Prime, Compress, Author

When I was in graduate school studying for my qualifying/comprehensive exams, I had to devise a way I could learn as much information as I could. For one, I'm a terrible timed-test taker, I do not compete well in this setting. Remove the time constraint and I'm much more competitive; probably the case with others as well. However, that is a luxury that is rarely afforded to students since the goal of timed examination is to determine how well an individual can recall and utilize concepts in comparison to their peers. Because of my poor test taking abilities I was obsessed with preparing the best I could and formulated a technique that works for me. I'm positive elements of this approach exist in more well known standard methods.

My reason for writing about this approach now rather than 10 years ago is that I've been spending a lot of my time learning new things in the fields of AI, neuroscience, and thermodynamics computing. Because I have no formal background in these topics, I needed to leverage my previous approaches that worked well for me, hence my method used for my comprehensive exams.

Overview

My method is primarily designed for learning from books, but could potentially be adapted for audio or video content. It involves three key steps:

1. Prime: Prepare your mind by uninterrupted reading.
2. Compress: Annotate key ideas in your own concise style.
3. Author: Write as if you were the author of a textbook.

This is what I refer to as the PCA method, which is somewhat ironic given that principal component analysis is a method of compressing data, yet retains a richness of information.

🧨 Prime

The goal here is to familiarize yourself with the concepts and ideas in the learning material without interruptions. During this initial exposure, do not take notes; focus solely on understanding the content. Additionally, avoid consulting supplementary materials, even if parts of the text are unclear. This maintains the focus and efficiency of the session.

For effective priming, shorter segments are generally more manageable than entire chapters, as chapters often cover broad topics with complex details. Engaging with smaller content blocks allows for a more focused approach.

At the end of reading you should feel like you have a general sense of the content and what is important.

📦 Compress

I find this step crucial as it solidifies the understanding of the material. Re-read the text from the Prime step and take notes using paper and pen in real-time. The format of your notes should reflect your personal preferences, whether as text chunks or bullet points, etc. Include diagrams and summaries of important figures in your own style. Add footnotes for references or important details to revisit later.

The aim is to distill the content into a personalized, dense, and informative resource. These notes are meant for your use, tailored to your understanding. Don't write them thinking you'll share them with others, so "being messy" is okay!

📚 Author

Now, adopt the mindset of an "Author" or "Teacher." Use the notes from the Compress step as a basis to expand into a clear, detailed, and presentable format. This step involves refining your notes into a resource useful to others. Aim for thoroughness and professionalism.

Initially, try to rely on your memory and notes to guide your writing, ensuring the content is conveyed effectively. However, feel free to refer back to the original material and your notes, research further as needed, and enhance your explanations with additional resources.

Sometimes I call this step the "Being Feynman" step, in reference to the Richard Feynman technique. You could say PCA is a form of the Feynman technique, but as Dirac's self-appointed student it would be blasphemous to do so, 😉.

Re-read & Polish (Optional)

If time allows, revisiting the original text can enhance the accuracy of your work. Review your notes and the final document, possibly consulting with peers to identify and correct misunderstandings. Recognizing errors independently can be challenging, making peer feedback invaluable.

Tools You May Want to Use

1. Prime: Utilize the primary text, audio, or video only.
2. Compress: Choose quality writing tools and your preferred type of paper; I favor plain white printer paper.
3. Authoring: Consider various tools based on your needs:

App	Scope
$\LaTeX$	Ideal for non-computational writing, with Overleaf providing a quick start.
Obsidian	Excellent for non-computational, markdown-based documentation.
Jupyter Notebooks	Best for including computations and visualizations.
Quarto	Similar to Jupyter Notebooks but with expanded document format options.

Footnotes

---

1. Dividing the Prime step into smaller segments simplifies the Compress step by reducing the volume of notes and easing cross-reference with previous annotations. ↩

Reuse and Attribution

Superconducting Materials Primer

To become more familiar, I am putting together a "crash course" from a materials science perspective on superconducting materials. It's a work in progress and far from complete. I am primarily jotting down notes from reading a few good books, such as the introductory book by Speller. It's a straightforward read and will get you started.

Warning

These are working notes, so there are bound to be errors and missing content.

