A digression on Hermite polynomials

Recently a friend of mine made a short document with regards to the Hermite polynomials and the quantum harmonic oscillator. He suggested me to publish it on my website(Spanish)[1], this motivated me to study a little about the Hermite polynomials and then make my own text about them.

Yesterday I uploaded to the arXiv my manuscript “A digression on Hermite polynomials“. Since I have mostly seen these polynomials applied to quantum mechanics, I decided to emphasize the document and this post on other areas of application with a small bias on its probabilistic usage. There and here, I survey some general properties of the polynomials and in the end, some applications to the theory of polynomials, probability, and combinatorics are shown. Most of the content is well-known, except for a few sections where I added my own work to the subject, nevertheless, everything is meant to be self-contained.

1. A general overview of the polynomials

We start with the definition of the polynomials and some details regarding the notation. Afterward, we pursue a construction and an explicit expression for them.

1.1. Definition

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Colombian University Mathematical Olympiad 2017 Final Round – Problems

Today took place the XXI Colombian University Mathematical Olympiad Final Round at the Universidad Antonio Nariño[1](Spanish). Since I wasn’t allowed to take home the problem set, the following are the unofficial statements from my notes. If I acquire the actual problem set in some future, I’ll update this post. Likewise, if I have some time, some solutions would be eventually posted.

Problem 1 (6 Points)

Let {s(m)} be the sum of all positive divisors of a positive integer {m}. Prove that for each positive integer {k}, there exist a positive integer {m} such that {s(n)>km}.

Problem 2 (6 Points)

It is true that,

\displaystyle \int_0^{1} \frac{\ln(1+x)}{x}=\frac{\pi^2}{12},

find

\displaystyle \int_0^{1} \frac{\ln(1-x^3)}{x}.

Problem 3 (6 Points)

Let {f: \mathbb{C}^{4}\rightarrow \mathbb{R}} be a four-variable complex function such that,

\displaystyle f(z_1,z_2,z_3,z_4)=\left|z_1z_3+z_1z_4+z_2z_3-z_2z_4\right|.

Find it’s maximum value in the set,

\displaystyle \left\{(z_1,z_2,z_3,z_4) \in \mathbb{C}^{4} | \left|z_1\right|\leq 1,\left|z_2\right|\leq 1,\left|z_3\right|\leq 1,\left|z_4\right|\leq 1\right\}.

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Three Physics Olympiad Problems

Last Friday took place the 1st Latin American and the Caribean University Physics Olympiad- OLUF hosted by the Observatorio Astronómico Nacional de Colombia -OAN and organized by the Cuban Physics Society [1](Spanish). In this post, I present three problems that I found interesting from the Olympiad. Since I wasn’t allowed to take home the problem set, the following are just a rough description of the statements from my memory and enhanced from various sources to my taste. If I acquire the actual problem set in some future, I’ll update this post. Right now I supplement with some context, articles or related problems that can be useful to appreciate and complement more these problems.

Problem 2 – Definition of the Kilogram (20 points)

The kilogram is currently defined as the mass of the international prototype of the kilogram, a platinium-iridium alloy right-circular cylinder of around 39 mm of diameter. However nowadays other base SI units are defined in terms of fundamental constants, for example one second is currently defined as the duration of 9192631770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom. Then if we denote {\nu_0} as the frequency of the transition, one second is,

\displaystyle 1\,\text{s} = 9\,192\,631\,770 \frac{1}{\nu_0}.

Similarly the meter is defined using the speed of light {c} as a fundamental constant, then we can define the meter as,

\displaystyle 1\,\text{m} = \frac{9\,192\,631\,770}{299\,792\,458} \frac{c}{\nu_0},

that is, the length of the path traveled by light in vacuum during a time interval of {1/299\,792\,458} of a second.

a) Consider now Planck constant as our fundamental unit, {h=6.6260X \cdot 10^{-34}}J s, where {X} is a number related to the precision of the constant. How could we use {h} as a fundamental constant to define the kilogram ? What physical interpretation can you give to this definition of the kilogram?

b) Using {X=9.57} and {\Delta X = 0.044} as the recommended value for the Planck constant find the uncertainty of one kilogram. Is this uncertainty bigger or smaller than the current kilogram uncertainty of the international prototype of the kilogram, which is about {10^{-8}}kg ? Continue reading “Three Physics Olympiad Problems”

An Infinite Sequence Problem-1

Today took place the XXI Colombian University Mathematics Olympiad at the Universidad Antonio Nariño[1](Spanish). The test had very difficult problems and other not so much. In the following post, I present a solution for one problem and at the end, I discuss a little about the generation of fractals in the complex numbers. Maybe in some future, I post other Olympiad problem’s solutions or perhaps only the problems themselves.

