Here I lay out an "official" style guide I follow when constructing topical pages for this site. These guidelines aren't followed in writing this very page or the top level blog-post style pages, such as the [landing page.](Welcome%20to%20The%20Quantum%20Well!.md) In addition, older pages that I haven't revisited in a while don't necessarily follow the guidelines established here, but will in the future. Note this article is long and rambly and may be split up in the future. I don't really expect anyone to really get much from reading this page. However, I feel as though this page is a good way for me to document an evolving format and style I aim to apply widely. The method described here as well as the broader discussion on the [knowledge management](Knowledge%20Management.md) page is mostly derived from the experience of populating this site with technical entries. Thus it is limited by my own experiences as well as whatever implicit stylistic lessons I've drawn from reading textbooks and academic articles. In the future, however, I plan to consult literature on scientific writing from more experienced mathematicians and physicists in order to incorporate more of what others have learned from engaging in technical writing. One thing I've noticed in general is that the bigger the body of writing, the more stringent and expansive the rules must be in order to maintain consistency in quality as well as accurate and consistent connectivity between topics. 🏗️ THIS PAGE IS UNDER CONSTRUCTION 🏗️ # Pages Here I refer to the _page_ as a webpage with a header. A topic that potentially warrants its own page is any _[allowed topic](The%20Quantum%20Well%20Style%20Guide.md#Allowed%20topics)_ that could be a technical term listed in a glossary or an index, a noteworthy example that could constitute an exercise or exam question, or be the subject of a research paper (either recent or historical). These pages are listed on [Indices.](The%20Quantum%20Well%20Style%20Guide.md#Indices) A topic may not become a page if it is overly narrow and of little significance outside of another specific connected topic. In which case, such a topic may warrant a [subsection](The%20Quantum%20Well%20Style%20Guide.md#Page%20Structure) within a page. ^6372bd ## Page Structure * Every page regardless of topic will include an introductory section where the header is the title of the page itself. * Many but not all pages are divided into multiple subsections and sub-subsections (or however deep the well goes with regards to sub$^n$sections). These sub$^n$sections are delineated with sub$^n$headings. * There are several categories of content covered by sub$^n$sections: * A [relatively narrow topic](The%20Quantum%20Well%20Style%20Guide.md#^6372bd) related to the larger page that isn't significant enough to be elaborated on its own page. This is usually a sub-topic that's given its own subsection in order to make the page easier to read and to maintain a [Zettelkasten-like](Knowledge%20Management.md#^32f125) organization. See the example below of a more narrow topic forming a subsection:  * A closely related concept that also has its own page. * In order to give a full introduction for a larger topic, it's appropriate to also introduce most sub-topics in sub$^n$sections to the overarching page describing that topic. For example, in a page about unitary transformations in quantum mechanics, every category of unitary transformation has its own subsection and such subsections contain mostly [block references](The%20Quantum%20Well%20Style%20Guide.md#Block%20references) from the page containing a more in-depth explanation of that topic followd by a link with the statement "... see more" prompting the reader to potentially read the main page being quoted in the block references. See the below example:  ^d51ca0 * If needed, one or two sentenses may be given for the sake of contextualization prior to a block reference. * A Prerequisite concept in an [prerequisite knowledge section.](The%20Quantum%20Well%20Style%20Guide.md#Prerequisite%20knowledge) of an [index](The%20Quantum%20Well%20Style%20Guide.md#Indices) * A Prerequisite concept in an [prerequisite knowledge section](The%20Quantum%20Well%20Style%20Guide.md#Prerequisite%20knowledge) of an [index](The%20Quantum%20Well%20Style%20Guide.md#Indices) that already has its own page. In this cause follow the guidelines for other [subsections linked to their own pages.](The%20Quantum%20Well%20Style%20Guide.md#^d51ca0) * I do not follow hard word limits, however, I prefer to keep sub$^n$sections to no longer than 100 to 150 words. If I see that a sub$^n$section is getting too long, I first check to see if it can be made more [concise](The%20Quantum%20Well%20Style%20Guide.md#Conciseness) or I check if I can divide the concepts being defined into sub$^n$concepts that can form new sub$^n$sections or new pages entirely. This is to preserve the brevity needed to follow [my Zettelkasten organizational paradigm.](Knowledge%20Management.md#My%20own%20implementation%20and%20why%20it%20works%20for%20me) * There is no hard limit on page length. This is kept entirely at my discretion, however, few pages warrant more than 1000 words besides the [Indices](The%20Quantum%20Well%20Style%20Guide.md#Indices), which will include long lists of links to other pages as well as multiple sections and