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Physics Resonance: November 2016 -->

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Wednesday, 30 November 2016

Problem set 33

  1. The radius of the Fermi sphere of free electrons in a monovalent metal with an fcc structure, in which the volume of the unit cell is a^3, is
    1. \left(\frac{12\pi^2}{a^3}\right)^{1/3}
    2. \left(\frac{3\pi^2}{a^3}\right)^{1/3}
    3. \left(\frac{\pi^2}{a^3}\right)^{1/3}
    4. \left(\frac{1}{a}\right)^{1/3}
  2. Deviation from Rutherford scattering formula for \alpha-particle scattering gives an estimate of:
    1. Size of an atom
    2. Thickness of target
    3. Size of a nucleus
    4. half life of \alpha-emitter
  3. If the critical magnetic field for aluminium is 7.9\times 10^3 A/m, the critical current which can flow through long thin superconducting wire of aluminium of diameter 1\times10^{-3}m is
    1. 5.1 A
    2. 1.6 A
    3. 4.0 A
    4. 2.48 A
  4. An elemental dielectric has a dielectric constant \epsilon=12 and it contains 5\times10^{18} atoms/m^3. Its electronic polarizability in uints of FM^2 is
    1. 4.17\times10^{-30}
    2. 8.34\times10^{-30}
    3. 6.32\times10^{-30}
    4. 12.64\times10^{-30}
  5. Mangetite (Fe_3O_4) has a cubic structure with a lattice constant of 8.4A^o. The saturation magnetization in this material in units of A/m is :
    1. 6.2\times10^5
    2. 6.2\times10^6
    3. 12.4\times10^6
    4. 12.4\times10^5

Monday, 28 November 2016

Problem set 32

  1. The Hamiltonian of a particle of unit mass moving in the xy-plane is given to be: H = xp_x- yp_y -\frac{1}{2}x^2 +\frac{1}{2} y^2 in suitable units. The initial values are given to be (x(0),y(0)) = (1,1) and (p_x(0),p_y(0) )=(\frac{1}{2}, -\frac{1}{2}). During the motion, the curves traced out by the particles in the xy-plane and the p_xp_y-plane are
    1. both straight lines
    2. a straight line and a hyperbola respectively
    3. a hyperbola and an ellipse, respectively
    4. both hyperbolas
  2. A static, spherically symmetric charge distribution is given by \rho(r) = \frac{A}{r} e^{-Kr} where A and K are positive constants. The electrostatic potential corresponding to this charge distribution varies with r as
    1. re^{-Kr}
    2. \frac{1}{r} e^{-Kr}
    3. \frac{1}{r^2} e^{-Kr}
    4. \frac{1}{r} (1-e^{-Kr})
  3. Band-pass and band-reject filters can be implemented by combining a low pass and a high pass filter in series and in parallel, respectively. If the cut-off frequencies of the low pass and high pass filters are \omega_0^{LP} and \omega_0^{HP} , respectively, the condition required to implement the band-pass and band-reject filters are, respectively,
    1. \omega_0^{HP}<\omega_0^{LP} and \omega_0^{HP}<\omega_0^{LP}
    2. \omega_0^{HP}<\omega_0^{LP} and \omega_0^{HP}>\omega_0^{LP}
    3. \omega_0^{HP}>\omega_0^{LP} and \omega_0^{HP}<\omega_0^{LP}
    4. \omega_0^{HP}>\omega_0^{LP} and \omega_0^{HP}>\omega_0^{LP}
  4. Non-interacting bosons undergo Bose-Einstein Condensation (BEC) when trapped in a three-dimensional isotropic simple harmonic potential. For BEC to occur, the chemical potential must be equal to
    1. \hbar\omega/2
    2. \hbar\omega
    3. 3\hbar\omega/2
    4. 0
  5. Linearly independent solution of the differential equation \frac{d^2y}{dx^2}+3\frac{dy}{dx}+2y=0 are:
    1. e^{-x}, e^{-2x}
    2. e^{-x}, e^{2x}
    3. e^{-2x}, e^{x}
    4. e^{2x}, e^{x}

