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 = ååg_ij dxⁱ dx^j

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 = g_ij dxⁱ dx^j

If you know g_ij, you can find every property of the surface. It is the most fundamental tensor for any surface.

3. Show that g_ij 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. U_ij 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 U_ii=0. Reduce more variables ® U_ij = -U_ji. Now, equate all the energies. You get two equations with three variables U_ij. Why is it useful and necessary to have more variables and how can you explain the consistency of operations performed to reduce U_ij 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.