Introduction to quantum mechanics #2:How it began and who let the cats out?

In the last article , I gave you some technical insight about how,in quantum physics , we dealt with randomness , and we saw how to calculate the position of an electron shot from an electron gun. But , alas , we never spoke of what randomness was. And so , we shall , in a few moments.

But , before we delve into any of that , let’s think about something .. let’s think about the when and where .

It might not have begun this way , but let’s believe there were two philosophers, Aristotle and Democritus, fighting over a lovely lady. Democritus said “Stay away from her , or I’ll cut you in half!” to which Aristotle replied , not very creatively ., “Well , you stay away from her , or I’ll cut you into 4 pieces!”, then Democritus said “I’ll cut you into 8 little pieces!”

..and this went on for some time , until finally Democritus said , like that annoying childhood friend we all have or had

“I’ll cut you into further indivisible particles called Atomos!”

Not cool , Democritus.

But then Aristotle was also an annoying childhood friend ,the kind that says “Yeah , well right back at you infinity + 1 times! ” and implied that we could go on halving and halving and cutting and slicing and dicing and stabbing..ahh murder :)..wait what were we talking about.. oh yes.. He argued that we could do this forever and ever and ever and ever. He said that there was a continuum in space that was absolute.

Phew.. that was close.

And so it all began. The argument that continued for thousands of years, and for the first 2000 years , Aristotle was winning…But not for long..In the 1800s , there was a subtle kind of revolution.

See , it was around this time that the alchemists gave up. They finally said “Screw this.Screw this all. We can’t produce gold from iron…This friggin thing is different , everything is friggin different…EVERYTHING IS UNIQUE AND DIFFERENT!”

Then came John Dalton, a  chemist, meteorologist and physicist, and put forth these statements in front of the faces of various chemists and physicists

a)Matter is made of atoms i.e indivisible particles
b)All atoms of a particular element are identical
c)Atoms of different elements have different weights
d)Atoms combine to form compounds

Aaaand then came J.J Thompson, who , with his famous cathode ray experiment , showed that , uhh well… Dalton was kinda wrong , and the atom looked like kind of a pudding filled with positive and negatively charged particles , because they deflected in the tube. He also ended up saying that the electron ,the negatively charged particles , that we all know and love was a great deal lighter than the proton, the positively charged particles.

Aand then radioactivity was discovered soon after , which , it was theorized , would then allow radioactive spiders to me made and spider-man to exist… yeah, it was kind of a disappointment , maybe , but we did find that there were 3 types of rays which were emitted. The positively charged α rays , the negatively charged β rays , and the ‘I-dont-give a-flying-damn-about-your-charge ‘ γ rays.

(thinkgeek.com)

Then , Rutherford came along and designed an experiment called the ‘gold-leaf’ experiment.. it was pretty simple , what he did was took something that made α particles and made them emit in a particular direction. In the path of those , he put a gold leaf, and it was found that the no. of alpha particles detected with or without the gold leaf were the same . And thus concluded that it was mostly empty space and a new model of the atom was born. That there was a teeny-tiny nucleus and the teenier-tinier electrons would revolve around it in orbits. And the no. of protons would determine what the element was

Now even before Chadwick discovered the neutron , there was a lot of unpleasantness.

Since it was known that unlike charges attract, there was a doubt in the mind’s of people. They sat together one night drinking mead and ale and asked “Wait a goddamn minute , why won’t the electron spiral into the nucleus and tear the atom apart ?!”

 

How it came to that ? Well , if it was moving in a circular path around the nucleus , it meant that it had an acceleration , and accelerating particles gave out radio waves…But wait… if it is giving out radio waves, it is losing energy! So it must spiral down into the nucleus indeed!

THIS MY FRIENDS IS WHY WE ARE HERE! THIS IS PRETTY MUCH WHY QUANTUM PHYSICS WAS INVENTED AT ALL.

OHMYGOD

Aaand it is now we come back to where we left the first article. The young’s double slit experiment had detected a ‘interference pattern’ when light had passed through two slits and ended up on a screen. It was what we would expect would happen if light were a wave. A crest meeting a crest or a trough meeting a trough would create a bright spot on the screen , but the places where a trough would meed a crest would cancel out and you’d get a dark dark spot on your screen.

It was thought at the moment that this only happened with waves and not particles of matter… But they were wrong.

Let’s do a small thought experiment now. Suppose I have infected you with my new virus that I have made because I am totally awesome and can do that.

What happens to you in the disease caused by the virus is that you change your skin colour in a specific order , say Red to Green to Blue and back to Red , every second.

So , you go R-G-B-R-G-B-R-G-B-R and so on. Imagine this as a physical law.

So , using basic math , I can figure out what colour you’ll end up being at a point t if i know what colour you started with.

Now , lets say , my murder associ….err I mean lab assistant injects you with a similar kind of virus , but , you change colours in the reverse order. Meaning you go B-G-R-B-G-R-B-G-R and so on and so forth.

Who wouldn't have fun with this , man?

Who wouldn’t have fun with this , man?

Now, I can safely say that if I with the first virus and let it run for 100 seconds , and then injected the second virus and let it run for 100 seconds , You’d be the same colour you started with.

Now , lets talk bout’ that randomness like I promised. The randomness here is such that , at an unpredictable moment ,for one second , you would stop changing colour. Or that you would skip a colour and move on to the next one at some point for just one second.

Now , I cannot safely predict what you will be after a 100 seconds. There will be a equal probability for you to be any of the 3 colours.

