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Physics Resonance: Problem set 10 -->

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Monday, 17 October 2016

Problem set 10

  1. The function f(z)=u(x,y)+iv(x,y) is analytic at z=x+iy. The value of \nabla^2u at this point is:
    1. 0
    2. undefined
    3. \pi
    4. e^{-\pi^2}
  2. A head of mass m slides on a smooth rod which is rotating about one end in a vertical plane with uniform angular velocity \omega. The Lagrangian of the system is :
    1. L=\frac{1}{2}m\left(\dot r^2+r^2\dot\theta^2\right)-mgr\sin\theta
    2. L=\frac{1}{2}m\left(r^2\dot\theta^2\right)-mgr\sin\theta
    3. L=\frac{1}{2}m\left(\dot r^2+\dot\theta^2\right)-mgr\sin\theta
    4. L=\frac{1}{2}m\left(r^2\dot\theta^2\right)+mgr\sin\theta
  3. A partition function of two Bose particles each of which can occupy any of the two energy levels 0 and \epsilon is
    1. 1+e^{-2\epsilon/kT}+2e^{-\epsilon/kT}
    2. 1+e^{-2\epsilon/kT}+e^{-\epsilon/kT}
    3. 2+e^{-2\epsilon/kT}+e^{-\epsilon/kT}
    4. e^{-2\epsilon/kT}+e^{-\epsilon/kT}
  4. A one dimensional random walker takes steps to left or right with equal probability. The probability that the random walker starting from origin is back to origin after N even number of steps is
    1. \frac{N!}{\left(\frac{N}{2}\right)!\left(\frac{N}{2}\right)!}\left(\frac{1}{2}\right)^N
    2. \frac{N!}{\left(\frac{N}{2}\right)!\left(\frac{N}{2}\right)!}
    3. 2N!\left(\frac{1}{2}\right)^{2N}
    4. N!\left(\frac{1}{2}\right)^N
  5. Five electrons (Fermions with spin 1/2\hbar) are kept in a one-dimensional infinite potential well with width a. (Ground state energy of single electron well =\frac{\hbar^2\pi^2}{2ma^2}). The first absorption line corresponds to energy:
    1. \frac{\hbar^2\pi^2}{2ma^2}
    2. \frac{5\hbar^2\pi^2}{2ma^2}
    3. \frac{7\hbar^2\pi^2}{2ma^2}
    4. \frac{11\hbar^2\pi^2}{2ma^2}

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