Once I am done with my notes, I likely won't follow up on this subject unless there is compelling interest. One small reason for this is that, upon reading various sources, I became bothered by the "hand-wavy" approach to describing the physics of high-temperature¹ superconducting materials. BCS theory, which describes low-temperature (i.e., < 30 Kelvin) superconductors, is not well-suited for high-temperature materials due to factors such as strong electron-electron interactions and the presence of other exotic states of matter. To my knowledge, there is no all-encompassing HTSC theory.

References

[1] Speller, Susannah. A Materials Science Guide to Superconductors and How to Make Them Super. Oxford: Oxford University Press, 2022.

Footnotes

---

1. High-temperature superconductors are meaningful only in the context of cryogenics, which involves the application of refrigeration to cool samples down; they operate at temperatures up to about -150 degrees Celsius. ↩

Reuse and Attribution

Materials Informatics Book

Should I write a technical "how-to" textbook?

Why am I asking this question, well, I've come to somewhat of a cross roads in materials informatics. I have digested a lot of research papers and review monographs [1-4], but constantly run into the following problem:

How do I implement the technique discussed, reproduce the results, and extend it towards my specific domain topic?

The answer over and over is, these resources aren't going to enable that or provide a way to do so. Yes in some cases there are Github repos that provide the source, but what I find is the data input pipeline is so complicated or convoluted that figure out how to get my data to work is too much effort. It would just be better to actually implement the model from scratch based on my specific data preprocessing/pipeline.

So why would me writing a address this challenge? Well, for one no one else has written such a book. You have three textbooks available on this topic:

• Isayev, O., Tropsha, A., & Curtarolo, S. (2019). Materials informatics: Methods, tools, and applications. John Wiley & Sons. url.
• Rajan, K. (2013). Informatics for materials science and engineering: Data-driven discovery for accelerated experimentation and application. Butterworth-Heinemann. url.
• Kalidindi, S. R. (2015). Hierarchical materials informatics: Novel analytics for materials data. Elsevier. url.

There are a few other monographs [3-4] that try to focus on specific subdomain areas of materials science. The books above are actually pretty good if your looking for a foundational understanding of data science and machine learning applied to materials science and engineering. The problem with those books is they are more reference text for people involved in materials informatics. They won't help you get going in front of a computer or information systems.

This is what is missing and I want to provide a solution. One reason is that I'll learn more by writing a book. The second reason is I think the tools are now available to make writing this type of book much smoother. Writing literate programming is a regular thing (e.g., Jupyter notebooks) and therefore writing while coding is straightforward, usually.

The dilemmas I face are what programming language to use, what framework to write in, how much to cover, and best examples/case studies to use. For the programming language its between Python and Julia. I'm torn because I prefer to use Julia but Python is more broadly adapted and has very mature and standard packages (e.g. scikit-learn, pytorch). For the framework its also a challenge. I'm favoring Quarto at the moment and it won't matter if I use Julia or Python. Similar case for Jupyter Books. I haven't used Jupyter Books and I'm not too interested in adapting the MyST. There are other options to explore as well, such as Books.jl which is geared towards PDF and website generation.

For the content the book would cover, I need to be very thoughtful. My rough outline would be something like:

1. What is data and information
2. Describing data: Probability, Statistics, and Visualization
3. Processing & Transformation of data
4. Pattern extraction and reduced representation
5. Regression, optimization, and prediction
6. Neural network models
7. Autonomous solution seeking

The first 3 chapters are probably self-explanatory, the chapters 4,5, and 7 would correspond to unsupervised, supervised, and reinforcement learning. This is a moving target so it would change based on my particular interest and focus.

The key point is that each chapter provides the background and the code to actually do something at the computer. My goal would also be to try to do as much as possible from scratch. Meaning, that if it made sense I would actually have a section in the chapter on neural networks that builds the layers, does forward and backward propagation, and trains using minimal packages (e.g. NumPy). Why do so if in the end we all are going to implement and deploy using pytorch or `tensorflow? Because for most the act of doing is what solidifies understanding and comprehension. After coding up a simple NN, when someone talks about backpropagation, you'll know what is actually being down, at least from the most minimal implementation ¹.

Status

I have yet to really start writing, but plan to have some kind of a draft by end of 2024. My goal would be to make the draft available online first. For the physical copy I intend to go the self-publishing route using Amazon Kindle services.

