mayusthings

Monday, 3rd April 2023

Quantum Notes - Time-Dependent Schrödinger Equation

Back at the notes this morning after a chilled weekend; I’m hoping to finish quantum today, as the quicker I make my way through notes and flashcards and onto proper practising, the better - fingers crossed for a productive day :)

It really depends on which question you're actually asking. Obviously we "generally" treat exponentiation as an algorithm, but most mathematicians with interest in foundations will tell you that integer exponentiation is counting something.

From a combinatorial perspective, we say a^b is the number of ways you can make a choice from a options b times in a row. And normally there are no ways to make choices from zero things (and thus 0^1 or 0^5 is 0), but if the number of choices you need to make is zero then it's not a problem. How many ways can you make an impossible choice zero times? One: you don't make any choices.

Oddly enough, this logic extends deep into fundamentals. A set theorist, category theorist, algebraist, or foundations person will tell you that A^B is the set of functions from the set B to the set A. If B is a finite set with b elements and A is a finite set with a elements, then A^B has a^b elements. (Why? To determine a function from B to A we need to decide where the first element of B goes, and where the second goes, etc. So we have to make b choices from a options.)

But this logic and notation extends to places you weren't really thinking about it. We use R^3 to represent triples of real numbers. But we can also think of that as a function from {1,2,3} to R! The first coordinate is f(1), the second coordinate is f(2), and the third coordinate is f(3). And that means we can write things like 2^N for the set of binary sequences, or R^N for the set of real sequences.

And in this framework, we still have 0^0=1. In general 0^n is 0, because you can't have any functions from a non-empty set to an empty set. But there is one function from the empty set to the empty set: the empty function. (This sounds like bullshit but it's a real thing and this is unambiguously right once you get into rigorous foundations stuff.)

---

Okay, so why isn't there a total consensus? If you do analysis (read: formal, fancy calculus), it's not that clear cut. Analysts don't generally want to count things; they want to make things continuous. And the original question makes total sense in an analysis framework.

If you look at the function f(a) = a^0, then f(a) = 1 for any non-zero a. To make this function continuous, you set f(0) = 1. But if you look at the function g(b) = 0^b, then this is undefined for b<0, and is 0 if b>0. For continuity, you'd want to set g(b) = 0.

Some group theory notes that I took today.

📚Summer Studying Challenge 📚

August 8 2021: What books are you currently reading?

🌟3rd August 2021 🌟

Summer Studying Challenge by @myhoneststudyblr

Q. What is your least favourite thing about the beach?

Probably getting sand in my hair lol

📷 The picture attached is my bujo spread for the first week of August.

Hey guys! I wanted to make a challenge that felt achievable to folks while still giving that sweet sense of accomplishment. My mental health has been rough lately so I kind of designed it for myself as I try to find balance between taking care of myself and taking care of my goals.

The rules: Every day for 14 days, study and post a picture of what you worked on. It can be for two minutes or for hours. Doesn’t matter—you did something! There are also mental health prompts for each day I’d love if you answered. They range from pretty light to more thoughtful. Do your best!

I’ll be tracking and reblogging from the tag #TwoWeekStudyChallenge. Please don’t worry about your pics not being aesthetic enough or that what you’re doing isn’t interesting enough. It is, I promise!

This is set up like the 100 Days of Productivity Challenge: no start/end dates. It’s a personal challenge that’s there when you need it!

Week One Prompts

Day 1: What self care did you practice today?

Day 2: What did you do that made you feel proud today?

Day 3: Have you experienced burnout from studying? How did you handle it?

Day 4: How can you improve on taking care of your mental health?

Day 5: What kind of support do you wish you received from your school/university?

Day 6: What popular self-care ritual does nothing for you?

Day 7: What small thing brings you the most comfort?

Week Two Prompts

Day 8: What’s your comfort song?

Day 9: Do you think your long term goals are achievable and healthy? Why or why not?

Day 10: What subject brings you the most joy to study, either for school or on your own?

Day 11: Do you take mental health days? Do you think you should?

Day 12: What’s your creative outlet? What’s a new one you’d like to try?

Day 13: What makes you feel inadequate? Would you think it made a friend inadequate?

I’m so tired today,Btw my august spread.

