Hello! This is Kelanu's Website

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Cool People (Who Also Have A Website)

UCLA Mel Carter Seth Eric Daniel Liyu Nathan Srinjina Kyle Hunter Logan Brian Ky
Lumitron Christine Mauricio
James Joy Jason
Co-op Silvia Eli
Other Aaron

People I Do Not Know But I Think Are Cool

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:

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.

$$\langle \psi_m | \mathcal{D}(\alpha) | \psi_n \rangle = \sqrt{\frac{m!}{n!}}(-\alpha^\ast)^{n-m}\exp(-\frac{1}{2}|\alpha|^2)L_m^{n-m}(|\alpha|^2)$$

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

$$\langle \psi_m | \psi_n \rangle = $$

This one is courtesy of Ryan.

Consider the following expansions

$$\mathcal{D}(1) = e^{a-a^\dagger}=e^{\gamma \hat{p}} = \sum_k \frac{\gamma^k}{k!}\hat{p}^k$$

$$\mathcal{D}(i) = e^{ia+ia^\dagger}=e^{\lambda \hat{x}} = \sum_k \frac{\lambda^k}{k!}\hat{x}^k$$

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

$$\langle \psi_n | x^k | \psi_j \rangle$$

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.

$$\frac{k!\sqrt{n!j!}}{\sqrt{2^k}}\sum_{m=0}^{\left \lfloor{\frac{k}{2}}\right \rfloor}\frac{1}{2^m m!}\frac{1}{(s-n)!(s-j)!(s-k+2m)!}$$

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 $\omega=2\pi$.