sub$^n$sections containing what I consider to be [prerequisite knowledge.](The%20Quantum%20Well%20Style%20Guide.md#Prerequisite%20knowledge) ### Paragraphs Paragraphs are deliberately kept short - no more than 3 or 4 sentenses and are considered the smallest individual units of information here. Every topic that either has its own [page or sub$^n$section](The%20Quantum%20Well%20Style%20Guide.md#Pages) should be reachable with a hyperlink on that paragraph. ### Hyperlinks Usually every direct mention of a topic covered in a [paragraph](The%20Quantum%20Well%20Style%20Guide.md#Paragraphs) (including when it is given a context-dependent alternate name) is to be hyperlinked to that page or subsection. The exact constraints that define what I mean by the word "usually" are described [here.](The%20Quantum%20Well%20Style%20Guide.md#Restrictions%20to%20hyperlink%20use) This applies also to references within a page to other sub$^n$sections in that same page, or the broader page. * [Paragraphs](The%20Quantum%20Well%20Style%20Guide.md#Paragraphs) within a page will often reference other sub$^n$sections in that page. This will result in hyperlinks between sections within a page. * e.g. If in the page titled "State vector" I have a subsection referring to "measurement" on that page, which describes the process of measurement as it applies to state vectors, I reference the broader concept of state vectors. In doing so I link back to the broader page from one of its subsections. * The reason to do this is if a sub$^n$section or [paragraph](The%20Quantum%20Well%20Style%20Guide.md#Paragraphs) is linked to another page ([via a block reference](The%20Quantum%20Well%20Style%20Guide.md#Block%20references)), and it is previewed when hovering over that page, I want all references to other topics in that subsection visible as well. This is for the sake a clarity and fast navigation. #### Block references A block reference is a clickable body of text that links to another [page.](The%20Quantum%20Well%20Style%20Guide.md#Pages) This will often be a mathematical expression as shown in this example:  Here the wavefunction of a quantum harmonic oscillator is invoked via a clickable mathematical expression. This lets the reader see where this expression comes from, and keeps me from repeating the same math that's already written on another page. This creates an [inheritance structure](Knowledge%20Management.md#Inheritance) analogous to a _class hierarchy._ Avoiding repetition also helps homogenize notation across topics and allows me to trace the source of a typo or mistake that might otherwise be allowed to propagate across other notes. #### Restrictions to hyperlink use ##### Concept Hierarchy ##### Terms and concepts that should also be obvious ### Tags ### In-line citations The text in this project will read like a textbook and thus in-line citations will not be required for most claims. However, accompanied with any reference to a particular person's discovery, an in-line citation to their work is required (see the section on [referencing individual people.](The%20Quantum%20Well%20Style%20Guide.md#^6b5b06)). The only other case where a citation may be warranted is if I've noticed an explicit contradiction within the academic literature. This is something that, however, will very rarely be encountered unless a particular topic is subject to current research. ### Recommended Reading The role of the Recommended reading section is as a guiding hand. Either to future me, who needs to collect hundreds of references and might not remember which has what, or anyone who comes across my notes and may want to dig further or check that what I have written is correct. ### Math Pages #### Properties of mathematical objects A _Property_ is a statement that can be made about a mathematical object that isn't part of the definition itself. A property may follow immediately from the definition or require an extensive [proof.](The%20Quantum%20Well%20Style%20Guide.md#Proofs%20and%20Examples) * Properties of a given mathematical object are given in their own section or sub$^n$section labeled "Properties of [X]" and these properties are enumerated where the exact enumeration is generally determined by the "distance" of this property from the definition. That is, it is determined by how many steps it would take to prove or derive this property. What this means is that simpler properties are listed first. * Properties that require short proofs (any proof one line or shorter) are discussed below the enumerated list with short proofs being given. #### Proofs and Examples ### Physics pages #### Proofs and Examples ### Incomplete pages Empty sections and sub$^n$sections are allowed. To the reader they indicate the placement of future content and to me they are a reminder of what directions I need to expand into. I understand that incomplete content may look unprofessional or unpolished. However, allowing for and even embracing incompleteness makes the writing process much quicker. ## Mathematical notation I cover a wide array of topics, but despite this I endeavor to maintain a standard notation across all pages that's as homogeneous as possible in order to have it widely applicable across subfields of physics and math. # Website Structure ## Indices Indices are pages that include links to other pages pertaining to a broad