The Boring History Of Quantum Theory

So, this boring history of quantum theory starts when scientists began realizing that not everything can be explained using classical mechanics (though even today i dont see how it describes things πŸ€”; nothing seems real). Actually, in a sense they were already aware of their ignorance but as the experimental observations began piling up and they couldn’t figure out why, they started doubting that there is more to this world than the mold they were trying to fit it all in. Too bad πŸ’”πŸ₯ΊπŸ˜‘… But life goes on…

If I remember correctly, there was this blackbody radiation experiment; it turned out that warm bodies are quite picky in what kind of radiation they want to emit (like i set aside anything that i don’t like in my plate, no matter how fresh or healthy it is) no matter how powerful or energetic. And then Max Planck came to the rescue and told us that the electromagnetic radiation is very picky and can’t be helped.

In a pile of pages, there was a paper explaining how matter particles ignore the number of photons coming to its rescue unless they really struck the right cord. How romantic ☺❀πŸ₯° After reading this paper that person (you know who 🀭) declared that not only the electromagnetic radiation but atoms themselves are very picky. There are teenii tiny bits inside them; these little babies are very nauseas about leaving their home πŸ₯ΊπŸ˜­πŸ’”

I don’t know if he woke up from a dream where he saw a swarm of photons shouting, We Are Bored…, We Are Bored…, We are Bored…!!! It was really cute, so he decided to do something about it. He thought why not announce a race between them. So, he made two holes in a paper, and… listen carefully, made them Real Close. “Get, Set, annnnnnddddd GO!!!”, he said. Cute, chubby, and yellow photons glittering everywhere around him. The Sun must be really happy today; its children, little pieces of its own heart are having a bllaaassstttt!!! Those yellow babies started racing faster and faster…. “me going fassttt πŸ₯Ί” … “nooo me going faster”… but they didn’t realize that they were all racing at the same speed. “It’s so fun to see these cuties mushmashing in there”, he thought. “I should do it more often”.

Sadly, when people found out what was going on, they weren’t happy at all πŸ˜‘πŸ˜. They said that’s not how a swarm of photons should behave! It’s against the law… blah blah blah. But Remember, the photons were already having a blast so they never listened to anyone, and they even started enjoying when humans were annoyed by their behavior. So they decided to do something bigger…. 🀭

One of them, who had just arrived on earth and was pretty bored travelling for millions of years, suggested. “hey buddies, i’ve heard they are going to install cameras in tomorrow’s race, to spy on us!” … “whooaaaa really!”, a little photon who just got up from its nap said. “I’m telling you, we need to really mess it up!”. And after explaining them what its plan was (author: which is always kept secret in movies so i’m not gonna tell you either) raising both of the hands said, “Are You With Me!??” And everyone shouted, “Yes We Are! We Are Bored… Yes We Are! We Are Bored”.

And again, no one was happy how the swarm of photons were behaving. Suddenly, they were famous all over the world. They kept making new plans for annoying the humans, (who were always so keen to have a control over it all) and enjoyed every bit of it. People started telling these stories generation after generation.

Not only humans, but there were others who were not happy about their fame. And they decided to beat ’em up. They were teeny tiny bits inside the atoms. But no matter how silently they talked about their plan, the atom was bound to find out. So they took it in their plan as well. 🀫

So now not only photons, but those teenii tiny bits and their atoms also became famous. But photons had their own charm and no one could ever take their place. πŸ₯°

Humans were always having a blast, destroying everything, and killing each other but it wasn’t enough so they decided to find out why and how the photons and atoms were having such a blast without killing each other πŸ€”. But NO ONE EVER FOUND OUT WHY OR HOW. 🀯

While photons were celebrating and having a blast, humans were getting mad, scribbling on papers, trying to find out how they can figure out the secrets of the universe. (🀫🀭There were no secrets at all🀭🀫).

But there were some photons… They leaked some of the information to a german guy…,”Hey buddy, what’s up? You know we have decided to have some privacy. Hope you people don’t mind… and blah blah blah”. And he got so happy that he couldn’t sleep all night and waited for the sunrise with a big smile on his face.