Friday, 25 November 2016

Problem set 31

  1. The two dimensional lattice of graphene is an arrangement of Carbon atoms forming a honeycomb lattice of lattice spacing a, as shown below. The Carbon atoms occupy the vertices.
    1. The Wigner-Seitz cell has an area of
      1. 2a^2
      2. \frac{\sqrt{3}}{2}a^2
      3. 6\sqrt{3}a^2
      4. \frac{3\sqrt{3}}{2}a^2
    2. The Bravais lattice for this array is a
      1. rectangular lattice with basis vectors \vec d_1 and \vec d_2
      2. rectangular lattice with basis vectors \vec c_1 and \vec c_2
      3. hexagonal lattice with basis vectors \vec a_1 and \vec a_2
      4. hexagonal lattice with basis vectors \vec b_1 and \vec b_2
  2. A narrow beam of X-rays with wavelength 1.5 A^\circ is reflected from an ionic crystal with an fcc lattice structure with a density of 3.32 gm cm^{-3}. The molecular weight is 108 AMU (1 AMU = 1.66 \times10^{-24} g).
    1. The lattice constant is
      1. 6.00A^\circ
      2. 4.56A^\circ
      3. 4.00A^\circ
      4. 2.56A^\circ
    2. The sine of the angle corresponding to (111)
      1. \sqrt{3}/4
      2. \sqrt{3}/8
      3. 1/4
      4. 1/8
  3. If an electron is in the ground state of the hydrogen atom, the probability that its distance from the proton is more than one Bohr radius is approximately
    1. 0.68
    2. 0.48
    3. 0.28
    4. 0.91

Wednesday, 23 November 2016

Problem set 30

  1. A particle is confined to the region x \ge 0 by a potential which increases linearly as u(x) = u_0x. The mean position of the particle at temperature T is
    1. \frac{k_BT}{u_0}
    2. \frac{(k_BT)^2}{u_0}
    3. \sqrt{\frac{k_BT}{u_0}}
    4. u_0k_BT
  2. A plane electromagnetic wave is propagating in a loss-less dielectric. The electric field is given by {\scriptstyle\vec E(x,y,z,t)=E_0(\hat x+A\hat z)\exp{\left[ik_0\left\{-ct+\left(x+\sqrt{3}z\right)\right\}\right]}} where c is the speed of light in vacuum, E_0 , A and k_0 are constants and \hat x and \hat z are unit vectors along the x- and z-, axes. The relative dielectric constant of the medium, \epsilon_r and the constant A are
    1. \epsilon_r=4 and A=-\frac{1}{\sqrt{3}}
    2. \epsilon_r=4 and A=+\frac{1}{\sqrt{3}}
    3. \epsilon_r=4 and A=\sqrt{3}
    4. \epsilon_r=4 and A=-\sqrt{3}
  3. In a system consisting of two spin-\frac{1}{2} particles labeled 1 and 2, let \vec S^{(1)} = \frac{\hbar}{2}\vec\sigma^{(1)} and \vec S^{(2)} = \frac{\hbar}{2}\vec\sigma^{(2)} denote the corresponding spin operators. Here \vec\sigma\equiv (\sigma_x ,\sigma_y ,\sigma_z) and \sigma_x ,\sigma_y ,\sigma_z are the three Pauli matrices.
    1. In the standard basis the matrices for the operators S^{(1)}_x S^{(2)}_y and S^{(1)}_y S^{(2)}_x are, respectively,
      1. {\scriptstyle\frac{\hbar^2}{4}\begin{pmatrix}1&0\\0&-1\end{pmatrix},\quad\frac{\hbar^2}{4}\begin{pmatrix}-1&0\\0&1\end{pmatrix}}
      2. {\scriptstyle\frac{\hbar^2}{4}\begin{pmatrix}i&0\\0&-i\end{pmatrix},\quad\frac{\hbar^2}{4}\begin{pmatrix}-i&0\\0&i\end{pmatrix}}
      3. {\scriptscriptstyle\frac{\hbar^2}{4}\begin{pmatrix}0&0&0&-i\\0&0&i&0\\0&-i&0&0\\i&0&0&0\end{pmatrix}},{\scriptstyle\frac{\hbar^2}{4}\begin{pmatrix}0&0&0&-i\\0&0&-i&0\\0&i&0&0\\i&0&0&0\end{pmatrix}}
      4. {\scriptscriptstyle\frac{\hbar^2}{4}\begin{pmatrix}0&1&0&0\\1&0&0&0\\0&0&0&-i\\0&0&i&0\end{pmatrix}},{\scriptstyle\frac{\hbar^2}{4}\begin{pmatrix}0&-i&0&0\\i&0&0&0\\0&0&0&1\\0&0&1&0\end{pmatrix}}
    2. These two operators satisfy the relation
      1. {\scriptstyle\left\{S^{(1)}_x S^{(2)}_y,S^{(1)}_y S^{(2)}_x\right\}=S^{(1)}_z S^{(2)}_z}
      2. {\scriptstyle\left\{S^{(1)}_x S^{(2)}_y,S^{(1)}_y S^{(2)}_x\right\}=0}
      3. {\scriptstyle\left[S^{(1)}_x S^{(2)}_y,S^{(1)}_y S^{(2)}_x\right]=iS^{(1)}_z S^{(2)}_z}
      4. {\scriptstyle\left[S^{(1)}_x S^{(2)}_y,S^{(1)}_y S^{(2)}_x\right]=0}
  4. The radius of ^{64}_{29}Cu nucleus is measured to be 4.8\times10^{-13} cm.
    1. The radius of ^{27}_{12}Mg nucleus can be estimated to be
      1. 2.86\times10^{-13} cm
      2. 5.2\times10^{-13} cm
      3. 3.6\times10^{-13} cm
      4. 8.6\times10^{-13} cm
    2. The root-mean square (rms)- energy of a nucleon in a nucleus of atomic number A in its ground state varies as:
      1. A^{4/3}
      2. A^{1/3}
      3. A^{-1/3}
      4. A^{-2/3}