As for the applications and case studies used, well, I really want these to be real in the sense that they have either been done in academia or industry. I want to try and avoid "toy problems" not because they aren't useful but because I want to avoid the issue with a creating a insurmountable barrier to applying whats in the book the readers specific interest/problem.

My hope for this potential book is that grad. students and researchers who want to get into this area, but don't have any hands-on experience, can more easily do so by working through the book. The book obviously wouldn't be at the forefront of research methods, but it would be as if you were taking a graduate level lab at a major university.

References

[1] K. Takahashi and L. Takahashi, "Toward the Golden Age of Materials Informatics: Perspective and Opportunities", J. Phys. Chem. Lett., vol. 14, no. 20, pp. 4726-4733, May 2023, doi: https://doi.org/10.1021/acs.jpclett.3c00648.

[2] C. Li and K. Zheng, "Methods, progresses, and opportunities of materials informatics", InfoMat, p. e12425, Jun. 2023, doi: https://doi.org/10.1002/inf2.12425.

[3] T. Lookman, F. J. Alexander, and K. Rajan, Information science for materials discovery and design. Springer, 2015.

[4] I. Tanaka, Nanoinformatics. Springer, 2018.

Footnotes

---

1. There are different numerical implementations to achieve backpropagation and the book I would write showing how to implement a NN would focus on the most basic approach. ↩

Reuse and Attribution

Not so hidden: A clearly written popular quantum physics book by a masterful communicator

$^\dagger$My Commentary

First of all, I am a very big fan of Sean Carroll and in general many of the fantastic science and technology communicators in the 21st century. As a society we are in debt to their efforts and time in aiding in understanding the fascinating and intriguing universe we live in. What I enjoy about Sean Carroll's approach to communicating physics concepts and topics is his unmatched clarity in delivery and pace of speech. If you've every listened to his podcast (Mindscape) or others that he has been a guest on, you know what I'm talking about. His most recent popular physics book titled "Something Deeply Hidden: Quantum Worlds and the Emergence of Spacetime" continues his trend of excellence in science communication.

In this book, Sean focuses on the foundations of quantum physics and how we ( the community scientist) have become complaisant with the "shut-up and calculate" mentality when dealing with the quantum realm. The issues stems from the early pioneers of quantum mechanics who could not make sense of the predictions from the mathematics and the experimental observations, namely, that the quantum object/information we call the wavefunction, does not manifest as described by the math when measured in the lab. Lets throw some math into the mix just to make things clear. In nonrelativistic quantum mechanics we have an equation which describes the relationship between the time evolution of the quantum state (i.e. wavefunction) to the evolution of energy content of a system given that quantum state; this is the what we are told the infamous Schrödinger equation:

$$ i\hbar \frac{\partial} {\partial t} | \Psi \rangle= \hat{H} | \Psi \rangle$$

The quantum state function, $\Psi$ is called the wavefunction because in many cases it has a functional form that reassembles wave-like characteristics. This wavefunction differs from our intuitive notion of classical waves in that the amplitude is a complex number.  Sean Carroll's argument is that the wavefunction is all information needed to describe a quantum system. Furthermore, he argues that we take this equation at face value for what it tells us, namely, that given a quantum wavefunction we can describe its evolution deterministically. This is a key statement, because in popular science you may hear that quantum mechanics is a probabilistic theory, that is only true  if we are concerned with knowing additional information about the system using the wavefunction. For example, if we want to know the position, momentum, or energy then we can only speak in terms of probabilistic outcomes of those observable. But the wavefunction always evolves deterministically via the Schrödinger equation. For example if we want to know the probability expectation for observing  or measuring the momentum of a wavefunction describing a particle, we would write something down like:

$$ \langle \Psi | \hat{p} | \Psi \rangle $$

this equation provides us with a mathematical result about the momentum representation of the wavefunction  in a probablistic manner. It is  beyond the scope of my intent for this blog but the reason we can't say that the observed momentum of the wavefunction  $\Psi$ is exact, is related to the fact that the wavefunction is a superposition of equally valid solutions in what is known as Hilbert space. 

