by Anthony Salib | Oct 2, 2017 | Pure Maths
As promised, we begin our quest today in understanding what a normed vector space is. We will begin with a vector space , and we would like to give it some structure similar to our metric spaces. Now we could of course give any metric to , but we already have two...
by Anthony Salib | Sep 30, 2017 | Pure Maths
We left the last post on a cliff hanger, namely that in any given metric space, Cauchy sequences need not converge. Now if you’re reading this, you have chosen to take the red pill, and I’m here to show you just how the deep the rabbit hole goes. We shall...
by Anthony Salib | Sep 27, 2017 | Pure Maths
So in the last post I talked about wave functions being vectors that live in Hilbert space. In this post, I would like to describe the mathematics of Hilbert space, but before we do that we have to go on a journey, a journey through space(s)! The first type of space...
by Anthony Salib | Aug 15, 2017 | Physics
In quantum mechanics, we cannot speak of the exact location of a particle, but rather, we talk about the probability of a particle being in a certain region of space. Mathematically, we describe this using a probability density function, known as the wave function,...
by Anthony Salib | Aug 3, 2017 | Pure Maths
Firstly, let’s consider what it means to have a sequence of functions by looking closely at All this is saying is that our sequence is going of to infinity. But this is just boring though, what we want to know is what happens when I go to infinity?...
by Anthony Salib | Aug 3, 2017 | Pure Maths
Some things are just too hard to prove directly, so in this case, we may need to take a more indirect route. One way in which to do this is a proof by contradiction, whereby we assume something, then do some logical deductions which (hopefully) produce some sort of...
by Anthony Salib | Aug 1, 2017 | Physics
The Curl of a vector field is defined to be The curl is a measure of the vorticity, or “swirliness”, of the vector field. So imagine that describes some fluid and has the vector field plotted below. Now this field shows that the fluid is...
by Anthony Salib | Jul 13, 2017 | Pure Maths
I’m doing a research project in the School of Physics at Monash University, looking at whether is really found in quantum mechanics, which was suggested by a paper published in 2015 where they presented a quantum mechanical derivation of the Wallis product....
by Anthony Salib | Jul 13, 2017 | Pure Maths
Not going to go into too much detail about it, instead we will jump straight into the proof. But just so you know what your dealing with, the theorem says that if and are continuous on some interval then: there are two linearly independent solutions () to and that if...
by Anthony Salib | Jul 12, 2017 | Pure Maths
Firstly let us define what this Wronskian thing is. Say you have 2 solutions to a second order homogenous equation, this is an important point, they have to be solutions to a second order homogeneous equation. Just a reminder, this is what a second order homogeneous...
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