Monday, 21 November 2016

Problem set 29

  1. Let \psi_{nlm} denote the eigenstates of a hydrogen atom in the usual notation. The state \frac{1}{5}\left[2\psi_{200}-3\psi_{211}+\sqrt{7}\psi_{210}-\sqrt{5}\psi_{21-1}\right] is an eigenstate of
    1. L^2 but not of the Hamiltonian or L_z
    2. the Hamiltonian, but not of L^2 or L_z
    3. the Hamiltonian, L^2 and L_z
    4. L^2 and L_z, but not of the Hamiltonian
  2. The Hamiltonian for a spin-1/2 particle at rest is given by H=E_0(\sigma_z+\alpha\sigma_x), where \sigma_x and \sigma_z are Pauli spin matrices and E_0 and \alpha are constants. The eigenvalues of this Hamiltonian are
    1. \pm E_0\sqrt{1+\alpha^2}
    2. \pm E_0\sqrt{1-\alpha^2}
    3. E_0 (doubly degenerate)
    4. E_0\left(1\pm\frac{1}{2}\alpha^2\right)
  3. For a system of independent non-interacting one-dimensional oscillators, the value of the free energy per oscillator, in the limit T\rightarrow0, is
    1. \frac{1}{2}\hbar\omega
    2. \hbar\omega
    3. \frac{3}{2}\hbar\omega
    4. 0
  4. If the reverse bias voltage of a silicon varactor is increased by a factor of 2, the corresponding transition capacitance
    1. increases by a factor of \sqrt{2}
    2. increases by a factor of 2
    3. decreases by a factor of \sqrt{2}
    4. decreases by a factor of 2
  5. A cavity contains black body radiation in equilibrium at temperature T. The specific heat per unit volume of the photon gas in the cavity is of the form C_v = \gamma T^3,where \gamma is a constant. The cavity is expanded to twice its original volume and then allowed to equilibrate at the same temperature T. The new internal energy per unit volume is
    1. 4\gamma T^4
    2. 2\gamma T^4
    3. \gamma T^4
    4. \gamma T^4/4

