1. Show that
zeta function defined by:
is equivalent to the integral
Show that it is equivalent to
Find the relation between
explicitly (i.e. without using these integrals). Then,
try to represent
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
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.
xi, i = 1,2,3,…. represents the coordinates
and we have used the chain rule of partial
Square this and show that ds2
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
= gij dxi dxj
If you know gij, you can find every property of
the surface. It is the most fundamental tensor for any
3. Show that gij transforms as:
represents new coordinates.
= 1 (for m=n)
0 (for m≠n)
to show that Laplacian operator defined by:
where superscripts represent the
indices, not powers.
the general form
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.
= 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
are different. u is the velocity of particle with
respect to frame and u¢
is the wave’s velocity. Show that uu¢=c2;
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
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
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 <
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
8. Black hole solution obtained first by Karl Schwarzchild
was due to following metric structure of space:
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
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
The determinant is called Jacobian.
10. Schrodinger equation is written in time dependent form
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
12. Try to obtain the equations of planetary motion using the Langragian formalism. Take kinetic energy to be ½mr¢2
(the rotational and translational kinetic energies;
apostrophe denotes time derivative) and the potential
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.