Reflection: My Graduate Experiences at Princeton University

Please note: there will be mentions of the current global public health crisis in this post. I have no background in medicine, public health, or closely-related fields. Please consult public health agencies and other governmental agencies for guidance regarding responses to this crisis, and please consult actual professionals as appropriate for individual problems in this context.

This post is the third in a series of three posts about the end of my time as a PhD student in Princeton University (in this post henceforth referred to simply as "the university" when there is no ambiguity). As a write this, I have successfully defended my PhD thesis! Furthermore, I will officially be graduating this coming weekend. This post follows the first, which was meant as a reflection of the events of this public health crisis that led to my premature physical departure from the university campus combined with a paean to the friends I made over the course of 6 years in the PhD program, and the second, which explained the experiences & thought processes that led to my decision to change careers from research in physics to transportation policy. This post is a broader reflection of my time and experiences at the university, with all of its ups and downs, and a message of gratitude toward the people in the university and elsewhere who meant so much to me during my time in the program; a lot of it is taken from the acknowledgments in my thesis, though for privacy reasons, I won't be giving explicit names. Additionally, there will undoubtedly be many comparisons over the course of this post to my undergraduate experiences at MIT, for which I wrote a post around the time of graduation 6 years ago. Follow the jump to see more.

Classical Damping of Gases and Oscillators

I was on vacation last week, and during some quiet time, I randomly happened to be thinking about explanations for damping in physical systems. I remember learning in ELE 456 — Quantum Optics, from last spring, that the phenomenological linear damping of a classical oscillator could be derived by coupling a quantum oscillator to a thermal bath of quantum oscillators; each linear oscillator is microscopically undamped, but by treating the bath through statistical thermodynamics, the coupling of the oscillator in question to a bath essentially produces a linear damping coefficient dependent on the spectrum of the bath (and the coupling too). Microscopically, the quantization of energy levels in a linear oscillator makes it easy to interpret how discrete excitations can move from one oscillator to another coupled oscillator, but I was wondering if quantum mechanics is really necessary to explain damping. Follow the jump to see an extremely rough sketch of ideas that may (or may not) justify the use of classical mechanics by itself. (Added after finishing: this turns out to be a rambling and possibly ultimately pointless post with a much clearer and more self-consistent explanation linked at the end, so for the time being, humor me.)

On Transitioning into Graduate Life, One Year In

This is a post that's more about what's going on in my life right now, so if you would have liked to see a software review or an otherwise more technical/generally topical post, fear not! That shall come in at least one more post this month. This post is more about some thoughts I've had about mentally and socially transitioning to life in graduate school after a little over a year in it, so I just hope that anyone going through a similar transition may find this even mildly interesting. Follow the jump to see more.

Reflection: 2014 Summer

This summer, by design, I was able to relax basically the whole time. I was able to attend graduation parties, visit relatives in India, attend a wedding in New York, spend time with family & friends, and not worry about work a whole lot. Of course I was able to get a bit of work for my old UROP done too, especially as I'd like to turn it into a paper, but I didn't really feel pressure to be working on it all the time. In fact, working on that and a few other projects was mainly how I filled my downtime, but I never let those things get in the way of relaxing and having fun. Anyway, this summer is about to end, and that would make it my last formal summer break ever. In two days, I will be moving to Princeton to start a PhD program in the Electrical Engineering department; it'll likely be about photonics, quantum optics, or Casimir physics, but I have a semester to figure out the details. I'm really excited to be starting that, and I hope the journey will be a good one overall (though I have no doubt that there will be both ups and downs). If you're starting school, college, graduate school, a new job, or any other sort of new venture, good luck!

Trying out Julia

This is a fairly quick post, though I previously considered making it longer and more trollish. A handful of my friends have told me about Julia, the amazing programming language made for numerical computations and other scientific computing uses. For the 14.15 — Networks final project this past semester, one of my group partners used Julia to simulate large ensembles of 10000-node random networks, and it worked far quicker than MATLAB. I vowed to get a bit more familiar with Julia (the programming language, not a woman [yet]) this summer. It was actually pretty quick to get used to, considering its syntactical similarities to MATLAB, to which I am more accustomed. I was even able to use it to port over the MATLAB code used for data analysis in 8.13/8.14 — Experimental Physics I/II to Julia. The only issue that I have consistently run into has been plotting. For some reason, the plotting packages that interface with Julia do not work in the ways that I want: Winston is too basic, Gadfly doesn't work at all (which is unfortunate because it has all the features I need and more), and Gaston being a frontend for Gnuplot while having to deal with the quirks of Julia's plot execution order means that I might as well use Gnuplot itself. Indeed, that is what I've done: I've been able to write Gnuplot scripts to plot processed data that Julia outputs into a file. Although Gnuplot's syntax is a little arcane, it is so powerful that I'm OK with using it from a script of commands and changing only a few things here and there as needed. Other than that, Julia works like a charm; its speed is fantastic, and I really like how much structure it brings compared to MATLAB (including things like types and indexing). Plus, it combines the great features of both procedural and functional programming. Given that course 18 has largely switched over to Julia, I wonder when course 8 will do the same....