Saturday, 19 November 2016

Problem set 28

  1. The solution of the differential equation \frac{dx}{dt}=2\sqrt{1-x^2}, with initial condition x=0 at t=0 is
    1. \begin{align*} x=\begin{cases} \sin{2t},\quad 0\leq t<\frac{\pi}{4}\\ \sinh{2t},\quad t\geq\frac{\pi}{4} \end{cases} \end{align*}
    2. \begin{align*} x=\begin{cases} \sin{2t},\quad 0\leq t<\frac{\pi}{2}\\ 1,\quad t\geq\frac{\pi}{2} \end{cases} \end{align*}
    3. \begin{align*} x=\begin{cases} \sin{2t},\quad 0\leq t<\frac{\pi}{4}\\ 1,\quad t\geq\frac{\pi}{4} \end{cases} \end{align*}
    4. x=1-\cos{2t},\quad t\geq 0
  2. Given a uniform magnetic field \vec B=B_0\hat k (where B_0 is a constant), a possible choice for the magnetic vector potential \vec A is
    1. B_0y\hat i
    2. -B_0y\hat i
    3. B_0(x\hat j+y\hat i)
    4. B_0(x\hat i-y\hat j)
  3. Consider a charge Q at the origin of 3-dimensional coordinate system. The flux of the electric field through the curved surface of a cone that has a height h and a circular base of radius R is
    1. \frac{Q}{\epsilon_0}
    2. \frac{Q}{2\epsilon_0}
    3. \frac{hQ}{R\epsilon_0}
    4. \frac{QR}{2h\epsilon_0}
  4. A Hermitian operator \hat O has two normalised eigenstates |1 > and |2 > with eigenvalues 1 and 2, respectively. The two states |u > =\cos\theta|1 > +\sin\theta|2 > and |v > =\cos\phi|1 > +\sin\phi|2 > are such that < v|\hat O|v > =7/4 and < u|v > =0. Which of the following are possible values of \theta and \phi?
    1. \theta=-\frac{\pi}{6} and \phi=\frac{\pi}{3}
    2. \theta=\frac{\pi}{6} and \phi=\frac{\pi}{3}
    3. \theta=-\frac{\pi}{4} and \phi=\frac{\pi}{4}
    4. \theta=\frac{\pi}{3} and \phi=-\frac{\pi}{6}
  5. The ground state energy of a particle of mass m in the potential V(x)=V_0\cosh{\left(\frac{x}{L}\right)}, where L and V_0 are constants (and V_0>>\frac{\hbar^2}{2mL^2}) is approximately
    1. V_0+\frac{\hbar}{L}\sqrt{\frac{2V_0}{m}}
    2. V_0+\frac{\hbar}{L}\sqrt{\frac{V_0}{m}}
    3. V_0+\frac{\hbar}{4L}\sqrt{\frac{V_0}{m}}
    4. V_0+\frac{\hbar}{2L}\sqrt{\frac{V_0}{m}}

Friday, 18 November 2016

Problem set 27

  1. The Fourier transform of f(x) is \tilde{f}(k)=\int_{-\infty}^{\infty}dx\:e^{ikx}f(x). If f(x)=\alpha\delta(x)+\beta\delta'(x)+\gamma\delta''(x), where \delta(x) is the Dirac delta-function (and prime denotes derivative), what is \tilde{f}(k)?
    1. \alpha+i\beta k+i\gamma k^2
    2. \alpha+\beta k-\gamma k^2
    3. \alpha-i\beta k-\gamma k^2
    4. i\alpha+\beta k-i\gamma k^2
  2. A particle moves in three-dimensional space in a central potential V(r)=kr^4, where k is a constant. The angular frequency \omega for a circular orbit depends on its radius R as
    1. \omega\propto R
    2. \omega\propto R^{-1}
    3. \omega\propto R^{1/4}
    4. \omega\propto R^{-2/3}
  3. The Lagrangian of a system is given by {\scriptstyle L=\frac{1}{2}m\dot q_1^2+2m\dot q_2^2-k\left(\frac{5}{4}q_1^2+2q_2^2-2q_1q_2\right)} where m and k are positive constants. The frequencies of its normal modes are
    1. \sqrt{\frac{k}{2m}}\sqrt{\frac{3k}{m}}
    2. \sqrt{\frac{k}{2m}}(13\pm\sqrt{73})
    3. \sqrt{\frac{5k}{2m}}\sqrt{\frac{k}{m}}
    4. \sqrt{\frac{k}{2m}}\sqrt{\frac{6k}{m}}
  4. Consider a particle of mass m moving with a speed v. If T_R denotes the relativistic kinetic energy and T_N its non-relativistic approximation, then the value of (T_R-T_N)/T_R for v=0.01\:c, is
    1. 1.25\times10^{-5}
    2. 5.0\times10^{-5}
    3. 7.5\times10^{-5}
    4. 1.0\times10^{-4}
  5. Two masses, m each, are placed at the points (x,y)=(a,a) and (-a,-a). Two masses, 2m each, are placed at the points (a,-a) and (-a,a). The principal moments of inertia of the system are
    1. 2ma^2, 4ma^2
    2. 4ma^2, 8ma^2
    3. 4ma^2, 4ma^2
    4. 8ma^2, 8ma^2