Problem 2 (5 Points)

Let {x_0=1} and {x_{n+1}=\frac{3}{16}+\frac{1}{2}x_n - x_n^2} for all natural number {n}. Find {\lim_{n\rightarrow\infty}x_n}.

Solution:

Let’s first prove a very useful result,

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FEM Intro: (Finite Element Method)

There are many methods to find approximate solutions to PDE( Partial Differential Equations). Maybe the most common is the Finite difference method (FDM), however there are more methods, some of them are Finite volume method (FVM)Finite element method(FEM) , etc. One of these methods might be more convenient than other depending on the particular application.

In the recent time I’ve been studying  FEM and using it in some applications, I’m going to start a series of blog posts about the theory and the computational implementation of the method. Some will be more mathematically rigorous than others, but in general an introductory course on vector calculus is enough to understand pretty much all.

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Local Gauge and CPT transformations invariance of the Schrödinger field

Recently I’ve been taking a seminar in Neutrino physics and while learning a little bit about symmetries, we showed the local Gauge and {\mathcal{CPT}} transformations invariance of the Dirac Lagrangian with electromagnetic coupling.

I’m aware that the Schrödinger equation in the classical framework of quantum mechanics is local Gauge invariant and in fact that the electromagnetic interaction is a direct consequence of the existence of a local gauge symmetry, this is well presented in [1],[2].

However this result is always presented within the framework of the Hamiltonian formulation of Quantum mechanics, I’m going to show here that the same invariance also appears in a modified Schrödinger field using the tools of the Lagrangian Field Theory. Though the equations of motion obtained in both theoretical frameworks are conceptually different, it is nevertheless interesting to see the great power in action of the symmetries in the Lagrangian Field Theory formulation.

UPDATE 1: I decided to add a section on the {\mathcal{CPT}} transformations of the Schrödinger field since its natural to show the invariance on the field with electromagnetic coupling. Furthermore, these invariances reaffirm the relation between the local gauge symmetry and the electromagnetic interaction by means of the transformed quantities.

0.1. Schrödinger field equation of motion

Its widely known [3] that the Schrödinger field equation of motion can be obtained by the methods of the Lagrangian Field Theory considering the following Lagrangian for a complex field :

\displaystyle \mathcal{L}_{schr}=\frac{\hbar^{2}}{2m}\nabla\psi\cdot\nabla\psi^{*}+V\psi^{*}\psi+\frac{\hbar}{2i}\left(\psi^{*}\dot{\psi}-\dot{\psi^{*}}\psi\right).

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Relation between Legendre polynomial Generating function and Legendre differential equation

In the last couple of days, I’ve had a little spare time, so I decided to admire a couple of mathematical tools used in the theory of Electrodynamics.

In the following post, I’ll describe the relation between the Generating function of Legendre Polynomials and the Legendre differential equation.

Remarks

  • 1.Most Electrodynamics books ([1] ,[2]) say that the Legendre Polynomials can both be obtained as the solution to the Legendre differential equation and that they are the coefficients of the Generating function of Legendre Polynomials. However, they left this discussion for the mathematical physics Books [3].
  • 2. The following post derives the relation between the generating function and the differential equation in the spirit of the following article [4]. In this article, there is a good discussion of the general theory of generating functions and some applications to Physics and Probability. Finally, at the end of this post, an alternative derivation of this fact is cited.
  • 3. Some additional optional subsections were added to familiarize the reader to some elementary properties of Legendre Polynomials.

1. The Legendre Polynomials

The Legendre Polynomials following [4] are defined as follows,

Definition 1 The Legendre polynomials {\{P_n(t)\}_{n=0}^{\infty}} are defined in the interval {-1\leq t \leq 1} and they satisfy the recurrence relation

\displaystyle (n+1)P_{n+1}(t)= (2n+1)tP_n(t) - nP_{n-1}(t) \qquad \textnormal{ for } \: n\geq 1, \ \ \ \ \ (1)

with {P_0(t)=1} and {P_1(t)=t}.

So the polynomials generated by this recurrence relation are the Legendre polynomials for us, now we will prove that the solutions of the Legendre differential equation indeed follow the previous recurrence relation.

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Potential energy of longitudinal wave in a bar

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Recently I was challenged to solve the following problem:

1. Problem: Find the Potential Energy of a Longitudinal wave in an elastic rod

(I solve the problem by two different ways then I relate it to Problem 7-24 French[2] and there are some final remarks about the general properties of wave phenomena)

We consider an elastic rod that changes its length via an incoming longitudinal wave as in the following figure shows,Imagen2

1.1. Solution 1:

Hint: Consider a simpler problem: find the potential energy of a clamped elastic rod of total length {L} extended a distance {\Delta L}.

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