Relativistic Quantum Mechanics

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.

What happens when cows move rapidly !

Cows not moving and moving fast. How does this difference impact the image in a modern digital camera?

Honestly I haven’t checked it with old day manual camera neither do I remember what impact motion brings into mages taken by such, eg does anyone remember when he/she took a picture of a friend standing in a platform and a train was coming, what happened to the image due to such motion?

Do you see where the fuzziness coming from, in the pictures, where the cows are moving?

Its coming from the relative motion between “objects being imaged” (cows and grass etc) and equipment of imaging (camera). This fuzziness is quite small when they are both still (the object and camera, wrt each other).

New forms of Uncertainty Relationships in Quantum Mechanics.

ΔE.Δt ~ h

Δp.Δx ~ h

ΔL.Δθ  ~ h

* Shouldn’t the constant above be ℏ/2 ? Let us first clear up some air of confusion.

The above relations are kind of vague even though look like canonically powerful ways to represent the formal concepts of the Heisenberg Uncertainty relationships. I have myself confused with these at times … with the added degree of confusion coming from h or h-cross?

But there is nothing to be ashamed of, if one makes such mistakes.

Does ℏ/2 come due to normalization of wave function or discrepancy in definition? eg do they come because variance (ΔE, Δx) and standard deviation (σx , σH ).

Uncertainty Principle and Photography !

why a moving object becomes fuzzy when you take its picture. Speed bears an uncertainty with momentum (hence energy ) just like time with energy and position with momentum. But for photons which are always ultra-relativistic we should not talk about its positions. Due to speed (relative motion of objects such as your and moving while other body parts being still) energy and momentum are uncertain. Hence position becomes uncertain. (Do not confuse between position of photon vs position/location on your image although its connected to wave-function collapse BEFORE or AFTER the observation ? is the question you should be asking, BEFORE the observation no sense of photon’s position, but AFTER collapse we do see only a particular outcome in terms of fuzzy images.)

Why is the helicity for a mass-less particle Lorentz invariant?

Result; now that photons are mass-less, their energy, momentum, speed, etc are no more variables, in the sense of arbitrariness. They are constants, taking only a few values, but constant in a given situation. But other particles have these properties; arbitrary. So electrons energy and momentum are not fixed, but arbitrary.

But as long as we are considering only elementary particles (that is, we are in a Quantum Zone) eg, electrons, protons, photons, and not nutmegs, soccer balls and airplanes and satellites there is another quantity that is of important consequence that is constant. Spin; whether a mass-less particle or not, spin has the same magnitude for them. that is spin is same for photon, its always 1. Spin for an electron is always 1/2. Spin for proton is always 1/2. Its for this reason photon is called a Boson**. Any thing with spin, 0, 1, 2, etc will be a Boson. Anything with spin 1/2, 3/2 etc will be called Fermion.