Wednesday, 16 November 2016

Problem set 26

  1. The free energy difference between the superconducting and the normal states of a material is given by \Delta F = F_s-F_N =\alpha |\psi|^2 + \frac{\beta}{2}|\psi|^4, where \psi is an order parameter and \alpha and \beta are constants such that \alpha > 0 in the normal and \alpha < 0 in the superconducting state, while \beta > 0 always. The minimum value of \Delta F is
    1. -\alpha^2/\beta
    2. -\alpha^2/2\beta
    3. -3\alpha^2/\beta
    4. -5\alpha^2/\beta
  2. Consider a hydrogen atom undergoing a 2P\rightarrow 1S transition. The lifetime t_{sp} of the 2P state for spontaneous emission is 1.6 ns and the energy difference between the levels is 10.2eV. Assuming that the refractive index of the medium n_0 = 1, the ratio of Einstein coefficients for stimulated and spontaneous emission B_{21}(\omega)/A_{21}(\omega) is given by
    1. 0.683\times10^{12} m^3J^{-1}s^{-1}
    2. 0.146\times10^{-12} Jsm^{-3}
    3. 6.83\times10^{12} m^3J^{-1}s^{-1}
    4. 1.46\times10^{-12} Jsm^{-3}
  3. In the scattering of some elementary particles, the scattering cross-section is found to depend on the total energy and the fundamental constants h (Planck’s constant) and c (the speed of light in vacuum). Using dimensional analysis, the dependence of \sigma on these quantities is given by
    1. \sqrt{\frac{hc}{E}}
    2. \frac{hc}{E^{3/2}}
    3. \left(\frac{hc}{E}\right)^2
    4. \frac{hc}{E}
  4. If y=\frac{1}{\tanh x}, then x is
    1. \ln{\left(\frac{y+1}{y-1}\right)}
    2. \ln{\left(\frac{y-1}{y+1}\right)}
    3. \ln{\sqrt{\frac{y-1}{y+1}}}
    4. \ln{\sqrt{\frac{y+1}{y-1}}}
  5. The function \frac{z}{\sin{\pi z^2}} of a complex variable z has
    1. a simple pole at 0 and poles of order 2 at \pm\sqrt{n} for n=1,2,3,\dots
    2. a simple pole at 0 and poles of order 2 at \pm\sqrt{n} and \pm i\sqrt{n} for n=1,2,3,\dots
    3. poles of order 2 at \pm\sqrt{n} for n=0,1,2,3,\dots
    4. poles of order 2 at \pm n for n=0,1,2,3,\dots

