Mathematics: God's gift to physicists!

It has taken me some time, but I have realized now what most physicists do at some point in their life: you can not possibly become a functioning physicist without having a strong hold on mathematics. Yes, experimental and theoretical alike! One might think that understanding the laws of physics must serve one well, but while that is the necessary condition it surely isn't a sufficient one (Pun #1). While understanding physics well itself requires a lot of mathematics, researching in physics is a whole another ball game. I say this because when you are involved in researching you attempt to expand the boundaries of our knowledge, and that, at least for a theorist, requires extreme mathematical rigor. Why? Because mathematics is the only tool known to mankind that one can use to justify one's hypothesis. You make assumptions based on the facts you already know (or believe) are true and based on those assumptions you derive the mathematical form of your idea. And it doesn't end there, your theory isn't proved until it can be proven experimentally, ask Peter Higgs! I had started this post to express how important mathematics is for a physicist, and have drifted to telling you about the plight of a theorist. What a subtle distraction indeed.

While physics answers the 'why' behind a physical phenomenon, mathematics answers the 'why' behind the formulation involved in answering the former 'why'. For example, while you might know about how quantum mechanics involves Hilbert spaces and Dirac notation and is able to precisely explain the quantum world, functional analysis tells you why use Hilbert spaces, among all others, in the first place. Mathematics is like a tool box a physicist must have, and knowing more doesn't hurt because you never know what tool may be required to solve what problem. When I was in secondary school, I always felt that matrices were a lame topic. You arrange numbers in a definite form to ease calculations. Only in college did I understand the vastness of the field. Think of it, matrix algebra, linear algebra to be general, is one of the most important topics for a physicist as all the basic concepts and subjects one learns involves these bad boys. 

And complex numbers, don't get me started on complex numbers. I can say this without even an iota of doubt that they are the most beautiful part of mathematics I have yet encountered (Pun #2). If you think about it, it is a very simple idea: what if we took the square root of -1? And there you have it, i, a letter that changed how people looked at the world. Moving from imaginary to some real stuff, no matter how much we get accused of being day dreamers, physicists work towards explaining things that are real. While we might get rigorous with complex numbers, at the end of the day, real numbers are what we return to. This itself makes it very important for a physicist to understand real numbers, and real analysis sure helps a lot in it.

I know what you are thinking, three paragraphs in and I haven't mentioned calculus! Well, do I need to? Calculus to me feels like a mathematicians' version of a theory of everything. Before you say it, I know calculus can not be used everywhere, but I say this because of the immense applicability of the subject. Well, I am sure Leibniz and Newton will be proud. 

Sometimes people claim that physics is just applied mathematics. While this may sound tempting to mathematicians, I will have to side with the physicists here. I know that a mathematician can very well derive the same laws that a physicist did and in probably a more elegant manner, but a mathematician might not be able to appreciate the meaning behind that law like a physicist can. The difference in what they appreciate is what differentiates between the two most amazing people in the word: the mathematician and the physicist.

Quantum Mechanics- First Impressions!

You know when you've heard about something being extremely mysterious, mind-boggling and makes people's world turn round, but you've never had the chance of getting introduced to it? That is what Quantum Mechanics was for me, until this January. I had read people say that if you do understand Quantum Mechanics, you don't know it well enough. Such a claim makes a curious mind anxious and excited at the same time. So, without anymore build up, let me tell you about my first impression of Quantum Mechanics after one month of study.

When one is introduced to QM, it does not take one much long to realize that QM is not a theory that illustrates human incompetency during experiments, but actually, one that sheds light on the 'spooky' way in which nature works at the smallest level, namely, the quantum level. What attracts some and repels others is the fact that initially, QM challenges intuition. But if you are exposed to it for long enough and in the right way, it redefines intuition and the way you see the world. Classically, we have been taught about discrete systems with definite states and properties. So, classical mechanics talks about certainty and some might stretch the laws to even talk about determinacy. Then comes QM which talks about probabilities and uncertainties. Not only that, the Heisenberg's uncertainty principle states that uncertainty is embedded in the working of the universe. In my opinion, this leap from classical to quantum in one's intuition is the difficult part when it comes to mastering QM. 

QM owes its success to its mathematical formulation. We can mathematically calculate the outcome of an experiment without even conducting it. And when we do end up conducting the experiment, the theoretical and the experimental outcomes are in complete agreement. Whenever this happens, it lightens up the day of a physicist. Knowing this, in my opinion, one needs to realize that the best way to understand QM is to approach it mathematically and master the mathematics that makes up the theory. Once you have accomplished that mighty task, you can begin with developing the intuition for QM such that the much talked about 'spooky' action now becomes 'expected'. 

At present, I am only in the beginning stages of exploring this vast field. I started by learning about vector spaces, Dirac notation including the ket and the bra spaces, operators and their eigenvectors and eigenvalues, and their action on kets resulting in quantum states. So, you see, I really haven't done much. But the much I have done makes me excited about what is going to come and what more can be done using QM. I am curious to know how the basic principles will give rise to mysterious concepts of quantum entanglement, etc. and how and where I will be able to use my knowledge in this field.

I read about Einstein's trouble with QM and justifiably so, given the fact that QM and General Relativity don't go hand-in-hand. This is a major problem in Physics, a problem I am already interested in. It does create a sense of zeal when trying to connect a theory that works at the largest scale to one that works at the smallest. This reminds me, I am also taking a course in Theory of Relativity this semester which aims to make me proficient in Special Relativity and its consequences, and also introduce me to the formulation and aspects of General Relativity. But, talking about that is a whole other post!