Books… hmmm…

if a book doesn’t make you smile, throw it away and actually throw away all of them until you find the right one β€πŸŒΌπŸ¦‹πŸŒΏ

Real Analysis by Terence Tao! Oh yeah!

Never thought, the day would come when I’ll read this masterpiece – Real Analysis. Today I’m going to review a few things that got stuck in my mind.

I should be sleeping now, my phone’s battery is just 6%. But I’ll just write the things I remember. Also I dont have the book in my phone. So, lets goooo!

Okay so it starts with “Why bother” section. A perfect start for people like me. In it, he said, Analysis is the Why of calculus. And most of us get satisfying feeling in knowing the Why even if knowing How does the job.

He then asks a lot of questions, like what is the smallest positive number after 0? And similar ones..

Then there are a lot of examples showing how the known methods of computation in calculus can lead to wrong conclusion.

These examples include convergence of series, for one.

Like, if you add 1 – 1 + 1 – 1 + …

like this, 1-(1-1)-(1-1)-.. then its 1.

but if you sum it like this, (1-1)+(1-1)+(1-1)+…=0

so which one of these is correct?

Another example was, swapping the rows and columns and finding that either way, the summation is the same. Though, it doesn’t work if the summation involved infinite number of elements.

There was also an example of swapping the variables in a second order partial derivative; the answer was different after swapping.

Similar example was shown for interchanging the limit and integration.

We have used these methods and techniques but have no idea how and why they work and most importantly as shown in all those examples, where they don’t.

He promised that we will get all these answers as we read more.

But I’m not very sure if I will.

I’d like to try atleast a few more chapters.

Seeya!


Disclaimer: I am neither a mathematician nor I strive to be one. Period.

Inner and Outer Product in Quantum Mechanics

Alright, This is NOT going to be a very revealing kind of post like was planning but here you go…

In quantum mechanics, the particles are represented as state vectors. These state vectors are part of a state space for a given system or a Hilbert Space.

Let’s take two state vectors, \psi and \phi from a given state space. Now, we can perform some operations on these, just like we do with coordinate vectors.

But to define the inner product of state vectors we’ll have to define a dual space too. Given a basis, we can write these state vectors as matrices. Kets are written as column vectors and they are elements of state space, whereas the Bra vectors are written as row vectors and are elements of the dual space.

So, a ket vector is written as | \psi> and a bra vector as < \psi|. Their inner product is defined as,

< \psi|\psi>

The resulting vector is either a scalar or a complex number.

If the inner product is one then, the vector is said to be normalized and if it’s zero, then they are orthogonal.

Whereas, the outer product results in a matrix or an operator and it’s written like this,

|\psi> < \psi|

I’m not really sure if all the outer products result in actual operators in quantum mechanics. Because in QM, they are mostly Hermitian Operators. I’d really like to think more about it in future. πŸ€” Hopefully πŸ˜„πŸ˜„

see ya!

Inner and Outer product in Linear Algebra

Let’s take two coordinate vectors, \textbf v and \textbf u. Both of them can be represented as column matrices (why not as row matrices?) of order m \times 1 and n \times 1.

The inner product of these two coordinate vectors is, \textbf v^T \cdot \textbf u.

\textbf v = \begin{pmatrix} a_1 \\ a_2 \\ a_3\\...\\a_m \end{pmatrix}

\textbf v^T = \begin{pmatrix} a_1 & a_2 & a_3 & ... & a_m \end{pmatrix}

\textbf u = \begin{pmatrix} b_1 \\ b_2 \\ b_3\\...\\ b_n \end{pmatrix}

\textbf v^T \cdot \textbf u = \begin{pmatrix} a_1 & a_2 & a_3& ... & a_m \end{pmatrix} \cdot \begin{pmatrix} b_1 \\ b_2 \\b_3 \\ ... \\b_n \end{pmatrix}

\textbf v^T \cdot \textbf u = ( a_1 b_1 + a_2 b_2 + a_3 b_3 + ... + a_m b_n)

Notice that it only works if m = n and it always results in a matrix of order 1 \times 1.

