# waves and particles

## Inherent ability = difficulty * accomplishment.

All of Physics is this “Inherent ability = difficulty * accomplishment”. Thats just intuitive but can easily be seen to correspond mathematically with the Principle of least action.

First the edifice: whats the problem? The problem is given you move in straight line when every direction is same around you, which direction will you chose? While you are waiting for a good answer from astrologers intelligent people already give a good hint. Think you have some inherent ability which is fixed.

fixed: which changes only if estimated wrong.

That inherent ability is actually action. Accomplishments are adjusted for difficulties, you waded through a swamp 5 meters you would have accomplished in sand 8 meters with that given inherent ability called action. Because action is abstract we have been sticking to time and path-length, but they are not as fundamental, they are merely specifics.

## What happens when squirrels do not move.

Stationary state and squirrels !

Okay so the squirrel isn’t moving around so much, giving a fuzzy image. We can call the instant during which the squirrel wasn’t moving akin to a “gross” stationary state.

But why is the railing of window fuzzy? It isn’t moving !

Diffraction.

Due to diffraction enough light is coming from the other side of the iron bar, to enable us to see whats there. That quadrangle is diffracting the light. But why its fuzzy?

Diffraction is by its own very nature even though purely classical in many sense, fuzzy towards energy if time window is small. [due to energy vs time uncertainty]

That is, full energy cycle isn’t available, because a small time window is chosen. If you time-lapse the photograph, diffraction will become quite insignificant. Although it might still be there depending on details.

## Waves, particles and Einstein !

Waves are something that have no mass and move at the maximum speed, mass m = 0. speed c = 1. So whats their momentum? p = m.v = 0? Right?

No. For pure waves; momentum does not come from mass. It comes only from motion.

(pure wave; they do not have mass)

For matter waves, on the other hand, momentum comes in two ways, mass as well as motion.

(impure, now they have mass)

Albert Einstein recognized this fact and derived his relation; $latex E = \sqrt {(pc)^2+(mc^2)^2}$

This relation is called as Einstein’s relativistic equation, also Einstein’s mass-energy relation. But more appropriately mass-energy-momentum relation.

Let us consider E as the hypotenuse, p and m; as base or perpendicular as is your choice.

triangle_copyThen $latex E = \sqrt {(pc)^2+(mc^2)^2}$ is Pythagoras Theorem; when p is momentum and m is mass.

For pure waves such as photon … the quanta of light, m = 0.

Hence the Pythagorean Triangle is now one, where the mass side is arbitrary small. Thus E = p.

## The time-energy uncertainty relation.

The time-energy uncertainty relation is a blessing in disguise which comes in handy to check various values that are quoted, so as to see if something is inconsistent or not. It’s very powerful in guiding to check if we are ourselves making something silly or not.

I have described in two recent articles — will link later, why.

1. One must be careful what energy and what time one is relating to, one just does not take any time and any energy and make a relation, in-fact one can see who is a good physicist from one who is a novice, by seeing how this relation is used by him.

This was joked by Landau: I can measure the energy and then look at my watch, time is just a parameter. But Einstein and Niels Bohr argued “during a very short time interval one must be careful what energy is allowed and what is not, there is a constraint on the windows of errors or uncertainties”.

2. Life-times are arbitrary variables as are energies, their means are not necessarily linked inversely as in case of the uncertainty relation itself, the latter gives a relation between the error-window which are linked inversely.

So watch out how much inconsistent description is given in an average article eg in Wikipedia and even in our text-books. These are training the future physicists very wrongly. One needs experience of solving good problems, one is to work in experiments of highest standard and understand them.