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[edit] Question 1: Quantum Mechanical Operator associated with Mass
In QM we know that there is an associated hermitian operator with every observable. Mass is an observable. What is its associated operator? The question extends to charge of particles also?
(A guess answer might be as follows: In introductory QM we have hermitian operators for dynamical observables (like E,T,p,x..). The rest mass of a particle on the other hand is an invariant quantity which can be expessed in terms of dynamic variables by relations like E2 = p2c2 + m2c4. Inserting the corresponding operators for energy and momentum we get the mass squared operator. But I do not know if this makes sense. If it does what would mass squared operator and its associated eigen values imply.
No clue abt the charge though.
[Himanshu Raj]
Your question is an interesting one. One point is, is mass a parameter or an observable? For example, Newton's law defines mass as follows: Second law states that force is proportional to acceleration and the constant of proportionality is the mass. So, here mass is a parameter. Other definition would be the quantity that enters while defining the momentum. Or, in a relativistic case, the quantity 
. In all these cases mass is not an observable.
Further, as far as observations go, one can measure energy and momentum directly but not the mass.
So, in quantum mechanics, there is no operator for mass. But when you do second quantization and introduce fields for particles, there is an operator for mass. If a field ψ represents a particle, the mass operator is 
( and similarly for charge ). But again here m appears as a parameter in the operator.
[phatak]
[edit] Question 2: A QUESTION ON INVISIBILITY
When I take photograph of a moving fan with my friend's camera (2 MP),the image does not show the still blades of the fan but are actually like a faded wheel. Even that is what I see with my eyes. Now let us consider the time taken by light to come reflected from the fan blades onto the camera lens or my eyes. It will be nearly about 2metres per 3*10^8 metres per second ; nearly of the order of nanoseconds. Then I may to an approximation assume that the speed of the blades of the fan is not so much high to literally change positions or at least complete a full rotation. Yet I see no distinct blades of the of the moving fan , neither with my naked eyes or with a camera. Perhaps and mostly, what I see is not due to change of Position but it is something limited in my detecting device that gives me this illusion. Well I have my persistence of vision and so I cannot see with my naked eye a completely still frame of the blades while the fan is moving. Probably the the same thing with the camera. It has a limitation to the number of frames it can capture per second , usually 15fps or 30 fps. But a careful thought induces a relative fact. When I see the still blades of the fan they are actually opaque to me. But in case of a moving fan I can see the the roof behind it. Has it turned TRANSPARENT? Perhaps no! There is no change in the nature of the fan. But the illusion really attributes to my EYES, it has a limited sensitivity. But every thing(device) has its own limitations. NOW
Can I have some opaque thing made INVISIBLE by just moving it very fast with a speed enough to hardly be detected by the optical detector?
A induction may be referred : The more is the speed of the fan the more is its transparency. THANX IN ADVANCE!!!! suryans
The answer to your question is in your statement of the question itself. The reason why the fan appears blurred and you see the seiling is because of persistence of vision. Our eye takes about a tenth of a second to recognize an object. If the object moves during this period, it appears blurred, that is, it appears (approximately) as sum total of images or the average image of the object over tenth of a second. Same is true for a camera because there is a finite shutter speed. Let us imagine that the angular span of the blade of the fan is θ and its velocity is nθ per second with n much greater than 10. In tenth of a second the angle traversed by the blade is much larger than the angular span of the blade. So, during this period, one is seeing the blade of the fan for part of the time and ceiling for rest of the time. What we see is sum total of these two things so the fan appears blurred and transparent. The blades haven't become transparent.
A related phenomenon is, when one sees the fan in tube light, the blades appear to be moving slowly and at some angular speeds of the fan they appear stationary or even rotating backwards. That's because the light coming from the tube is not continuous (in time) but is emitted intermittently so the fan is illuminated with light which is emitted with certain frequency (this is not the frequency of light waves but the frequency at which light is emitted).
[Phatak]
[edit] Question 3 :THE COSMIC BACKGROUND RADIATION AS AN EVIDENCE TO BIG BANG
The CBR is estimated to have radiated after 3*10^5 years of BIG BANG which itself is estimated to be an event 15*10^9 years ago. But the CBR is also detected now. If the CBR traveled at speed of light then how it can reach us now? For that EARTH has to travel with a speed of about 0.999999 c Is this expectable???? [suryans]
The cosmic background radiation is understood (can be considered as an evidence of) big bang cosmology. In this model the universe is considered to be uniform, homogeneous and isotropic everywhere, at least on a large scale. Isotropy means it appears to be same in all directions and homogeneity means it is same at all places at a given time. It began with a big bang (it can be called as the origin of time or zero of time for our time measurement) and has been evolving since then. At certain time since the big bang it became transparent to light and light that was present at that time is what we see as background radiation. This transparency happened at all places in the universe at the same time (as a consequence of homogeneity assumption) and the photons which were present at that time have been traveling with velocity of light in all directions and at a given point, they are reaching from all directions. The cosmic background photons we are seeing now were not emitted here but at a very far away places in different directions. On the other hand, the photons which were emitted here have traveled a long distance and have reached far away places by now. We cannot see them.
The isotropy of cosmic background radiation shows that universe is isotropic as well as homogeneous. Isotropy is obvious. Homogeneity follows from the fact that the radiation coming from two direction is same so the places which have always been well separated (to the extent that they could not have communicated with each other) had same conditions long back and hence homogeneity.
[Phatak]
