Quantum mechanics is probably the greatest intellectual accomplishment or discovery , if you will , of the last century , but it is also the least understood , and the most unique of all sciences. It’s something you do after which you take a deep breath , look up at the skies and say with a loud sigh “You’re screwing with me , aren’t you?”
So , needless to say , this part of physics is the LEAST understood, but the best part is that EVERYTHING is a manifestation of quantum mechanics and there are many problems left to be solved! More than the problems there are in a Michael Bay movie ! There are still mysteries or puzzle pieces to be put together and I have no reason to believe that you couldn’t be the one to do it!
However, I won’t pretend to understand all of it either . I’ll give you what is needed to sound smarter and be the life of the party
So , we begin with a quote from a person who hated the very idea of Quantum Mechanics….Einstein.
Yeah , when I heard it first I went all ‘Say whaaaat?’ too , but it’s true..He had famously said that “God does not play dice”
(To which Neil’s Bohr said “Einstein , don’t tell God what to do ” )
Anyway , so what we are saying here is that the subject is full of uncertainty. You can generally calculate how far your car will go with the amount of fuel left in it but when it comes to quantum mechanics , you never know how far you will go , the amount of fuel you have left or even which car you drive on occasions.
So , to tackle this problem , we lean on probability , that’s right , we write probability distributions about the fuel in the tank , the car , the road and other blah blah. And on a number of such distributions we get a very narrow result of answers where the probability of some event to happen is almost 1! However , doing this is hard.
For eg. take an electron… go ahead take it….you have one now? good. Now , the probability distribution that the electron will react with the electromagnetic field is 1 , however , the electromagnetic field is always always fluctuating .. So , you don’t know what state it is in and as a consequence you do not know what your electron will do, so you rely on probability.
Now , if you are still thinking classically , you might be thinking ‘ The hell ? That’s impossible to say ! ‘ and well , that would be the case if you were calculating the air resistance at every point in space if the air fluctuated like the electromagnetic field does while on the trajectory of my fist getting to your face. But it is relatively easier to calculate probability distribution in quantum mechanics thanks to the methods or equipment we have today. But , it is mathematically very complex.
Now , back to the probability . The way physicists calculate probability is a little unique.. that’s because they calculate Probability Amplitudes.. that is to say
P = |A|^2 , where A is the amplitude and also a complex number . ( A complex number is something that has both a real and an imaginary part .. the imaginary part is whatever which is with the number ‘i’ .. ‘i’ is the square root of -1 .. which doesn’t exist… hence the word imaginary…jeez )
Now , lets see how this works.. take this electron gun…..go on , take it..don’t be shy.
Now , scare Young away by firing it on his double slit experiment apparatus.
Except , now instead of light going through those slits , it’ll be the electrons that you will fire.. since playing C.O.D or battlefield doesn’t make you really good with guns , the electrons you fire will be scattered like sheep without a shepherd.
(BTW , you will understand why we don’t treat electrons like actual bullets in later posts if you don’t know already , stay tuned 😉 )
Lets name those narrow slits S and T , and a random point on the screen as X
so , P(X) = | A(S) + A(T) | ^2
I.e amplitude of the probability of going through path S added to amplitude of probability of going through path T , squared.
P(x) = |A(S)|^2 + |A(T)|^2 + |A(S)A*(T)| + |A*(S)A(T)|
So , what a complex conjugate is , is that the sign of the imaginary part is opposite… thats all.. dont let the words throw you away
So , Lets math it a wee bit more
P(X) = P(S) + P(T) +2Re(A(S)A*(T)
Hmm… maybe this will help?
And now , if we look at P(S) and plot it on a graph , it’ll kinda look like a bell shaped curve. Ofcourse , P(T) will look the same , except it’ll be on the other side of the origin……. Kinda like the image above :P.. Except they’ll be broad bell curve.. and they will intersect at some point near the origin. Now , lemme just tell you what the amplitudes are .
A(S) = | As | e^iøs
A(T) = |At| e^iøt
and ,at the area where you cannot distinguish whether the curve is a by S or T which is the centre of the screen of the double slit apparatus
(It is kinda by both , so you get the idea )
|As| = |At| … well…. approximately. Since it is the root of amplitude and you can’t exactly tell the difference between the two.
So , see rejoice… things are getting easier?
Look , coz there is this undeniable probability at the centre of the screen , right? Coz , both the paths have that included in them… also , since they are nearly equal , we will consider one path to bugger off .
So , the combined probability becomes
P(X)= 2P(S) + 2|As)|^2 Re e^i(øs-øt)
Wait.. recognize that as the probability?
It becomes simpler now
P(x) = 2P(S)[ 1+ cos(øs-øt)]
I hear you loud and clear.. enough math for one post.. So , lets put our pieces back into place..
In classical sense , what we would get is the first part of the result we did get.
2P(S) , and the other part of it , [ 1+ cos(øs-øt)] , is the quantum mechanical interference. Meaning with that piece of ‘Mumbo Jumbo’ , you get more or less the position of something that is incomprehensibly small. And yet , the world around you would collapse if you couldn’t do this. The pc or phone or your tablet you are reading this with would not work the way it does if we couldn’t do this. Most things around you will be used as a paperweight .
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Stay tuned 🙂
– Rishikesh Jani