The outer product of these two vectors is \textbf v \cdot \textbf u^T.

\textbf v \cdot \textbf u^T = \begin{pmatrix} a_1 \\ a_2 \\ a_3 \\ ... \\ a_m \end{pmatrix} \cdot \begin{pmatrix} b_1 & b_2 & b_3 & ... & b_n \end{pmatrix}

\textbf v \cdot \textbf u^T = \begin{pmatrix} a_1 b_1 & a_1 b_2 & a_2 b_3 &...&a_1 b_n\\a_2 b_1 & a_2 b_2 & a_2 b_3 &...& a_2 b_n\\a_3 b_1 & a_3 b_2 & a_3 b_3 &...&a_3 b_n\\...& ... & ... & ... & ... \\a_m b_1 & a_m b_2 & a_m b_3 & ... & a_m b_n \end{pmatrix}

If you have a column vector of order m \times 1 and a row vector of order 1 \times n respectively, their outer product will be a matrix of order m \times n. Here m need not be equal to n.

The significance of inner product in quantum mechanics becomes clear right away, but what about the outer product?

To be continued in the next article: Inner and Outer Product in Quantum Mechanics

A complex-looking simple problem

I would like to share the solution of a very simple problem. I tried solving it during an exam. It looked really complex (to me) at first but then I finally did it!

Problem: There is a square of side r. There are two circles drawn with radius equal to the side of the square. What is the probability of a particle to be found in the shaded (yellow) region?

Usually when we solve the probability problems (not that I am an expert), the formula that we use is, (number of desired items e.g. red balls) divided by (total number of items e.g. red and yellow balls). Similarly, here we can use, (area of shaded region) divided by (total area). To calculate it, we need to figure out the area of the yellow region.

It can be seen, that there are two circles here. And the area of circle is, \pi r^2, but the circle inside the square is just \frac{1}{4}^{th}, so area of first circle inside the square is, \frac{1}{4} \pi r^2. The area of square itself will be r^2. Now if we subtract the area of the circle from the square, we get the area of one purple region. r^2 - \frac{1}{4} \pi r^2 = r^2(1 - \frac{\pi}{4} ). The area of second purple region will also be the same, and hence the total area is, 2 [r^2(1 - \frac{\pi}{4})]. In the end what we need to do is, subtract this from the area of square to get the area of yellow region. Which is going to be, r^2 - 2 [r^2(1 - \frac{\pi}{4})] = -r^2 + \frac{\pi}{2}r^2.

Finally, we can get the probability of the particle to be found in the yellow region, which is, \frac{[-r^2 + \frac{\pi}{2}r^2]}{r^2} = \frac{\pi}{2} - 1.

Remembering Hawking

The series “In to the Universe with Stephen Hawking”, when first premiered world wide in 2010 really kindled my interest in him. He introduced me to Black holes, and made me aware of my existence in this universe.

I have read (or at least tried reading) most of his books. Especially “A brief history of time” really made me realize why I was studying anything in the first place; to understand the universe we live in. The George book series he wrote with her daughter Lucy was also interesting; perhaps my first ever sci-fi series (if we exclude “The little Prince” which I read in Hindi). George’s parents were basically environmentalists and used to eat most of the things grown in their garden. But George was not very supportive of that fact. The drawings were also really beautiful and made the story more alive. The first time George accidentally goes to Eric’s house, trying to find Freddy, his pig; there he meets Annie and her parents Susan & Eric. On his very first interaction with Eric, they talk about how a charged scale attracts water towards it. He also gets to see Eric’s magical computer, Cosmos. It has this feature of opening a portal to any part of the universe. And later, Annie and George secretly go on many adventures. Like their rather reckless trip to Saturn’s moon Enceladus; they even rescue Eric when he falls into a Black hole. In another book of the series, Annie, George and Eric sit back and witness the creation of whole universe unfolding through the portal created by world’s smartest computer, Cosmos.