category. Indices include: * a list of links to pages that belong in that category, * followed by a list of sub-indices * followed by [subsections containing prerequisite knowledge,](The%20Quantum%20Well%20Style%20Guide.md#Prerequisite%20knowledge) that are under a section titled "Basic concepts." * Lastly, a [bibliography](The%20Quantum%20Well%20Style%20Guide.md#Bibliographies) of all sources consulted when writing every note listed in the index and its sub-indices. Indices will always be titled with the name of the category they encompass followed by the word "index" in parentheses. E.g. "Algebra (index)." ### Defining Indices ### Prerequisite knowledge Under a section titled "Basic concepts," what I may consider to be prerequisite knowledge needed in order to dive into the pages listed on the index is included. * Prerequisite knowledge may be often what most people with an intent to go to college or university would encounter in high school/secondary school or, once in college, their first year of undergraduate math or physics. * In addition I may include [block references](The%20Quantum%20Well%20Style%20Guide.md#Block%20references) to other advanced topics as prerequisite knowledge. * For example, some entries in set theory may be referenced in the Analysis index as a prerequisite for that topic. Same could be said for many pages on linear algebra and their relation to the whole field of quantum mechanics. ### Bibliographies Each bibliography entry will eventually link to a [Literature note.](The%20Quantum%20Well%20Style%20Guide.md#Literature%20notes) ## Literature notes ### Links to other topics from literature notes A Literature note may only link to [indices,](The%20Quantum%20Well%20Style%20Guide.md#Indices) pages that directly cite that note, pages that link to that literature note through a [recommended reading section](The%20Quantum%20Well%20Style%20Guide.md#Recommended%20Reading), and other literuature notes. This is in order to avoid clutter in backlinks to main topic entries. For example, if I have an entry on the Schröndinger equation, I don't want or need to link to every paper that invokes it since there are many thousands of papers that ultimately invoke it that might not be useful or relevant to anyone wanting a general understanding of that equation. # Writing Style An attempt is made to present information [concisely](The%20Quantum%20Well%20Style%20Guide.md#Conciseness), [precisely](The%20Quantum%20Well%20Style%20Guide.md#Avoiding%20vague-sounding%20statements) and [objectively](The%20Quantum%20Well%20Style%20Guide.md#Objectivity) and there are particular rules I impose in order to do this. In addition, there is a certain degree to which I have to be very exacting with every word I use. Exactly what words to use in a given statement or definition can be a matter of opinion, but it is an inherent challenge to good academic writing. Naturally when a lot of technical jargon is included it can be surprisingly easy to flat out have wrong information due to an imprecise choice of words. In addition, I'm also mindful of all the times I've felt misled by definitions and statements I didn't quite understand and thus I've aimed to minimize statements that may be technically correct, but easily misleading to someone who isn't an expert. ## Conciseness Conciseness is required by the constraints presented in the description of the [general page structure.](The%20Quantum%20Well%20Style%20Guide.md#Page%20Structure) and is also aided by the [avoidance of certain topics.](The%20Quantum%20Well%20Style%20Guide.md#Topics%20I%20avoid) Generally speaking wordiness is also avoided. Thus, simple sentences are preferred, simple vocabulary is preferred, and definitions and explanations are kept to minimal lengths. The emphasis on simple vocabulary is also helpful towards the construction [of precisely worded statements.](The%20Quantum%20Well%20Style%20Guide.md#Precisely%20worded%20statements) One way if keeping things concise is to avoid filler, besides avoiding certain topics, this also means avoiding [analogies and metaphors.](The%20Quantum%20Well%20Style%20Guide.md#The%20use%20of%20analogies%20and%20metaphors) ## Objectivity Objectivity is encouraged by [the avoidance of certain topics.](The%20Quantum%20Well%20Style%20Guide.md#Topics%20I%20avoid) Additional rules are given below. * No subjective statements are given anywhere except the [recommended reading](The%20Quantum%20Well%20Style%20Guide.md#Recommended%20Reading) sections. Here I limit myself to only giving my opinion on which texts are introductory, which are brief and concise, and which are lengthy and thorough, and which are good for whom (e.g. is the text more appropriate for a physicist or a mathematician? Is it for a graduate student or an undergraduate? Advanced undergraduate?). * This is in order to align this section with its [purpose,](The%20Quantum%20Well%20Style%20Guide.md#Recommended%20Reading) which is as a guide rather than as a mere source of information. * Subjective statements are anything that could be considered the opinion of an author and must be avoided. * This potentially comes at the cost of not emphasizing that some topics are more important than others, but I believe it's for the reader to determine what they think is important given their own personal needs. My hope is the importance of a