Reflection: My Undergraduate Experiences at MIT

Commencement is a few days away, so I don't have too much more time on campus. I've finished all four years of my undergraduate education. It has been a really wild and amazing ride, and now that things are marginally quieter, I think I could use a little reflection on those 4 years (or, at least, the highlights, learning experiences, and more recent parts that I remember). I am no poet, so a lot of this may sound repetitive, awkward, or stilted; believe me when I say this is really how I feel. Follow the jump to read more.

Done with 8th Semester!

I'm done with my eighth and final semester of my MIT undergraduate semester! (Actually, I was done on Sunday, May 18 around 3pm upon completion of my last problem set, but I didn't get around to writing this until today.) It was extremely satisfying to see a bit about nanoparticle scattering of infrared light in a new UROP project and write about that in my thesis, along with getting excellent results for my ongoing photonic crystal UROP project and writing about that too. My thesis gave my the most trouble in the two weeks leading up to its submission on May 9, though I started writing during spring break itself. In terms of classes, I had the most trouble in 8.334 — Statistical Mechanics II (Statistical Field Theory), as the problem sets and exams alike were quite challenging, and the final project was an 18-hour marathon on Friday, May 16. Also annoying was 14.15 — Networks; it wasn't taught or organized very well, and the final project gave me and my group partners a fair amount of stress too. More manageable was 8.962 — General Relativity, which only had problem sets, and most of those were quite reasonable and straightforward. Anyway, I don't have any final exams this semester (by design), so I'm really done, and all I need to worry about now is commencement! (I will have a longer post reflecting on my time at MIT in the coming days, so stay tuned for that.) After commencement, I plan to spend most of my summer time relaxing and picking up small projects at home; I may also travel for a bit too.

Thesis and Papers and Projects, Oh My!

I realize I haven't been able to post anything in...a month, actually. That's because most of my time has recently been devoted to finishing my undergraduate thesis (due in 1.5 weeks), 2 final projects (due in 2.5 weeks), the work for a potential paper for my UROP (hopefully soon), problem sets (all the time), and exams (sporadically, though thankfully I have no final exams). I hope to have more posts (including a few reviews) out in the coming weeks when I'm a little more free. In the meantime, enjoy this nugget of crazy physics: apparently it's possible to derive asymptotic freedom in QCD from classical statistical field theory.

Green's Functions and Correlations

I had the idea of writing this post a couple of weeks ago, but I didn't feel like I had enough stuff to write here at that time. Now I do, so here goes. (Also, here's hoping that inputting LaTeX into this post works once more.)

When I took 18.03 — Differential Equations in 2010 fall, one of the topics covered was linear time-invariant systems. The general system of interest was $Lu(t) = f(t)$ where $L$ is a linear time-invariant operator. The technique of course is to find a weight function $w(t)$ where $Lw(t) = \delta(t)$, and once that is done, the solution is $u(t) = \int_{-\infty}^{\infty} f(t') w(t - t') dt'$ which is a convolution of the input $f$ with the weight $w$. The professor mentioned that it is essentially akin to inverting the operator $L$, but while I could see the general utility in this method, I never quite understood why it might be considered inversion on any deeper level.

Last semester, I took 8.07 — Electromagnetism II, and there we discussed Green's functions a little more in the context of electromagnetism & electrodynamics. In a static situation, the Green's function comes up in solving the Poisson equation $\nabla^2 \phi = -\rho$. In this case, $\nabla^2 G(\vec{x}, \vec{x}') = -\delta(\vec{x} - \vec{x}')$ is solved by the familiar potential of a unit point charge $G(\vec{x}, \vec{x}') = \frac{1}{4\pi |\vec{x} - \vec{x}'|}$. I started to see a little more clearly why this worked, because if a general charge distribution was some superposition of point charges, then a general potential distribution should be the same superposition of point charge potentials. However, it still wasn't entirely clear to me how this was "inversion" per se. Follow the jump to see what changed.