Monday, 14 November 2016

Problem set 25

  1. The energies in the ground state and first excited state of a particle of mass m =\frac{1}{2} in a potential V(x) are -4 and -1, respectively, (in units in which \hbar = 1 ). If the corresponding wavefunctions are related by \psi_1(x)=\psi_0(x) \sinh x, then the ground state eigenfunction is
    1. \psi_0(x)=\sqrt{sech~ x}
    2. \psi_0(x)=sech~ x
    3. \psi_0(x)=sech^2~ x
    4. \psi_0(x)=sech^3~ x
  2. The perturbation \begin{align*} H'=\begin{cases} b(a-x),&-a < x < a\\ 0&\text{otherwise} \end{cases} \end{align*} acts on a particle of mass m confined in an infinite square well potential \begin{align*} V(x)=\begin{cases} 0,&-a < x < a\\ \infty&\text{otherwise} \end{cases} \end{align*} The first order correction to the ground-state energy of the particle is
    1. \frac{ba}{2}
    2. \frac{ba}{\sqrt{2}}
    3. 2ba
    4. ba
  3. Let |0> and |1> denote the normalized eigenstates corresponding to the ground and the first excited states of a one-dimensional harmonic oscillator. The uncertainty \Delta x in the state \frac{1}{\sqrt{2}}\left(|0>+|1>\right) is
    1. \Delta x=\sqrt{\hbar/2m\omega}
    2. \Delta x=\sqrt{\hbar/m\omega}
    3. \Delta x=\sqrt{2\hbar/m\omega}
    4. \Delta x=\sqrt{\hbar/4m\omega}
  4. What would be the ground state energy of the Hamiltonian H=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}-\alpha \delta(x) if variational principle is used to estimate it with the trial wavefunction \psi(x)= Ae^{-bx^2} with b as the variational parameter? [Hint: {\scriptstyle\int_{-\infty}^{\infty} x^{2n} e^{-2bx^2}\:dx = (2b)^{-n-\frac{1}{2}}\Gamma\left(n+\frac{1}{2}\right)}]
    1. -m\alpha^2/2\hbar^2
    2. -2m\alpha^2/\pi\hbar^2
    3. -m\alpha^2/\pi\hbar^2
    4. m\alpha^2/\pi\hbar^2
  5. A given quantity of gas is taken from the state A \rightarrow C reversibly, by two paths, A\rightarrow C directly and A\rightarrow B\rightarrow C as shown in the figure below.

    During the process A\rightarrow C the work done by the gas is 100 J and the heat absorbed is 150 J. If during the process A\rightarrow B\rightarrow C the workdone by the gas is 30 J, the heat absorbed is
    1. 20 J
    2. 80 J
    3. 220 J
    4. 280 J

Saturday, 12 November 2016

Problem set 24

  1. The Hamiltonian of a simple pendulum consisting of a mass m attached to a massless string of length l is H = \frac{p_\theta^2}{2ml^2}+ mgl(1- \cos\theta). If L denotes the Lagrangian, the value \frac{dL}{dt} is:
    1. -\frac{2g}{l}p_\theta\sin\theta
    2. -\frac{g}{l}p_\theta\sin{2\theta}
    3. \frac{g}{l}p_\theta\cos{\theta}
    4. lp_\theta^2\cos{\theta}
  2. Two bodies of equal mass m are connected by a massless rigid rod of length l lying in the XY-plane with the centre of the rod at the origin. If this system is rotating about the z-axis with a frequency \omega, its angular momentum is
    1. ml^2\omega/4
    2. ml^2\omega/2
    3. ml^2\omega
    4. 2ml^2\omega
  3. An infinite solenoid with its axis of symmetry along the z-direction carries a steady current I. The vector potential \vec A at a distance R from the axis
    1. is constant inside and varies as R outside the solenoid
    2. varies as R inside and is constant outside the solenoid
    3. varies as \frac{1}{R} inside and as R outside the solenoid
    4. varies as R inside and as \frac{1}{R} outside the solenoid
  4. Consider an infinite conducting sheet in the xy-plane with a time dependent cunent density Kt\hat i, where K is a constant. The vector potential at (x, y,z) is given by \vec A=\frac{\mu_0K}{4c}(ct-z)^2\hat i. The magnetic field \vec B is
    1. \frac{\mu_0Kt}{2}\hat j
    2. -\frac{\mu_0Kz}{2c}\hat j
    3. -\frac{\mu_0K}{2c}(ct-z)\hat i
    4. -\frac{\mu_0K}{2c}(ct-z)\hat j
  5. When a charged particle emits electromagnetic radiation, the electric field \vec E and the Poynting vector \vec S = \frac{1}{\mu_0} \vec E\times\vec B at a large distance r from the emitter vary as \frac{1}{r^n} and \frac{1}{r^m} respectively. Which of the following choices for n and m are correct?
    1. n = 1 and m=1
    2. n = 2 and m=2
    3. n = 1 and m=2
    4. n = 2 and m=4