topic can also be more directly quantified by the structure of this site. E.g. the importance of a particular [page](The%20Quantum%20Well%20Style%20Guide.md#Page%20Structure) is highlighted by the number of back-links to it in addition to the length of that page. * In a derivation, it should never be stated that a step "simply follows" or, as is common in mathematics textbooks "is trivial." * Generally if a step really is actually trivial, it would normally only require an additional line of text or a single line of math to present it. Thus, it should be shown. This may be a simple as stating "we plug in X into Y in order to obtain Z by evaluating the expression," but no simpler, since knowledge of elementary algebra is assumed in my chosen writing style. * Otherwise, another common reason a mathematician or a physicist may call something trivial or simple is if a step requires a piece of knowledge that may seem really basic in that context. However, broadly speaking whether something is actually basic is subjective. It may also be alienating to a beginner or someone who may only have recently learned the "basics" or has gaps in their knowledge if a presentation skips certain steps. * The solution here is to reference the relevant mechanics of a basic step through a hyperlink. Unfortunately not every idea can be covered and certain terms don't warrant always being linked-to. Thus we are constrained by the guidelines offered [here.](The%20Quantum%20Well%20Style%20Guide.md#Terms%20and%20concepts%20that%20should%20also%20be%20obvious) * The choice to use any particular [analogy or metaphor](The%20Quantum%20Well%20Style%20Guide.md#The%20use%20of%20analogies%20and%20metaphors) is inherently subjective and may convey the biases or cultural background of the writer. See also note on [analogies within topics in math and physics.](The%20Quantum%20Well%20Style%20Guide.md#Analogies%20within%20topics%20in%20math%20and%20physics) Thus, no analogies or metaphors are allowed. ## Avoiding vague-sounding statements ### Precisely worded statements #### Words with variable meanings or connotations There should absolutely be no question over the meaning of any particular word or phrase. Of course, this can only be a guiding principle since language is fluid and there are slight regional variations in English. * An example is the word "in general." In common speech, "in general" is taken to mean "usually," however in mathematics and physics it may be used to mean "always" unless an additional qualifier is given. There is a slight discrepancy here between meanings, thus, I discard the phrase all together. * See the following example where the usage of the phrase "in general" may cause some confusion:  * _All_ baryons have half-integer spins and the first sentence in this excerpt is by a certain standard correctly written since there are no qualifiers and we can take "in general" to mean always. However, some readers, such as myself might read that first sentence and interpret it as saying that most but not all baryons have half integer spins. For this reason, for cases like this, I'd be more explicit by saying "all" in this case and words like "usually" and "most" in cases where something isn't _always_ true. If I were to use a word like "usually" or "most" I will then be obligating myself to immediately followup with a [precisely worded](The%20Quantum%20Well%20Style%20Guide.md#Precisely%20worded%20statements) statement concerning the implied exceptions.^[There is actually a more clear-cut mistake in this excerpt as well. It states that protons are the lightest hadrons when actually they're the lightest baryons. This was the subject of a [discussion on Twitter.](https://twitter.com/QuarkWilliams/status/1514539550718279680)] ### The use of analogies and metaphors In general the usage of analogies and metaphors may ground someone's understanding. Where they are most useful in my opinion is in the creation of mental associations that can help a reader memorize a particular concept. This can be very helpful for many learners, however, memorization is not the same as understanding and I've never had much of a proclivity towards understanding ideas through the lens of metaphor or analogy. #### Analogies within topics in math and physics Often times there are different topics and ideas that appear to be analogous to each other within the scope of [allowed topics.](The%20Quantum%20Well%20Style%20Guide.md#Allowed%20topics) If this is the case, the first question for me to ask myself as a writer is, if any given pair of topics are actually _merely_ analogous to each other or if there is a connection that's arguable either via a mathematical proof, derivation, or scientific data. Consider the following statement: "Schrödinger's equation is analogous to Newton's 2nd law." This is a commonly used statement in both popular media and introductory lecture courses. However, I'd say it's incomplete to the point of being potentially misleading, even if it is absolutely true in some sense. Indeed, both Newton's 2nd law and Schrödinger's equation are equations of motion and serve a particular role as being a fundamental formulas for their respective theories. But are they even fundamental in the same ways within their respective theories? Someone who may know only about Newton's laws but not Schrödinger's equation won't know in what way they are analogous. Also, clearly the Schrödinger equation doesn't "look" like Newton's 2nd law. Does it? I won't examine its particular resemblance here, but I can imagine someone who may only know of Newton's [2nd law](https://en.wikipedia.org/wiki/Newton%27s_laws_of_motion) as $F=m\mathbf{a},$ will draw a different connection between the two in their minds than someone who thinks of Newton's 2nd law as $F=\frac{d\mathbf{p}}{dt}.