Now the main focus of the book is on an alternative understanding for the measurement catastrophy in quantum physics, that is to say, when experiments are performed on quantum systems we don't get the entire probability distribution of the wavefunction for an observable/expectation as an output. What we get is a single data point from that sampling space. If we conduct enough experiments, then of course we recover the distribution. But why don't we get the entire wavefunction probability when we measure? The historical and mainstream thought on this is that something happened by which when an observer (e.g., the human eye or a digital sensor) measured the quantum state/wavefunction so that it "collapased" into a single value. Now you say "What do mean? What forced the function to collapse?", yes this is indeed a strange phenomena. The Schrödinger equation nor any other mathematical interpretation tells us anything about a wavefunction "collapse". For many years I never really thought about this, but more recently it really is bothersome that a quantum state just "collapses" to a single value as if something forced it for which we don't know anything about. Albert Einstein's thought on this was that the forced collapse is due to local hidden variables; things we are unable to identify as being part of the system and are locally causal.

Now Sean Carroll's approach is more epistemic, we know we have this quantum mathematical object called the wavefunction and every thing in the universe can be described by it, so what happens when the quantum system I am describing interacts with an observer who is also treated as being a quantum system. The outcome is that we get  parallel quantum states that are very much deterministic and in existence, but having different probabilistic outcomes (I think this is how I understand it?). In other words, the act of quantum systems interacting produces many outcomes that in a sense occur in parallel worlds, hence the many worlds. To be clear we don't need to think of the same physical space being occupied, but rather that in some abstract representation of many isolated outcomes have occurred with validity. This approach goes by the Everettian or Many Worlds interpretation.

The main tenants of the the many worlds argument are 1.) we should not try and interpret the meaning of the Schrodinger equation but just follow the mathematics as providing what is real and 2.) don't select which systems behave quantum mechanically, assume every physical object in the universe can be described by a quantum state.

Although I very much appreciate Sean's insight and excellent introduction to this foundation quantum physics vantage point, my own human bias doesn't want to agree. Its not that I don't think its a valid understanding of the outcomes of the Schrodinger equation, but more that it leaves me wondering about the other "branches" of the wavefunction. For example, can one roll-back time to traverse a new branch? This should be possible since the time evolution operator in the Schrödinger equation is a unitary operator.

There are two things I should mention, 1.) I'm not a quantum physicist by training so my understanding of the topic could have gone wary, 2.) I haven't finished the book.

$^{\dagger}$ I haven't finished the book yet so this is a partial commentary.

Reuse and Attribution

An op-ed for a new STEM professional degree

I have recently read a few articles about how the Ph.D. process is inadequate in non-academic workforce preparation and how it needs to be revamped. I agree that its time to revamp workforce training, but not necessarily the Ph.D. degree itself. The following post is just my thoughts and opinions, and takes a US and STEM centric focus. I’m just thinking out loud here — not trying to persuade anyone —  so feel free to disagree since I’m interested to hear what others think.

In my opinion it is unfortunate that the Ph.D. degree has been miss represented. The Ph.D. degree is not a degree, in principal, intended to be tailored towards workforce preparation in the same way as professional degrees like a M.D. , J.D., D.D.S, Pharm. D., etc. A Ph.D. is a basic or applied research degree, which means you are being educated on how to expand the knowledge in a given domain by formulating a hypothesis or problem statement and conducting the necessary tasks to provide an explanation or answer. This skill set can be extremely valuable to employers if that is what they are looking for, however, in my opinion many non-academic employers are really looking people who are STEM educated and have a high-level expertise in specific technical skills, which is usually a byproduct of the Ph.D. process. However, I don’t believe a Ph.D. is necessarily the best or only option for this.

I believe a new terminal degree, say we call it Masters (or Doctor) of Technical Skill in ABC, can be established with direct input from non-academic entities. This would be a none-research professional degree.  Something like this probably already exist within a Masters of Engineering or a non-thesis M.S option, but these programs are typically structured around lectures and seminars with minimal practical hands-on training. There is also the obvious issue that most of these kind of programs don’t have modern infrastructures and facilities as well as exist with limited funding levels.

The goal of such programs should be focused on workforce preparation towards becoming an expert in the knowledge, application, modification, and operation of sophisticated software or hardware. From my perspective and background, I could easily see programs such as Masters of Technical Skill in Materials Characterization, whereby students would become extremely skilled in understanding the principals/applications and advance hands-on operation/modification of selected instrumentation, for example SEM, TEM, XPS, Auger, FTIR, Raman, etc. You might argue that this is just a technician degree, but I would argue that these types of systems (or software) require a significant amount of scientific and engineering education prior to becoming a verifiable expert. We can parallel this with how M.D. students after graduation are expected (its more accurate to say required if the person is going practice medicine) to take on a postgraduate residency training program even though they are already highly educated in medicine and life sciences.