Thursday, 10 November 2016

Problem set 23

  1. A diode D as shown in the circuit has an i-V relation that can be approximated by \begin{align*} i_{_D}=\begin{cases} v^2_{_D}+2v_{_D},&\text{for }v_{_D}>0\\ 0,&\text{for }v_{_D}\leq 0 \end{cases} \end{align*}
    The value of v_{_D} in the circuit is
    1. (-1+\sqrt{11})
    2. 8V
    3. 5V
    4. 2V
  2. The Taylor expansion of the function \ln{(\cosh x)}, where x is real, about the point x= 0 starts with the following terms:
    1. -\frac{1}{2}x^2+\frac{1}{12}x^4+\cdots
    2. \frac{1}{2}x^2-\frac{1}{12}x^4+\cdots
    3. -\frac{1}{2}x^2+\frac{1}{6}x^4+\cdots
    4. \frac{1}{2}x^2+\frac{1}{6}x^4+\cdots
  3. Given 2\times2 unitary matrix U satisfying U^\dagger U=UU^\dagger = 1 with \det U = e^{i\phi}, one can construct a unitary matrix V \:( V^\dagger V = VV^\dagger= 1) with \det V=1 from it by
    1. multiplying U by e^{i\phi/2}
    2. multiplying any single element of U by e^{i\phi}
    3. multiplying any row or column of U by e^{i\phi/2}
    4. multiplying U by e^{i\phi}
  4. The value of the integral \int_C \frac{z^3dz}{z^2-5z+6}, where C is a closed contour defined by the equation 2|z|- 5 = 0, traversed in the anti-clockwise direction, is
    1. -16\pi i
    2. 16\pi i
    3. 8\pi i
    4. 2\pi i
  5. The function f(x) obeys the differential equation \frac{d^2f}{dx^2}-(3 - 2i)f = 0 and satisfies the conditions f(0) = 1 and f(x)\rightarrow \infty as x\rightarrow 0. The value of f(\pi) is
    1. e^{2\pi}
    2. e^{-2\pi}
    3. -e^{-2\pi}
    4. -e^{2\pi i}

Wednesday, 9 November 2016

Problem set 22

  1. Given the usual canonical commutation relations, the commutator [A,B] of A=i(xp_y-yp_z) and B=i(yp_z+zp_y) is
    1. \hbar(xp_z-p_xz)
    2. -\hbar(xp_z-p_xz)
    3. \hbar(xp_z+p_xz)
    4. -\hbar(xp_z+p_xz)
  2. The entropy of a system, S, is related to the accessible phase space volume \Gamma by S = k\ln \Gamma(E, N,V) where E, N and V are the energy, number of particles and volume respectively. From this one can conclude that \Gamma
    1. does not change during evolution to equilibrium
    2. oscillates during evolution to equilibrium
    3. is a maximum at equilibrium
    4. is a minimum at equilibrium
  3. Let \Delta W be the work done in a quasistatic reversible thermodynamic process. Which of the following statements about \Delta W is correct?
    1. \Delta W is a perfect differential if the process is isothermal
    2. \Delta W is a perfect differential if the process is adiabatic
    3. \Delta W is always a perfect differential
    4. \Delta W cannot be a perfect differential
  4. Consider a system of three spins S_1, S_2 and S_3 each of which can take values +1 and -1. The energy of the system is given by E = -J\left[ S_1 S_2 + S_2 S_3 + S_3 S_1\right], where J is a positive constant. The minimum energy and the corresponding number of spin configurations are, respectively,
    1. J and 1
    2. -3J and 1
    3. -3J and 2
    4. -6J and 2
  5. The minimum energy of a collection of 6 non-interacting electrons of spin-\frac{1}{2} and mass m placed in a one dimensional infinite square well potential of width L is
    1. 14\pi^2\hbar^2/mL^2
    2. 91\pi^2\hbar^2/mL^2
    3. 7\pi^2\hbar^2/mL^2
    4. 3\pi^2\hbar^2/mL^2

