Transitioning from microscopic to macroscopic and quantum to classical regimes

I recently read two things that were of interest to me having previously worked in physics. One was an article in The New Yorker magazine [LINK], in which the author does a good job of going over the successes of mathematical modeling in the physical sciences and contrasting this with the limitations of mathematical modeling in public health (showing, for example, how many models of the spread of contagions fail when governments & societies take fast & drastic collective actions to limit the spread), the failures of mathematical models in social sciences where the outputs of those models can create feedback loops with public sentiment (for example in political polling), and the way that many people who use machine learning models in different domains expect the fancy curve-fitting of those models to represent fundamental understanding when that might not really be so. The other was a journal article published in Physical Review Letters [LINK] about how it can be possible to test the extent to which a massive (as opposed to massless) object which exhibits the dynamics of a simple harmonic oscillator and prepared in a quantum coherent state can be tested for deviations from classical behavior using a protocol that does not depend on the mass of the object (although I question this given that the protocol depends on timed measurements that depend on the frequency of oscillation, and in many physics contexts the frequency does depend on the mass as \( \omega = \sqrt{k/m}\), but this is somewhat of a quibble). These two things got me to think about something that I realized I never got out of many years of formal undergraduate & graduate education in physics. This can be illustrated with the following example.

In introductory physics classes that focus on Newtonian mechanics, a prototypical problem involves a block, modeled as a point mass, sliding (with or without friction) down a fixed triangular incline in the constant gravitational field of the Earth. In the context of those classes, instructors will be careful to note that this is merely a model, and corrections could come from the inclusion of the variation of the Earth's gravitational field & surface curvature, the technical possibility of moving the triangular incline (which must be much more massive than the block in question), the shape of the block, variations in the touching surfaces, air resistance, et cetera. In later classes, instructors may point out corrections due to special relativity (i.e. the speed of light) and general relativity (as it relates to the Earth's gravitational field).

However, in later classes about quantum mechanics & statistical mechanics, instructors explain how different the models are from models of Newtonian mechanics at human scales, but they often promise that appropriate treatments of aggregates of microscopic constituents can consistently recover results from Newtonian mechanics, yet this promise is almost never fulfilled. In particular, wavefunctions that describe pure states of single microscopic particles are quite far removed from the simple dynamical variables describing blocks on inclined planes, although statistical mechanics can probabilistically describe the solid states of the block & inclined plane as well as the gaseous state of the surrounding air, it is not usually extended to describe the dynamics of the block sliding down the inclined plane. For example, if a block sliding down a fixed inclined plane of horizontal angle \( \theta \) in a uniform gravitational field is described as having equations of motion \( m\ddot{x} = mg\sin(\theta) \) where the \( x \)-axis is defined as pointing downward parallel to the slope of the inclined plane for increasing \( x \) and the \( y \)-axis points outward in the normal direction from the inclined plane, then I wish to see corrections of the form \( m\ddot{\vec{x}} = \sum_{\mu = 0}^{\infty} \sum_{\nu = 0}^{\infty} \hbar^{\mu} k_{\mathrm{B}}^{\nu} \vec{f}^{(\mu, \nu)} \) where the lowest-order term is \( \vec{f}^{(0, 0)} = mg\sin(\theta)\vec{e}_{x} \). I have never seen these sorts of quantum or statistical corrections to Newtonian equations of motion in simple (in the context of Newtonian mechanics) systems. Similarly, it is rare to see how quantum or statistical mechanical systems can, in appropriate limits, reproduce classical systems; I can only think of the quantum coherent state of the simple harmonic oscillator as well as how the Moyal bracket in the phase space formulation of quantum mechanics reduces to lowest order in \( \hbar \) to the Poisson bracket, and in the latter case, intuitive construction of the quantum phase space quasiprobability function is made more difficult (compared to construction of a classical phase space probability density function, as I did in a post [LINK] from a few years ago) by the fact that unlike the classical phase space probability density function, the quantum phase space quasiprobability function cannot be arbitrarily localized in phase space, it can take on negative values for certain wavefunctions, it is compressible in phase space with respect to its own evolution over time, and it is not obvious how it should look for a system of many particles constituting a macroscopic object like a block (in contrast to a classical phase space probability density function, which for such a system could just be a product of Dirac delta functions localizing each microscopic constituent to a point in phase space).