Just in materials science and engineering I can imagine several different tracks and sub-focuses. For example,  another program could be a Masters of Technical Skill in Ceramic Synthesis. Students would become highly knowledgeable in powder preparation, green body forming (i.e. relevant wet chemistry), hot isostatic pressing, spark plasma sintering operation, etc. The list can go on and its all about tailoring the program focus towards an industry sectors needs, although to be honest, industry in some cases should be much more specific and realistic about what they are looking for in such students.

The level of training and time will depend on the complicated nature of the skills, systems, or software. I would imagine that the mean time to completion would be around 2-3 years, with some programs probably suitable to be completed in < 2 years. An example of a potentially more challenging program would be Master of Technical Skill in Quantum Computation, here students would have to master both the theory and mathematics of quantum computation/information  and learn about operating and interfacing with complicate hardware.

The two quotes for this post are:

"Education is what remains after one has forgotten what one has learned in school."
– Albert Einstein

"The more that you read, the more things you will know, the more that you learn, the more places you’ll go.” 
– Dr. Seuss

Reuse and Attribution

Discussion on Quantum States & Corresponding Electrons

Possible Conceptual Issues

My intent of for this blog post was to explore pedagogical approaches to explaining solutions (i.e.  wavefunctions) for the time-independent Schrodinger equation and how electrons are associated with those solutions. In its present form, the blog is not polished and may have erroneous statements, so keep in mind the blog just follows my train of thought. I apologize in advance for confusion or mistakes. For clarity I will try to avoid any gobbledygook (i.e., unnecessary nomenclature).

Quantum States$^\dagger$: A Train of Thought

The first thing to discuss is that we are interested in explaining the motion and interactions of our – humans that is – notion of particles that make up matter. The main confusion is that our intuition/experience tells us to think of the world as particles, however, the whole premise of quantum mechanics is that the motion and interactions cannot be described by classical point particles but rather by "wave-like"$^\mp$ states which are solutions to quantized operators/transformations. It behooves us then to concede that with our current mathematical tools (i.e. equations of quantum mechanics) we are told that the material world, at its smallest scales, is in  "wave-like" states. What does "wave-like" states even mean? It means that the mathematics used to treat the motion and interactions of what we call particles$^\ddagger$ within the theory of quantum mechanics (QM) looks similar to the mathematics used to describe the motion of waves. The difference from the classical picture of waves arises in the nature of the quantum mechanical solutions, that is, the "wave-like" states and their amplitudes exist in a complex number space.

The essence in interpreting the physical meaning of the "wave-like" state solutions, is an additional layer of struggle and falls within the philosophical interpretation of quantum physics. For example what is the meaning of the "wave-like" state if when I interact with the physical world I always observe or measure particles — this is what experiments show us. This bewildering problem still exist even though it has been nearly 100 years since the conception of QM. So why don't we know the meaning? I'm not sure. Its not to say that many great minds haven't proposed insightful ideas, but that there is no clear victor. The most widely accepted and used approach is commonly referred to as the Copenhagen interpretation, which was primarily pushed by Niels Bohr. The main premise is that it is only the magnitude of the complex numbered "wave-like" state that we can assign meaning to as a kind of probability about our [human] intuition of where particles could be. Then there are the many-worlds theory initially proposed by Hugh Everett, pilot-wave theory of David Bohm and De Broglie, and the spontaneous wavefunction collapse theory (GRW). It seems to me that the complexity of trying to explain the philosophical implications of the quantum "wave-like" state is something to admire given that it eludes some of most brilliant of minds. The general theme in physics seems to focus on QM (i.e., wavefunction) predictive description of reality not the inference of its existential meaning.