$ Someone who knows that the [Euler-Lagrange equation](https://en.wikipedia.org/wiki/Euler%E2%80%93Lagrange_equation) alone describes Newton's laws of motion will have yet another perspective. Finally the person who may have the most accurate interpretation of what it means to refer to the Schrödinger equation as an analog to Newton's second law (or actually all three of Newton's laws of motion?) may be someone who's aware of the [Hamilton-Jacobi equation](https://en.wikipedia.org/wiki/Hamilton%E2%80%93Jacobi_equation). And indeed the classical equation that the Schrödinger equation most closely resembles is the Hamilton-Jacobi equation, which can be used to derive (or more accurately argue for) the validity of the Schrödinger equation in a way that can't be done with a more basic understanding of Newton's laws. Thus it is, in my opinion, better to simply state that the Schrödinger equation is an equation of motion (where I've also defined the general notion of equations of motion in another page) - that way no one is primed to draw any potentially inaccurate or incomplete connections to prior knowledge. More often than not (if not always), relationships between topics are not coincidental and it is often the case that two analogous-seeming concepts are actually unified by a more general concept and pursuing this more general concept and elaborating on it is always going to be the preferred route. With graph connections I like to be able to give the reader the opportunity to explore these connections while allowing me to be as precise as possible with all of my definitions and descriptions. Thus there will never be the phrase "[X] is analogous to [Y]" within these notes. # Allowed topics The following topics are to be covered here. There is an explicit bias in the choice of topics based around what's useful for a physicist and particularly a quantum physicist (as [I've already alluded to in the landing page](Welcome%20to%20The%20Quantum%20Well!.md)). * Mathematics (both pure and applied) * Information theory * as well as computer science (excluding all information pertaining to particular programming languages or software tools) * Physics * as well as applied physics * and a wide range of topics that may be considered engineering topics, such as particular experimental tools. However, only operational principles will be included. There won't be, for example, a page describing how to use the [Inspire™ IR Automated Ultrafast Optical Parametric Oscillators,](https://www.spectra-physics.com/f/inspire-ir-ultrafast-opo) however a page or series of pages describing the operational principles for different types of OPOs are perfectly valid entries. See [edge cases](The%20Quantum%20Well%20Style%20Guide.md#Edge%20cases) for more detail. The particular writing style and formatting I follow is optimized for these allowed topics and I think I'm already trying to do way too much at once with this project. Therefore, I think these boundaries are important. ## Edge cases There may be some additional scientific or mathematical topics that are directly established by the [above topics](The%20Quantum%20Well%20Style%20Guide.md#Allowed%20topics). Topics that may be viewed as not being purely physics or math will only be included if they are of particular interest via references to the above topics in academic literature. Consider the following edge case. * This recent studying showing [how bird retinas use quantum spin dynamics in order to sense magnetic fields](https://www.nature.com/articles/s41586-021-03618-9) could absolutely form the basis for a valid page or sub$^n$section since it invokes particular phenomena established from physics. However, any technical information needed to describe this phenomenon that isn't based on physics or mathematics won't have its own pages or subsections and would only be elaborated on with external links if that is deemed necessary. * For example, there should be no pages or sub$^n$sections about particular proteins, chemical processes, or say, the theory of evolution, or the concept of a retina since those topics aren't established using the principles laid out by physics or mathematics. * A similar standard will be applied to topics in chemistry, astronomy, neuroscience, psychology, or quantitative social sciences - all areas of research that include topics grounded in pure mathematics and even methods from physics, but also incorporate observations that are highly emergent from underlying laws and thus aren't described by the language of physics or mathematics. There will consistently be a focus on certain areas of physics and mathematics that I've studied already. Thus edge case aren't likely to arise. However, this project, with its links and notes is fractal in nature, and at the end of the day, boundaries need to be set in order to prune the tree of knowledge. ## Topics within math and physics