Random Thoughts

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.

Books… hmmm…

| Cosmoshivani

if a book doesn’t make you smile, throw it away and actually throw away all of them until you find the right one ❤🌼🦋🌿

Real Analysis by Terence Tao! Oh yeah!

| Cosmoshivani

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.