These considerations reminded me of a discussion I had last year with friends from college, who also did course 8 (physics) with me. We came to a consensus that while people who do not become physics majors should, as usual, get exposure to Newtonian physics and the basics of electricity & magnetism, people who become physics majors should have a curriculum over 3-4 years that exhibits a sensible conceptual progression. In particular, after seeing Newtonian mechanics, such students should then be exposed to Lagrangian & Hamiltonian formulations of classical mechanics. The Lagrangian formulation of classical mechanics should then be used to develop intuitions about mechanical waves, which in turn can lead to introductions to classical field theory and development of classical electromagnetic theory as a rich example of a classical field theory. (I would also personally recommend using the introduction of mechanical waves to introduce the linear algebraic treatment of waves and then reintroduce the linear algebraic treatment of waves into the treatment of linear classical field theories in general & linear classical electromagnetic theory in particular.) The Hamiltonian formulation of classical mechanics should then be used to develop intuitions about probability distributions in classical mechanics, which in turn can be used to develop intuitions about statistical mechanics. Optionally, at this point, the Hamiltonian formulation of classical mechanics can also be used to develop intuitions about nonlinear dynamics & chaos theory, but while this is good for the broader education of physics students, it is less immediately relevant for the introduction of quantum theory to come soon after (because quantum mechanics is linear). Finally, only after these things happen should quantum theory be introduced, such that there are clear connections of the wavefunction formulation of quantum mechanics to mechanical waves, the phase space formulation of quantum mechanics to classical phase space probability distributions, and the linear algebraic framework of quantum mechanics to linear algebraic treatments of classical field theories (including linear classical electromagnetic theory); this will ensure that students understand how ideas like superposition, interference, rotation through a Hilbert space, statistical uncertainty, and related ideas are not unique to quantum mechanics (which is unfortunately too often a consequence of the way quantum mechanics is typically introduced in undergraduate curricula, at least in the US). We also came to a consensus that in each course, there should be clear explanations of what prototypical systems are analytically solvable, what prototypical systems are not analytically solvable, and why (in each case).

Taking a Class After 3 Years of Full-Time Research

This spring semester, I'm taking a class; as the title explains, this is the first class I've taken in 3 years, during which time I've engaged in full-time research as a graduate student and have been a TA for 3 semesters. This class is in a very different field from my current area of research, as I'm exploring other fields for opportunities after graduation. After 2 weeks of class, I've been considering how taking a class now feels different than it did in high school, college, and the first two years of graduate school.

In high school and college, my main focus was on classes, and I wanted to make sure that I challenged myself as much as I felt I could and got good grades in those classes. This mentality stayed with me through the first two years of graduate school, which is why I felt like I could do pretty well in graduate classes but had a harder time initially finding my footing in research while I remained mentally so focused on classes above all else. I felt quite relieved when I finished my course requirements 3 years ago so that I could renew my focus on research. Since then, I do feel like I've been able to establish a pretty good track record with my research, and given that I'm approaching the end of the PhD program and want to explore other fields, I am comfortable taking this class with fresh eyes and without worrying about grades; in particular, I can really feel like I'm taking this class purely to satisfy my own curiosity and am willing to accept that I'll get out of it exactly what I put into it. Moreover, for the classes I took until 3 years ago, I was fairly engaged with the instructor during lectures, frequently asking questions whether for clarification or edification; now, especially because the others in my class are all undergraduate students, I feel more comfortable letting them take the reins with their own education, and will only ask questions about points that I feel need urgent clarification.

Having been a TA for 3 semesters, I now have a much greater appreciation for the amount of work even instructors whose lectures are of average quality have to do with respect to preparation and delivery of a lecture, fielding questions from students during and outside of class, and grading assignments. Concomitant with that, I especially appreciate the instructors from my past who were particularly good at clearly communicating concepts in the class to as many people in the class as possible in an engaging way, and realize that I was truly lucky to have had so many great class instructors in high school, college, and graduate school. At the same time, my patience for instructors who do a poor job is even less than it was before, because I feel like such instructors are in some sense neglecting the responsibilities to their students fundamental to their job; while I recognize that not everyone develops skills for or interest in teaching immediately, I would hope that such instructors at least put some effort into developing such skills knowing they are responsible for educating young citizens.