Monday, 7 November 2016

Problem set 21

  1. Four charges (two +q and two -q) are kept fixed at the four vertices of a square of side a as shown
    At the point P which is at a distance R from the centre ( R > > a), the potential is proportional to
    1. 1/R
    2. 1/R^2
    3. 1/R^3
    4. 1/R^4
  2. A point charge q of mass m is kept at a distance d below a grounded infinite conducting sheet which lies in the xy-plane. For what value of d will the charge remains stationary?
    1. q/4\sqrt{mg\pi\epsilon_0}
    2. q/\sqrt{mg\pi\epsilon_0}
    3. There is no finite value of d
    4. \sqrt{mg\pi\epsilon_0}/q
  3. The wave function of a state of the hydrogen atom is given by {\scriptstyle\Psi=\psi_{200}+2\psi_{211}+3\psi_{210}+\sqrt{2}\psi_{21-1}}, where \psi_{nlm} is the normalized eigenfunction of the state with quantum numbers n,l and m in the usual notation. The expectation value of L_z in the state \Psi is
    1. 15\hbar/16
    2. 11\hbar/16
    3. 3\hbar/8
    4. \hbar/8
  4. The energy eigenvalues of a particle in the potential V(x) =\frac{1}{2}m\omega^2x^2-ax are
    1. E_n=\left(n+\frac{1}{2}\right)\hbar\omega-\frac{a^2}{2m\omega^2}
    2. E_n=\left(n+\frac{1}{2}\right)\hbar\omega+\frac{a^2}{2m\omega^2}
    3. E_n=\left(n+\frac{1}{2}\right)\hbar\omega-\frac{a^2}{m\omega^2}
    4. E_n=\left(n+\frac{1}{2}\right)\hbar\omega
  5. If a particle is represented by the normalized wave function \begin{align*} \psi(x)=\begin{cases}\frac{\sqrt{15}\left(a^2-x^2\right)}{4a^{5/2}}&{\scriptstyle\text{for} -a < x < a}\\ 0 &\text{otherwise} \end{cases} \end{align*} the uncertainty \Delta p in its momentum is
    1. 2\hbar/5a
    2. 5\hbar/2a
    3. \sqrt{10}\hbar/a
    4. \sqrt{5}\hbar/\sqrt{2}a

Sunday, 6 November 2016

Problem set 20

  1. Let v, p and E denote the speed, the magnitude of the momentum, and the energy of a free particle of rest mass m. Then
    1. dE/dp=constant
    2. p=mv
    3. v=cp/\sqrt{p^2+m^2c^2}
    4. E=mc^2
  2. A binary star system consists of two stars S_1 and S_2, with masses m and 2m, respectively, separated by a distance r. If both S_1 and S_2 individually follow circular orbits around the centre of mass with instantaneous speeds V_1 and V_2 respectively, the speeds ratio V_1/V_2 is
    1. \sqrt{2}
    2. 1
    3. 1/2
    4. 2
  3. Three charges are located on the circumference of a circle of radius R as shown in the figure below. The two charges Q subtend an angle 90^o at the centre of the circle. The charge q is symmetrically placed with respect to the charges Q. If the electric field at the centre of the circle is zero, what is the magnitude of Q?
    1. q/\sqrt{2}
    2. \sqrt{2}q
    3. 2q
    4. 2q
  4. Consider a hollow charged shell of inner radius a and outer radius b. The volume charge density is \rho(r) =\frac{k}{r^2} (k is a constant) in the region a < r < b. The magnitude of the electric field produced at distance r > a is
    1. \frac{k(b-a)}{\epsilon_0r^2} for all r > a
    2. \frac{k(b-a)}{\epsilon_0r^2} for a < r < b and \frac{kb}{\epsilon_0r^2} for r > b
    3. \frac{k(r-a)}{\epsilon_0r^2} for a < r < b and \frac{k(b-a)}{\epsilon_0r^2} for r > b
    4. \frac{k(r-a)}{\epsilon_0a^2} for a < r < b and \frac{k(b-a)}{\epsilon_0a^2} for r > b
  5. Consider the interference of two coherent electromagnetic waves whose electric field vectors are given by \vec E_1 = \hat i E_0 \cos{(\omega t+\phi)} and \vec E_2 = \hat j E_0 \cos{(\omega t+\phi)} where \phi is the phase difference. The intensity of the resulting wave is given by \frac{\epsilon_0}{2}\left < E^2\right >, where \left < E^2\right > is the time average of E^2. The total intensity is
    1. 0
    2. \epsilon_0E_0^2
    3. \epsilon_0E_0^2\sin^2{\phi}
    4. \epsilon_0E_0^2\cos^2{\phi}