Eighth Semester at College

I'm at the home stretch! This is my eighth and last semester as an undergraduate at MIT. Classes start tomorrow. I'll be taking 8.334 — Statistical Mechanics II (which is really statistical field theory), 8.962 — General Relativity, 14.15 — Networks, and 8.THU — Undergraduate Physics Thesis. The cool thing is that 8.334 — Statistical Mechanics II and 14.15 — Networks will have a bit of overlap in some places, as both discuss graph theory, collective phenomena, and phase transitions to varying degrees. More importantly, 8.THU — Undergraduate Physics Thesis is basically going to be my UROP, formalized into credits contingent on me producing a thesis at the end of it. That's also how I can start a new UROP project on nanoparticle absorption and scattering of infrared radiation. Even though I'm only taking 3 classes besides my UROP and [as far as I can tell] none of them have final exams, the semester will still keep me quite busy, but this will be the last semester where I can take more random classes that I want to take, as graduate school will likely only let me take classes related to my research interests. Here's hoping that my last semester of my undergraduate career turns out to be the best one yet, and good luck to everyone else for the new semester!

Reflection: 2014 IAP

This IAP was quite a bit more hectic near the end of it. I was starting to wrap up my current UROP project on photonic crystal enhancement of spontaneous emission and start learning about a new project on nanoparticle absorption & scattering of infrared radiation. Also, especially in the last week, I was doing a lot for making a video for the MIT-K12 project. Finally, there was organization to be done for the SPS Lightning Lectures on the last day of IAP. Overall, it was quite productive. At the moment, I am still awaiting graduate admission results (except for one positive one so far). And I await and anticipate one last semester of classes and research as an undergraduate at MIT!

FOLLOW-UP: Gibbs Entropy and Two-Level Systems

As a follow-up to this post, I'm going to briefly discuss what two statistical mechanics professors (who shall remain nameless) I talked to about this had to say. For those who don't remember or are too lazy to read through, the issue is that a new paper publicized by the MIT news office claims that by adopting a view of entropy as per Gibbs as opposed to Boltzmann, negative temperature can be removed from statistical mechanics. I pointed out many issues I had with the arguments for that, and I would thereby cast doubt on the paper and premise as their wholes. Follow the jump to see what information I was able to learn after talking to those professors. (It appears that rendering LaTeX on this blog no longer works right after the takedown, so I'm enclosing any useful LaTeX formulas in dollar signs for you to copy and paste into a LaTeX renderer, if you so choose. The rendering of LaTeX in past posts is inconsistent, just as a heads-up.)

Electromagnetism Basics in 1 or 2 Dimensions

This was a post that I had been thinking of doing for a while, but I couldn't get around to it until now. A lot of introductory electricity & magnetism problems constrain charges to only move in 1 or 2 dimensions, but in reality the constraint existed within a 3-dimensional space. I thought that would cover the bases for electrodynamics in 1 or 2 dimensions, but then I saw that in cylindrical coordinates, the order-0 multipole moment outside a line charge is $\phi \propto \ln(r)$ as opposed to $\phi \propto \frac{1}{r}$. That made me realize that there is in fact a distinction among 1 or 2 or 3 dimensions. In all of the following, I will make use of the conventions and relations \[ x^{\mu} = (ct, x, y, z) \\ \partial_{\mu} = \left(\frac{1}{c} \frac{\partial}{\partial t}, \frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z}\right) \\ \eta_{\mu \nu} = \begin{bmatrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \\ F^{\mu \nu} = \begin{bmatrix} 0 & E_x & E_y & E_z \\ -E_x & 0 & B_z & -B_y \\ -E_y & -B_z & 0 & B_x \\ -E_z & B_y & -B_x & 0 \end{bmatrix} \\ \partial_{\nu} F^{\mu \nu} = \frac{4\pi}{c} J^{\mu} \\ \epsilon_{\mu \nu \zeta \xi} \partial^{\nu} F^{\zeta \xi} = 0 \\ \mathbf{F} = q\left(\mathbf{E} + \frac{\mathbf{v}}{c} \times \mathbf{B}\right) \] in 3 dimensions, with Einstein summation and CGS implied (with more on that last point nearer to the end), with Latin indices representing only spatial components, and with Greek indices representing spacetime components. Also note that the fully antisymmetric tensor $\epsilon$ has $n$ Latin indices in $n$ spatial dimensions and $n+1$ Greek indices in $n+1$ spacetime dimensions; for example, in 2 spatial dimensions, the antisymmetric tensor over only space looks like $\epsilon_{ij}$, while over spacetime it looks like $\epsilon_{\mu \nu \xi}$, and I will frequently switch between the two as needed. Follow the jump to see what happens.

