Hello! This is Kelanu's Website

Cool People (Who Also Have A Website)

UCLA	Mel	Carter
Lumitron	Christine	Mauricio
James	Joy	Jason
Co-op	Silvia	Eli
Other	Aaron

People I Do Not Know But I Think Are Cool

Isreal Gelfand
Anyone with an interest in mathematical physics or AQFT has no doubt come across the name Gelfand of the famous "Gelfand Triple/Sandwich" or GNS construction. I found, after reading more about him, that he was also influential in molecular biology, ran a correspondence school for those who didn't have access to a university, and believed in animal rights and was a vegetarian then vegan towards the end of his life.
Paul Erdos

Alexander Grothendieck
Below is a picture of Grothendieck with his students in North Vietnam in 1967.

About Me

My name is Kelanu. I study physics and mathematics at UCLA. I was previously an engineering intern at Lumitron Technologies from June 2021 to April 2024. I now work for Professor Saltzberg at UCLA as an undergraduate research assistant.

I am currently taking:

Physics 115C: Quantum Mechanics
Physics 180Q: Quantum Optics Laboratory
ECE 279AS: Special Topics in Physical and Wave Electronics: Physics of Quantum Information and Computation
ECE 271: Classical Laser Theory

I have a spreadsheet of every TV show and movie I have or plan on seeing; feel free to add comments.

If you want to contact me my email is kelanucr at g dot ucla dot edu or kranganath at physics dot ucla dot edu.

Once a month for the past four years I make a collaborative playlist, previously this has been on Spotify, this year I am trying YouTube Music. If you would like to add to the playlist this is the link

I have a YouTube channel.

Interesting Formulas

Formulas Involving Displacement Operators

The following formulas are useful when dealing with displacement operators.

$⟨ ψ_{m} | D (α) | ψ_{n} ⟩ = \sqrt{\frac{m!}{n!}} (- α^{*})^{n - m} \exp (- \frac{1}{2} | α |^{2}) L_{m}^{n - m} (| α |^{2})$

This one I found on Stack Exchange but I forgot where and only have the original link.

$⟨ ψ_{m} | ψ_{n} ⟩ =$

This one is courtesy of Ryan.

Consider the following expansions

$D (1) = e^{a - a^{†}} = e^{γ \hat{p}} = \sum_{k} \frac{γ^{k}}{k!} {\hat{p}}^{k}$

$D (i) = e^{i a + i a^{†}} = e^{λ \hat{x}} = \sum_{k} \frac{λ^{k}}{k!} {\hat{x}}^{k}$

We know however, that a displacement by a purely real alpha is kick in position and a purely complex displacement is a kick in momentum. We see then that displacements in momentum are equivalent to position operators and vice versa.

Perturbation Theory

The first order correction to an anharmonic oscillator requires evaluating

$⟨ ψ_{n} | x^{k} | ψ_{j} ⟩$

First we notice that if we expand $x^k = \gamma^k (a + a ^\dagger)^k$ then we can evaluate this by hand by drawing trees. It would be nice if we could instead have a formula for this.

$\frac{k! \sqrt{n! j!}}{\sqrt{2^{k}}} \sum_{m = 0}^{⌊ \frac{k}{2} ⌋} \frac{1}{2^{m} m!} \frac{1}{(s - n)! (s - j)! (s - k + 2 m)!}$

Riesz Representation Theorem

Gelfand Triple

LuaLaTeX

I am a big fan of LuaLaTeX, I've been using it to compile my LaTeX documents for close to two years now. In that time I've developed a set of scripts and enviroments that transform psuedo-code into well defined LaTeX macros for ease of use -nothing takes you out of the flow quite like aa curly bracket am I right fellas.

There are still some big bugs to work out before I can release it, but when I do I'll put a link here.

Solutions to Selected Problems

Quantum Computing

These are selected problems from Physics 245 at UCLA with Professor Hudson. A lot of the problems are computational and use a package called QuTip. I've selected some of the non-QuTip problems that I think have an interesting solution.

"Ramsey" spectroscopy with a QHO?!

Suppose at $t=0$ a vacuum state is displaced by a displacement operator with $\alpha = 1$. Then after a variable time $t_w$ a second displacement operator with $\alpha = -1$ is applied. Plot the probability of being in the vacuum state as a function of $t_w$, assume that $m=1kg$ and $\omega=2\pi$.