BSc Mathematics SEM V 2016 17 2016-17 Phisics Paper I Mathematical & Statistical Physics Question Paper - Mumbai University | munotes

Questions asked in this paper

• (2) Figures to the right indicate maximum marks

1. Q1 (A) Attempt any one : 10 marks
   • (a) Solve
   • b) Solve y=2cosx > (ii) Discuss the method of solving the second order homogeneous linear 10 ordinary differential equations with the constant coefficients
   • (B) Attempt any one = (i) Test the following equation for exactness and find its solution 5
   • (ii) The equation of motion of a damped simple harmonic oscillator is 5 Find its solution
2. Q2 (A) Attempt any one: 10 marks
   • (i) Expand the following function in Fourier series Hence show that Graphically represent f (x) in the internal [-z, 2] and outside
   • (i) (a) the complex form of Fourier series and hence describe formal 49 development of Fourier transform pair
   • (b) Find Fourier transform of f(x) given by
   • (B) Attempt any one
   • (i) Show the change of interval of Fourier series from to 5 Extend the interval to involve all values of x and obtain the Fourier integral formula
   • (ii) Find sine transform of 5
3. Q3 (A) Attempt any one:
   • (i) Show that, for an infinitesimal general interaction, TdS = dU+ PdV. 10
   • (ii) Consider a system in contact with a heat reservoir at temperature T Find the probability that the system is in quantum state r of energy = Define partition function z and show that
   • (B) Attempt any one
   • (i) Describe Gibb's free energy (G) of a system. Discuss its variation during thermal interaction and at the equilibrium state
   • (ii) Discuss the concept of phase space. Hence find number of quantum states lying between p and p+dp. Here p is the momentum
4. Q4 (A) Attempt any one:
   • (i) | What are fermions? Derive Fermi-Dirac distribution law. Hence 10 discuss the Fermi energy
   • (ii) State and explain Boltzmann distribution law ofenergies. 49 law to obtain equations for root mean square speed, most probable speed and average seed of the molecules
   • (B) Attempt any one :
   • (i) Show that the average energy of a quantized oscillator is
   • (ii) Describe the concept of a priori probability and thermodynamic probability in relation to distribution of N balls in k cells. Hence 5 write equation for the total probability
   • (A) Attempt any one : ‘The equation of motion of a body falling under gravity in 4 medium is —+bv=g Solve this equation for v if the body starts from rest = (ii) Solve at = - AN(t) by method of separation of variables. 4
   • (B) Attempt any one:
   • (i) Get the Fourier transforms of first order and second order derivatives 4
   • (ii) State Dirichlet's theorem and explain Dirichlet's conditions
   • (C) Attempt any one:
   • (i) Relative probability of the two states of particle inasystemise?at 4 temperature 410 Calculate energy difference between the two
   • (ii) Consider a system of six spin half particles uniform magnetic 4 field B. Each particle has magnetic moment } associated with it Find various possible macrostates for the system and number of microstate in each macrostate. Which macrostate is most probable?
   • (D) Attempt any one :
   • (i) Calculate number of modes of vibrations per unit 3 ‘body cavity for the wavelengths between 6000 AU and 6010 AU
   • (ii) Find number of possible arrangements of 6 particles in 8 energy cells assuming that they obey : (i) B-E statistics (ii) F-D 3

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