30th July 2020

🌿What would be your perfect summer day? 🌿

30th July 2020

🌿What would be your perfect summer day? 🌿

july 16, 2021

Hey everyone! For the past few days (and since I found out about energy management) I’ve actually been super productive and getting all my work done! It’s very satisfying tbh. And I’ve already started planning out my senior year again. I’m aiming for all A’s this semester! And it’s gonna be difficult because all of my courses are either honor level, accelerated honor level or college level courses because of the school I go to (besides stuff like PE, student life courses, etc) but I’m confident that I can do it! :)

16th July - Do you have any summer traditions?

oh okay i like this. and of course i have favorite theorems, mostly based on how useful they are. here you go :

fermat's theorem (for congruences)

lagrange's theorem

fundamental theorem of finite abelian groups

density theorem

fundamental theorem of calculus (the one which uses riemann integrals)

wilson's theorem (for congruences)

Heya! I finished chapter 12 of Analysis 1.It was I think the toughest chapter so far but it's not that tough that I'll loose my sleep over it.I also got the above book from Amazon today.The title is sexist but so is history 🤷🏻‍♀️I started from the last chapter and made some notes on it.I also revised chapter 8 today.Otherthan that it was a pretty chill day.I also answered an ask on Tumblr.I thought it was important to answer the question correctly and so I tried to the best of my ability 🐰

Now Nighty Nighty you guys

Hey!

I just want you to know that mathematics is difficult at the University level but it is not more difficult than any other university course.It is just different.📚

So how is mathematics different and why you shouldn't get discouraged by it?

There is an entire book written on it.It is called 'Alex's Adventures in Numberland' written by Alex Bellos.There is one particular page that I really like.It talks about how are brain perceives the world around us on a logarithmic scale but mathematics is linear and that is why we have to put the extra effort into it (To convert logarithmic to linear) but according to me this extra effort is far less than wading through hundreds of textbooks often contradicting each other that many other university courses require. Here is the cover page of the book I recommended if you would like to read it.

Apart from having a positive attitude towards it I think it'll be great if you could start studying it before your university begins.

So from where should you start?

If you know where you'll be going for your university studies then you can check if your university has uploaded their notes online and start accordingly.Some old students also usually post their notes,so see if you can get your hands on them.If these two options fail then there are two more.Oxford university has made all their mathematics notes avalilable.You can start with their Introduction to mathematics and complex numbers course (I'll put the link at the end).A student from Cambridge University named Dexter Chua also has made his notes available.You can start with Numbers and sets if you use his notes (Again I'll put the link at the end).

Oxford notes:

https://courses.maths.ox.ac.uk/overview/undergraduate

Dexter's notes:

https://dec41.user.srcf.net/notes/

You should be fine with these but the beginning steps into mathematics needs a little more help.You can use the book called as 'How to prove it:A structured approach' by Daniel J. Velleman.If you can read this with the first course that you are taking (your university notes or Oxbridge notes) then I think you'll enjoy the course more just like I did.Here is the image of the book

If you want to know more then I'm just a message away so feel free to ask your doubts.

Heya!!

So today I just revised the first four easy chapters of analysis 1 and then I started chapter 12 (Analysis 1).Chapter 12 is about tests for convergent series like the leibinz test for alternating series,comparison tests (there are two of them) and a lot more which I will be doing tomorrow.I am feeling very tired today.I didn’t even do much.I just had 2 hours of statistics exam and have been feeling tired ever since.Maybe I did not rest enough after the exam or maybe I just have to go and sleep now.

I did place an order for some multivitamin pills though.This is the first time I’ll be using the brand so let’s just see how that goes and apart from that I telephoned two of my school teachers.Here in India it’s guru Poornima (Basically Teacher’s Day according to the Hindu calendar) and it was lovely talking to them.I really do miss them a lot.I also put a sheet mask on in the evening hoping that it’ll make me feel better.It did but I’m still tired so I’m just gonna go sleep.

Aww look at you cutie reading everything hehe I hope your day went better^_^

Hi everyone, here’s the folder :)

It contains PDFs of A-level textbooks for:

OCR Physics A 

OCR Chemistry A

Edexcel Further Maths - Mechanics

Edexcel Further Maths - Core Pure 1 & 2

Edexcel Further Maths - Decision

Edexcel Maths - Pure 1 & 2

Edexcel Maths - Stats/Mechanics 1 & 2

july 21, 2021

Hello! Here’s a picture of some of the books in my bookshelf that I haven’t read yet :). Today is kind of ok, besides the stomach cramps I’m having ;-;. But I’m trying not to let them ruin my mood anyways :)

21st July - What is an unpopular opinion that you have about summer?

Heya!!

Today I revised all the topics I covered in Analysis 1 so far and did chapter 8 and 9.

Chapter 8 is about the algebra of limits.In there I learnt how to write proofs using ε-N method.There were quite a number of cases (with examples!).Most of them were good and could be deduced easily from previous theorems and axioms but I did find 4 proofs a little challenging.I have noted them down and I’ll be discussing those with my prof soon.

Chapter 9 was about monotonic sequences.In there I learnt about the monotonic sequence theorem and the proof of this theorem is very cute.This theorem's proof is very intresting because it connects to the theorem where square root of 2 is proved as irrational but it is more general.So it patches all the gaps on the real number line quit nicely.There were some examples too which I liked and I also learnt the steps of proving a theorem when monotonicity is involved.

It was a pretty cool day.

I hope you had a good day too^_^

***

The Mayuriit Project

Will I make it? Stay tuned to find out!