There is a kinda similar randomness when it comes to quantum mechanics. For eg. when individual electrons pass through one of the slits , there is an element of  ‘randomness’ that it will be kicked upwards or downwards.

This is where things get freaky.. If I were to reverse the physical law right now , and cause the electron to trace it’s path back , it would go back inside the electron gun

Almost as if the randomness didn’t matter at all. You started out as one colour and you still ended up the same colour when the different virus was inserted and allowed to run for some time. But the randomness was still there!

Wait.. it gets weirder… and I mean waaay weirder… This only happens if we don’t observe the electron at all. If we do watch it , without interfering , it doesn’t go back to the gun if it was knocked upwards or downwards.

And , if you let it through to both the slits , it forms an interference pattern.

What exactly is going on here?

If I had to explain it in the least words I’d say

‘Look : Particle , Don’t look : Wave ”

That is how the electron doesn’t spiral into the nucleus , and that’s how it overcomes the randomness! It’s a wave when we don’t look at it, and a particle when we do.

This is called ‘The wave-particle duality’ .

(http://www.swellpath.com/assets/Warning-Math.jpeg) But it is important if you wanna know what is going on further in

So , what does a wave look like… mathematically?

Well , we generally represent a wave like this: (In one dimension)

ψ = Ae^i(kx – ωt)

Which , if you know the famous Euler form, can be written like

Ψ = A [Cos(kx – ωt) + iSin(kx – ωt) ].

Which shouldn’t be surprising because all we have done is represented the wave in terms of sine waves and cosine waves. So , this is clearly a wave term , with the amplitude of A.

 

Now , if we differentiate that very same wave , keeping time constant

dψ/dx = ik Ae^i(kx – ωt)

wait a sec.. thats ‘ikψ’

OMG that was so much fun , lets differentiate it again

I am , trust me.

I am , trust me.

So , whats d^2ψ/dx^2 ?

d^2ψ/dx^2 = (ik)^2 Ψ

but hey , what’s k? k = 2π/λ

( You would know this if you have studied waves before, but if you haven’t ,k is called the wave number and measured in radians per metre) is  divided by the wavelength λ in metres)

If you know that momentum of the wave (p)  = h/λ

then p= hk/2π

and h/2π is the new planck’s constant , so

p = h(bar)k

and surprise surprise

k = p/h(bar)

So , what do we have now?

d^2ψ/dx^2 = -p^2/h(bar)^2 ψ

And this can be written as

-h(bar)^2 * d^2ψ/dx^2 = p^2 ψ

Wait.. are you there? Are you paying attention ?

EMMA WATSON AHEAD!

 

 

 

In my experience , this always works.

In my experience , this always works.

Let’s remind ourselves that Total energy = kinetic energy + potential energy

So , E = p^2/2m + u (Just a different way of writing it )

But , Schrödinger wasn’t happy with this equation. Neither was his cat. It used to hiss whenever it saw this equation.. And I imagine European cats having an accent so , i’d say the hiss sounded like ‘Psi’

 

Well , I tried.

(SMBC) Well , I tried.

So , Eψ= p^2ψ/2m + uψ

Wait…thats :

Eψ = -h(bar)^2/2m  * d^2ψ/dx^2  +uψ

And that , my friends , is called the time independent Schrödinger wave equation.

Eψ(x) = -h(bar)^2/2m  * d^2ψ(x)/dx^2  +u(x)ψ(x)

We can find out by remembering a simple fact that

E= h(bar)ω = hf and ‘math-ing’ that a little bit

we find that Eψ = ih(bar) dψ/dt (We differentiated with respect to time , this time )

So , if we put this in the Schrödinger time independent wave equation , we find

 ih(bar) dψ(x,t)/dt =  -h(bar)^2/2m  * d^2ψ(x),t/dx^2  +u(x)ψ(x,t)

And this is called the time dependent Schrödinger wave equation in one dimension…. for a non relativistic particle.

However , you can tell that this is not how you generally see it … the upside down triangle is missing here , isn’t it? That’s called the Laplace operator , which is basically the  divergence of the gradient.

So , finally becomes

 ih(bar) dψ(x,t)/dt =  -h(bar)^2/2m ∇^2  * d^2ψ(x),t/dx^2  +u(x)ψ(x,t) .

(NOTE: This is often depicted using a Hamiltonian, we shall get to what the Hamiltonian is later )

ENOUGH MATH!

What does this mean? This mathematical chaotic mumbo jumbo

Well , this is not often called the greatest breakthrough in quantum mechanics for nothing!

The cat guy knew what he was doing , and he was doing it well! In one master stroke, in an combination of brilliant syntax and semantics , he gave the world something that gives the interaction between particles and the relative fields between the particles! One line that describes everything it can do. Truly mind-blowing.

There’s quite a bit more to come. Infact , i’m determined to make you an expert in this subject before…..I die.. so don’t worry , we have time.

Oh and btw , we have an app!

Download it here!

And , give us a like at our facebook page for constant updates here!

See you! Over and out! 😉

-Rishikesh Jani

 

 

 

 

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Introduction To Quantum Mechanics #1

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.

And then there’s signs are this one.

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)|

by some math. The ‘*’ thing is called a complex conjugate QMCK

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?

Maybe if you get an A , she will agree to be your girlfriend....Maybe.

Maybe if you get an A , she will agree to be your girlfriend….Maybe.

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

and

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 )

You get

|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?

No no , I’m serious!

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)

Easy my ass.

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 .

If  you have any doubts about anything at all , email it to us at captainknowledge1@gmail.com

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Stay tuned 🙂

– Rishikesh Jani