Gibbs Entropy and Two-Level Systems

Today, I was browsing through the MIT news page when I saw this article about how two mathematicians claim to have disproved the notion of negative temperature. My heart sank, because one of the coolest things I remembered learning in 8.044 — Statistical Physics I was the notion of negative temperature existing, being hotter than hot, and being experimentally realizable. I also became confused when the article referred to Gibbs entropy, because the definition I thought was being used for Gibbs entropy was \[ S = -\sum_j p_j \ln(p_j) \] which is exactly equivalent to the Boltzmann entropy \[ S = \ln(\Omega) \] where \[ p_j = \frac{1}{\Omega} \] in the microcanonical ensemble. I figured this would mean that the Gibbs entropy would exactly reproduce negative temperature results in systems with bounded energies such as two-level systems. I wasn't able to read the most recent paper as discussed in the news article, because it is behind a paywall, but I was able to read this article by the same authors, which appears to lay the foundational ideas behind the most recent paper. It seems like on my end, the misconception appears to hinge on what one would call the Gibbs entropy. The formula \[ S = \ln(\Phi) \] appears to be the correct one for the Gibbs entropy, where $\Phi$ is the total number of states with energy not greater than $E$ and $\Omega = \frac{d\Phi}{dE}$ is the number of states with energy exactly equal to $E$ quantum mechanically (or the number of states with energy within a sufficiently small neighborhood of $E$ in the classical limit). With this in mind, follow the jump to see how this might work for a two-level system and explore the other implications of this new definition of statistical entropy. (UPDATE: Note that in all of this, $k_B = 1$.)

Done with 7th Semester!

It finally happened! The end of the semester rushed in and washed over just as quickly. My classes — 8.07 — Electromagnetism II, 8.09 — Classical Mechanics III, 8.333 — Statistical Mechanics I, and 14.12 — Economic Applications of Game Theory — were all together a bit more challenging than I anticipated. On top of that, I worked a lot on my UROP, and of course I had to submit graduate school applications by last weekend. Thankfully, my classes and graduate school applications are done. Now I can go home, relax, enjoy the company of family and friends...and probably work on my UROP. Of course, I'll be continuing my UROP over IAP, but I hope to be transitioning into a new project then, so I hope to get a fair amount of my current project done during the break. Happy holidays everyone!

Classes, UROP, and Applications Galore

I know I haven't posted here in a while. That's because this is around the time that a lot of graduate school applications are due, so I've been busy getting those done. At the same time, my UROP has been getting busier as I'm trying to wind down my current project, and classes are of course ever-present in the background. Anyway, my applications and classes will be done in about 3 weeks, so at that time I should have more time to write here. Meanwhile, happy Thanksgiving!

Seventh Semester at College

How did I become a senior? It doesn't feel like orientation and freshman year happened that long ago.
Tomorrow is the first day of class for the 2013 fall semester. I'll be taking 8.07 — Electromagnetism II, 8.09 — Classical Mechanics III, 8.333 — Statistical Mechanics I (a graduate class), and 14.12 — Economic Applications of Game Theory. I'm looking forward to all of these classes along with continuing my UROP (which may transition sooner or later into a new project as I wrap up my current one). The bigger things I have to deal with though are graduate school applications and the Physics GRE. The latter will be over in a few weeks. The former will be going on until around the beginning of December, but I hope to be done a while before that. Hopefully this semester goes well. Good luck to everyone else for the start of their school year/job/whatever else!

Particles in the Continuous Quantum Field

The last thing I discussed in the last post was about the energy eigenstates of the continuous field. The ground state $|0\rangle$ classically corresponds to there being no displacement in the chain at any spatial index $x$ and quantum mechanically corresponds to each oscillator for each normal mode index $k$ being in its ground state, while the first excited state $|k\rangle = a^{\dagger} (k)|0\rangle$ for a given $k$ classically corresponds to a traveling plane wave normal mode of wavevector $k$ and quantum mechanically corresponds to only the oscillator at the given normal mode index $k$ being in its first excited state (and all others being in their ground states). The excited state $|k\rangle$ has energy $E = \hbar v|k|$ above the ground state and overall momentum $p = \hbar k$ above the ground state. This post will discuss what the second and higher excited states are. Follow the jump to see more.

Operators and States of the Continuous Quantum Field

In my last post about intuiting and visualizing quantum field theory, I discussed the diagonalization of the Hamiltonian and overall momentum and how they become operators. In this post I'm going to discuss more the meanings of the operators and associated quantum states of this field. Follow the jump to see more.

Diagonalizing and Quantizing the Continuous Field Hamiltonian

In my previous post I discussed the intuition behind the classical acoustic field in one dimension. Now I'm going to talk about diagonalizing the Hamiltonian and making the step into quantum field theory. Follow the jump to see what it's like.