876 [Total No. of Pages : 3 [3722]-23 M. Sc. PHYSICS PHY UT - 603 : Statistical Mechanics in Physics (Old Course) (2005 Pattern) (Sem. - II) Time : 3 Hours] [Max. Marks : 80 Instructions to the candidates : 1) Question No. 1 is compulsory. Attempt any four of the remaining questions. 2) Draw neat diagrams wherever necessary. 3) Figures to the right indicate full marks. 4) Use of logarithmic tables and electronic pocket calculator is allowed. Constants : 1) Boltzman constant kB = 1.38 10 23 J/K. 2) Planck s constant h = 6.623 10 34 Js. 3) Avogadro s number N = 6.023 1023 cgs units. 4) Mass of electron ms = 9.1 10 31 kg. 5) Velocity of light C = 3 108 m/s. 6) Gas constant R = 8.314 J/mole/ K. Q1) Attempt any four of the following : a) 1 The energy levels of harmonic oscillator are given by En = n + ! . 2 Find the ratio of probability of the harmonic oscillator being in the first excited state to the probability of it being in the ground state. [4] b) For Helium gas at room temperature (300 K) and atmospheric pressure; calculate N/V. [4] c) Energy states of a particle moving in a rigid cubical box is given by 2 nx + 2 ny + 2 nz 2 ml 2 E = = 14 . 2! 2 Determine the number of microstates accessible to the particle. [4] P.T.O. d) If F = E TS = kTluZ Show that E = ( F ) . [4] e) A particle of unit mass is executing simple harmonic motion. Determine its trajectory in phase space. [4] f) State and explain the concept of equal-a-priori probability. Q2) a) [4] For canonical ensembles show that the probability of finding the system e Er in a particular microstate r having energy Er is given by Pr = . e E r r [8] b) Show that the fluctuation in number of particles in a system in grand canonical ensembles is given by ( N )2 = KT N . Q3) a) b) Q4) a) b) Obtain the partition function of a photon gas. Hence derive Planck s radiation formula. [8] State and prove equipartition theorem. [8] What is Gibb s paradox? How is it resolved? [8] Write the partition function for Bose-Einstein statistics and hence obtain B.E. distribution in the form n r = 1 e (E r ) 1 Where is chemical potential. Q5) a) [8] Obtain Maxwell s velocity distribution and hence show that the ratio of root mean square velocity rms to mean velocity to the most probable ~ ~ velocity is given by : : 3 : 8 : 2 . [10] rms [3722]-23 [8] -2- b) Q6) a) b) According to Pauli s paramagnetism, show that paramagnetic susceptibility is independent of temperature but strongly dependent on the density of the gas. [6] Obtain the expression for mean energy of fermions at T = O K. Show that for classical monoatomic ideal gas having N particles contained in volume V, the number of states (E) of the system in the energy range E and E + E is given by (E) = BVNE3N/2 Where B is constant independent of V & E. Q7) a) [6] [10] Write a short note on Statistical Ensembles . [4] b) Compare the basic postulates of B.E. and F.D. statistics. [4] c) Use canonical distribution to obtain the mean energy E . [4] d) Explain the concept of microstate and macrostates. [4] () rrrr [3722]-23 -3-

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Additional Info : M. Sc. PHYSICS, PHY UT - 603 : Statistical Mechanics in Physics (Old Course) (2005 Pattern) (Semister - II)
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