It'll be interesting to see how my thoughts on taking a class shift as the semester progresses, and how useful it ends up being with respect to my exploration of other fields. At the very least, I do hope to learn more about how to teach well (and how not to teach poorly) by applying what I've learned from being a TA to my observations of instruction in this class.

Book Review: "The Structure of Scientific Revolutions" by Thomas Kuhn

I've recently read the book The Structure of Scientific Revolutions by Thomas Kuhn. This is a classic treatise from 1962 expounding Kuhn's view of scientific progress not as cumulative and incremental but instead as comprising paradigms in each field and discipline which drive most scientific research while being subjected to drastic changes from time to time; this is the book that popularized the notions of scientific paradigms and shifts therein. It starts with a description of what "normal science" (in the sense of science comprising and being driven by existing paradigms) is, defining the notion of a "paradigm" in the context of science, and how people do science in that framework. It then moves onto the notion of a scientific crisis, and shows how that may or may not develop into a fully-fledged scientific revolution. Finally, it shows how new paradigms may take root and how scientific revolution may ultimately be resolved in one way or another.

While Kuhn did not perform serious sociological research for this treatise (though the book seemed to me like an informal sociological review of the scientific community at large), and while he later in life turned his attention more to fundamental questions of scientific philosophy, he was a historian of science and identified most strongly as that; I feel this may have helped shape this book into something far more clear and engaging for a layperson like myself than what I may have expected from a book about the philosophy of science, as the book is chock-full of relevant and easy to understand references to the history of science (though it may also have helped that Kuhn, having been a theoretical physicist before becoming a historian of science, focused almost exclusively on the historical development of theoretical frameworks in the physical sciences). Moreover, because this was meant as an extended essay, this book is not particularly long, though it is reasonably well-referenced with illuminating footnotes too; in fact, the chapters are called sections, as would befit an essay/treatise. One question to which I kept returning through the book was about how to distinguish between discoveries that answer open questions within a paradigm versus those which more fundamentally threaten the existence of such an established paradigm; Kuhn masterfully addresses the various aspects of this question in a clear progression over the course of the book, to the extent that I almost felt like he was speaking directly to me in order to answer my questions as I read the book. I do have a few criticisms of the book, though these should themselves be taken with a grain of salt and subjected to criticism too, as I am a layperson in the context of the philosophy of science; follow the jump to read those. Beyond that, though, I think this is a really interesting and valuable perspective on the practice of science at the level of groups/communities, and would be useful for anyone interested in how the sausage of science is made, discarded, and remade.

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.

A Month into Graduate School

I realized I haven't posted anything here for this month, so the least I could do would be to provide a quick update. I've settled into my apartment nicely. Classes are going decently: I'm taking ELE 511 — Quantum Mechanics with Applications, PHY 504/514 — Electromagnetism/Statistical Physics, respectively, and ELE 568 — Implementations of Quantum Information. More exciting though has been attending seminars that professors have given about their research, and being able to talk to those professors one-on-one as well. I even have a small side-project that could lead into full-time research with one of the professors with whom I'm interested in working! All in all, this is shaping up to be an exciting semester, and I can't wait to find out more about the research opportunities in the department and ultimately pick an advisor.

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.

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!

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, $k_B = 1$.)

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.

Classical Discrete and Continuum Fields

I've been reading various documents about quantum field theory over the last several weeks, specifically about the canonical quantization of quantum fields. In doing so, I've come to realize that quantum mechanics has a lot of crazy math and even crazier physical interpretations, and I just took that for granted, but now those things are coming back to haunt me in quantum field theory. It is very hard for me to wrap my head around, and I feel like I could use a lot more help in visualizing and intuiting what certain concepts in canonical quantization mean. This will be the first of a few posts which are outlets for me to gather my thoughts and put them out there for you all to see and correct; this one will be about classical fields.

I feel like the easiest quantum field system to study is the phonon. It is a spin-0 bosonic system, so it can be described by a scalar field. Furthermore, said field can be restricted to one dimension, which simplifies the math even further. This means that taking the continuum limit becomes a bit easier than in three dimensions. Follow the jump to see how it goes.