I was very excited when I read the explanation of Hawking radiation in brief history but sadly it’s flawed. The discovery certainly is really interesting. There is a somewhat nice and detailed description by Ethan Siegel but unfortunately, we can never fully comprehend or appreciate the concept until we read and understand quantum field theory and general theory of relativity in detail.

I was also really fascinated by the nautilus shown in “The Grand Design”. The way these books described things was so different from any text book I had ever read before. Mainly the things about origin and fate of the universe, like why do we exist and where do we come from struck me so deeply at that time and still do.

Blogroll: 2020

Why most Hacktoberfest PRs are from India : The author of this post has tried to show all of his frustration throughout this article. Its rather long, but very well written. I’m glad that someone took the trouble to type it all out. I don’t really want to add anything but would like to say that I agree with the views of the author.

The Butterfly Life Cycle : I came to know a very rare aspect of Adonis Blue butterfly through this post. Usually, the caterpillars form a chrysalis attached to the plant (to make it look like a leaf), but the caterpillars of Adonis Blue take very different route. Their caterpillars are adopted by ants! Ants take them to their nest because they smell and sound like the ant larvae. But sadly, when these caterpillars are there, they eat ant’s larvae too. There are similar ant-butterfly interactions found in nature all over the world.

Math reading challenge 2020

So, I found this article by Evelyn lamb in her blog Roots of Unity at the end of 2019. It had some prompts to help you find math related books. I wasn’t able to complete all of them but I did try some.

I read three books for the prompt, “A work of fiction in which a main character is a mathematician“. The very first book I read was, “The devotion of suspect X“. It was beautiful. Set in Japan, it’s about a mathematician Ishigami, who is a High school teacher. One day, when he was trying to take his life, someone rings the doorbell. His new neighbors, Yasuko and her daughter Misato. Without knowing, they save his life. And from that point, he does everything to protect this family next door. It’s mysterious, and very emotional at times. Next was, “The Housekeeper and the Professor“. Due to a terrible accident in 1970, a mathematician is now only able to remember last 80 minutes of his life. Like a hard disk which has only 80 minutes of memory limit; anything new is recorded on top of that. So whenever he wakes up in morning, he doesn’t remember a single thing that happened yesterday. Many housekeepers are assigned to take care of him but no one lasts. But it’s about to change. The tenth housekeeper, who is the youngest in the company takes the job and then starts a story of friendship between the housekeeper, her son and the professor. They go on many trips and he tells them interesting things like the importance of square root, amicable numbers, perfect numbers, triangle numbers etc. But the good days come to an end. His 80 minute memory tape is also broken and he cannot retain even a minute of new memories. Sadly, in the end, he is transferred to a new care facility. Then I read “Uncle Petros and Goldbach’s Conjecture“. Its story revolves around a mathematician who is obsessed with the Goldbach’s conjecture (Every even number greater than 2 can be written as a sum of prime numbers). And how he puts in so much effort to discourage his nephew from pursuing higher mathematics, because he doesn’t want him to get mad and obsessed like himself. It also has a very tragic ending.

For the next prompt, “A graphic novel about math or mathematicians“, I read “Logicomix: An Epic Search for Truth“. It had really awesome script and beautiful drawings. I read about Russel’s paradox for the first time in it, which is introduced by Bertie Russell himself. It goes something like this, if a set consists of all the sets that do not contain themselves, then does that set contain itself as an element? If it does then it does not follow the property that it should not contain itself; If it doesn’t contain itself then it does follow the property and thus it belongs to the set but then again if it does it shouldn’t and if it doesn’t then it should.

For the most part, the story follows the life of Bertrand Russell. There are many events of his life from childhood to a university student; how he started questioning everything that others took for granted. It also contains many dialogues and fictional meetings between famous mathematicians, which makes it even more interesting.

logicomix
Logicomix

So basically, all I read was mathematical fiction. But I did learn some concepts in number theory and logic here and there.