A voyage through space(s) (Part III)

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...

A voyage through space(s) (Part II)

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...

A voyage through space(s) (Part I)

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...

Collapse that wave function!

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,...

Why won’t my sequence of functions converge?

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?...

Curl and divergence of a vector field

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...

Derivation for the Wallis Formula for pi

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....

The Basis Theorem

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...

Wronskian Theorem

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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