Learning Quantum Mechanics

Why the Dirac Notation Works?

| Cosmoshivani

When we first learn Dirac notation (the bra and ket thing to represent state vectors), the notation for the inner product becomes so succinct.

But why? Why does it become so succinct? I know there are a lot of questions that are meaningless or they need not have an answer for the theory to work but there should be an explanation for it. Is there a theory or something deep inside that makes it work. Or it is a mere coincidence.

Are there other such notations that promise to make the notations look more succinct like the Dirac notation or is it unique?

I like to think about QM sometimes…

| Cosmoshivani

My ramblings – part one

I woke up this morning and just randomly opened a chapter in the quantum mechanics book. I could see all those paragraphs, some underlined, things written here and there. “Yes I’ve been here before”, I thought. I don’t really wanted to read but then I started reading unwillingly. Questions started popping in my head. Its like when you read a thing but when you come back to it after a long time, you ask something to yourself and then the whole paragraph makes sense, like why it was written in the first place. “Oh, so that’s what it was trying to say!”. Now you understand that thing because you asked the right question in your mind. And you were curious to know why. The answers were just sitting there (like all of the truth) but what was missing is the right question.

And that right question was, “How do you solve a time dependent schroedinger equation?”. I have only focused on how the time independent one is solved that I forgot why we were doing it in the first place. You get different solutions for different energy values, so called infinite solutions. But then you can always form a linear combination of all of them to get the more general solution, or the solution of time dependent shroedinger equation.

My ramblings – part two

In mechanics, finding x and t is just the beginning. After that we have velocity, acceleration, force etc. to calculate. But calculating psi in quantum is such a difficult task.

In mechanics there are so many situations you have to apply those formulas to. But in quantum mechanics its so hard to even get past the first step. It’s just one of the many problems during qm learning process.

And another thing that’s even more daunting is that, the whole edifice of QM is based on the Hamiltonian formulation of classical mechanics (whatever that means 😕).

Probably because QM is so hyped of being mysterious and fundamental; that’s why there is an awe when we read these things, even if it’s hard. But in that sense, mechanics or statistical mechanics is as much mysterious and fundamental. But it still doesn’t feel that way (to me). I don’t know why.

The Boring History Of Quantum Theory

| Cosmoshivani

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 🤔; 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 💔🥺😑… 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 ☺❤🥰 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 🥺😭💔

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 🥺” … “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 😑😐. 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. 🥰

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 🤔. 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.

Inner and Outer Product in Quantum Mechanics

| Cosmoshivani

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. 🤔 Hopefully 😄😄

see ya!