I avoid In the spirit of both [conciseness](The%20Quantum%20Well%20Style%20Guide.md#Conciseness) and [objectivity](The%20Quantum%20Well%20Style%20Guide.md#Objectivity) I follow the following constraints even with regards to topics that may fall under the list of [allowed topics.](The%20Quantum%20Well%20Style%20Guide.md#Allowed%20topics) * Historical or biographical notes within pages or sub$^n$sections are not allowed. * There will never be a page or sub$^n$section that's simply the name of a person, even if it is to simply list technical publications. * I will refer to a particular person or groups of people by name only if they are credited as authors of a technical work that lays out in its entirety (or almost entirety) the topic being described in a given page or sub$^n$section. ^6b5b06 * There will be absolutely no discussion of any non-scientific or non-mathematical concept. * Any topic that may be speculative will necessarily have to meet the criteria required for it to be a hypothesis or conjecture. There must be peer-reviewed scientific literature to back up any given hypothesis or conjecture and this has to be from a "reputable" journal in mathematics, computer science, or the physical sciences. What is considered reputable is somewhat subjective, but there are lists of predatory journals or questionable low-impact journals that can be found online that may be consulted if the need arises. * There will be absolutely no discussion of philosophy - even if it may seem entirely called for by the nature of some foundational topics in math and physics. * There will be no scientific topics outside of [edge cases.](The%20Quantum%20Well%20Style%20Guide.md#Edge%20cases) ### Why I choose to avoid certain topics I simply do not believe that the lives of the people many equations and theorems are named after are relevant to what I want to do here, and I consider biographical information to be a distraction and subject to embellishment that doesn't fit with the scientific rigor I care about. I also avoid anecdotal stories surrounding particular experiments, publications, or proofs for the same reason. I believe historical information while helpful, since it can provide a pre-existing chain of deductions, may also bias our understanding in unhelpful ways. It is my belief that the more helpful picture is that which is the most general rather than the one that's informed by the history of a topic. When discoveries are made, it's usually to answer a specific question or done within a more specific context. These questions and contexts are definitely presented here, however, they aren't necessarily central to a given topic in my opinion. The advantage of focusing on more general pictures is information compression. %%Part of what informs my choices here is my rejection of the cults of personality that have formed around certain physicists and science popularizers such as Richard Feynman or Albert Einstein, while I believe other figures such as [John Bardeen](https://www.smbc-comics.com/comic/2010-12-24) or [Lise Meitner](https://xkcd.com/896/) are severely underappreciated by the general public. Frankly, I'm bothered by the fact that our historical views and who we collectively admire, ignore, or decide to explicitly name as being underrated bleeds into how students choose to prioritize topics or view physics or math as a whole. Moreover, typically, numerous nameless and nearly forgotten students, assistants, and rookie research fellows are behind every discovery, especially in current research. Therefore I think the most honest and complete presentation of physics and math is the physics and math itself followed by citations of technical authored works.%% For similar reasons, I avoid philosophy. I consider philosophy to be tangential and not necessarily grounded in the rigor of mathematical proof or scientific empirical observation. Philosophy is also simply not relevant to most topics on here, and like with which scientific figures we choose to give the most credit to, it can be biased by our culture. Where it may be tempting to include it is in the foundations of quantum mechanics as well as in the foundations of mathematics, such as set and number theory. However, there will be no philosophy regardless. # Typsetting All notes are written in markdown with a built in [$\LaTeX$](https://www.latex-project.org/) environment for all mathematical expressions where in-line expressions are placed in a $\LaTeX$ environment by bordering them with single dollar signs (e.g. \$F=ma\$) and centered expressions use double dollar signs (e.g. \$\$F=ma\$\$). Markdown with the ability to include $\LaTeX$ expressions is a core part of the [Obsidian](https://obsidian.md/) text editor used to build this site. A comprehensive list of mathematical symbols and their corresponding syntax in $\LaTeX,$ may be found [here.](https://ctan.math.washington.edu/tex-archive/info/symbols/comprehensive/symbols-a4.pdf) Note that not every symbol listed here is available on [Obsidian](https://obsidian.md/), however quite a few are and I've yet to run into any need to use math symbols not included with Obsidian. --- _This is one of several blog-post style pages that's not part of the [indexed notes](Welcome%20to%20The%20Quantum%20Well!.md#Indices) that constitute what I consider to be the core content to this site._