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 EXERCISE-LEVEL II

 LEVEL  II 1. Show that z(n), zeta function defined by:                 is equivalent to the integral Let Show that it is equivalent to Find the relation between z(n) and m(n) explicitly (i.e. without using these integrals). Then, try to represent z(n) in terms of integral of an inverse trigonometric function. Can you go further and generalize z(n) as an algebraic function of n? You will need some fundamental inverse trigonometric relations to solve this. 2. Metric tensors evaluate the length on a curved space. You are well aware that Pythagoras Theorem is not applicable on curved space.Let the length S(t) be parameterized by t.Then, ds/dt=S(ds/dxi)(dxi/dt) where xi, i = 1,2,3,…. represents the coordinates and we have used the chain rule of partial differentiation. Square this and show that ds2 = åågij dxi dxj We can use Einstein’s summation convention, which says that you drop the sigma notation and sum through the indices which appear both in subscript and superscript. So, the above expression can be written as ds2 = gij dxi dxj If you know gij, you can find every property of the surface. It is the most fundamental tensor for any surface. 3. Show that gij transforms as:                 where x¢ represents new coordinates. If g¢mn =   1 (for m=n)                         0 (for m≠n) then,                 Use it to show that Laplacian operator defined by:                 where superscripts represent the indices, not powers. has the general form                 4. Consider a particle in which wave property is just in time direction (particle is stationary). A = A0sin2pv0t ; v0 is rest frequency. Suppose particle starts moving with velocity u. Then, t ® g(t – vx/c2)                 where g = 1/(1 - v2/c2)1/2 Put this transformation in the wave equation; compare it with standard wave equation                 A = A0sin[2pv(t-x/u¢)]. Use it and Einstein’s equation to get de-Broglie wavelength for this wave. Here the two velocities u and u¢ are different. u is the velocity of particle with respect to frame and u¢ is the wave’s velocity. Show that uu¢=c2; hence u¢³c. 5. Consider the Schrodinger equation:                                 If U(x)=x, try to solve the equation. Look for Airy functions in a standard text for differential equations for more details. 6. Relativity of energy seems a rather strange concept. Consider three bodies, each of mass m1, m2, m3. In frame of the first, the other two are moving with velocities v12 and v13. Consider Newtonian approximation. So, for the first, the net energy of the system is U11+U12+U13+(1/2)m2v122+(1/2)m3v132. Uij is the potential energy of the body j with respect to the body i. For the second, the energy is U21+U22+U23+(1/2)m1v212+(1/2)m3v232. Similarly, for the third, the energy is U31+U32+U33+(1/2)m1v312+(1/2)m2v322. Assume that Uii=0. Reduce more variables ® Uij = -Uji. Now, equate all the energies. You get two equations with three variables Uij. Why is it useful and necessary to have more variables and how can you explain the consistency of operations performed to reduce Uij variables? Generalize the result for an n-particle system. 7. Consider again the Schrodinger equation. Let U(x) be a constant. Solve it for E > U and again, for E < U. If it were a classical particle, wave function would be zero for E < U. But, the presence of finite solutions means that particle can be there. It is called Tunnel effect. 8. Black hole solution obtained first by Karl Schwarzchild was due to following metric structure of space:            At , things become awkward. Such condition forms black hole. This radius is called Schwarzchild radius. What is inside the black hole? Is the radius metric negative there? Check out if the transformation r®1/r can help us. If it can, we will have strange space structure. From 0 to 2GM/c2, we have 1/r instead of r for coordinate system. After that, we have r going normally. 9. Jacobians help in transforming from one coordinate to another. Consider a set of coordinates → (x1, x2, …xn). Let the new coordinates be (y1, y2, …yn). Show, using chain rule of differentiation and property of determinants, that                          The determinant is called Jacobian. 10. Schrodinger equation is written in time dependent form as:                                                 Then, what is this half differentiation? Show that                                 Use Gamma function instead of factorial notation and try to find the Eigen value of such momentum operator. 11. Consider a uniform positive charge distribution on a rod. In a frame moving relative to the rod, the rod appears shortened because of the length contraction. Faster the frame moves, shorter the rod appears to be. Then, the charge density must increase and reach a limit where repulsion will make the system unstable. But, the rod is still stable in rod’s frame. How can the principle of equivalence be saved if such condition arises? 12. Try to obtain the equations of planetary motion using the Langragian formalism. Take kinetic energy to be ½mr¢2 + ½mr2q¢2 (the rotational and translational kinetic energies; apostrophe denotes time derivative) and the potential to be -GMm/r. Now, consider a small body which is capable of rotating about its own axis in the outer space. Suppose it starts rotating with angular velocity (about its axis) depending on its distance from the centre of the Earth. Suppose the body has to go in a straight line to Mars and it started at a distance r0 from the Earth’s centre, find the dependence of angular velocity on the distance from the centre of the Earth using Langragian (because it is the easiest way). If the body takes energy from the Sun (assume that potential due to the Sun remains constant in the motion), find the energy required to reach Mars whose distance is large compared to r0. Take r0 to be (say) 40 km more than the Earth’s radius.

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