Okay, lets shift back and think about some examples where waves describe the behavior of a classical physical system. The two most simplest to start with are a vibrating violin string  and drum membrane. In both cases a medium (the string and drum membrane materials) acts as a field — a property given at a point in space, here the property is the elasticity of the material. It turns out that because the string and drum head are pinned down at the edges, only certain characteristic vibrations can exist, these are called modes or in mathematics eigenstates (eigen in German means self/characteristic). The characteristic vibrations are a state that the system (i.e., string or drum head) can exist in, but they do not necessarily always exist, it depends on how the system is excited (i.e. plucked or banged). In many instances the overall vibration of the string or drum head is a combination of characteristic vibrations, an analogy one could think of is how we combine the colors red and green to get yellow. We can combine characteristic vibrations to get a new vibration, this is commonly called linear superposition or combination. 

So now that we've discussed a little about waves, lets revisit the "wave-like" aspect of quantum mechanics. We start by stating that the total motion and interactions of a quantum system is described by a "wave-like" state or wavefunction, $$\Psi\left(\chi_1,\chi_2,\chi_3,\cdots,\chi_N\right)$$,

where $\chi_i$ is a single state function that will depend on position, time, and the type/number of particles.  When the time dependence is removed from the description of the system, that is the system is time-independent, it described by the equation:

$$\hat{H}|\Psi(\chi_i,\cdots)\rangle=E|\Psi(\chi_i,\cdots)\rangle $$

This is the Schrödinger time-independent non-relativistic wave equation which is an eigenvalue problem. Lets describe each term in the equation in  more details. The term $\hat{H}$ is called the Hamiltonian operator named after William Rowan Hamiltonian and describes the dynamics, interactions, and energy components of the system. In Newtonian mechanics we describe a system in terms of the acting forces, in Hamiltonian's representation we use momentum and energy. For a quantum system we can write a generic electronic Hamiltonian as,

\begin{align}
H &= K.E. + P.E. \\
&=\frac{-\hbar}{2m_e}\sum_{i}^{N}\nabla_{i}^2+\frac{1}{2}\sum_{i,j}^N V(\mathbf{r}_i,\mathbf{r}_j)
\end{align}.

The first term representing kinetic energy (momentum) and potential energy (field interactions). In this representation the K.E. and P.E. are said to be operators — I tend to think of these as being equivalently equal to transformations —  that are "quantized" in their representation. These operators act on the "wave-like" state, $\Psi$, of a system producing a representation of the system in terms of K.E. and P.E. At this point the representation is a linear combination of characteristic "wave-like" states that the system could be in.

But what happens if I ask the question "What energy is associated with a particle in a given state and where can that particle be in physical space?", more specifically what can I know about the particle that makes sense or has meaning to me, the human. See at this point we just know a spectrum of energies associated with the "wave-like" solution states, which doesn't have a clear comprehensible meaning to us/me yet. It is at this junction where the "reality" of outcomes of Q.M. falls in the the realm of philosophical physics, or in other words, subjected to interpretation. In quantum physics pedagogy the stance is we need to make an expectation or guess on what the outcome is to be. To do this we say we have to project/collapse/measure the "wave-like" states onto a known meaningful platform, a set of states, to get a understanding of where a particle might be. It turns out that this projection is best interpreted as information about the probability or probability of transition of  "wave-like" state; that is with respect to the state its in and the state I know. This is typically what is understood as the expectation value or measurement collapse argument (also more commonly called the Copenhagen interpretation), which is necessary for us (humans) to relate the math or experimental results of quantum mechanics to our notion of particles. To write this out we would have:

$$ <\Psi|H|\Psi> = E <\Psi|\Psi> $$

I use this equation to try and understand the measurement outcome in quantum experiments, that is, the act of observing in an experiment is akin to enforcing a projection onto a known state/solution.

So lets recap, we mentioned that quantum systems can be treated mathematically similar to wave dynamics. We said that these quantum systems have characteristic "wave-like" solutions, e.g., eigenvalues and eigenvectors. Then an equation relating the operation on or transformation of the quantum "wave-like" state resulted in the same state multiplied by the eigenenergies, but we said it only tells us about the energy spectrum and the states the quantum system could be in. Finally, we tried to understand this by projecting or collapsing the "wave-like" solutions onto known solutions (I forgot to mention these could initial states we used prior to applying any operators or transformations).

All this discussion has so far abstracted the "wave-like" states from actual matter that occupies the states. This is akin to our string or drum which we known has vibrating solutions regardless of if its actually been plucked or banged. The next step is to understand how our notion of particles or matter is associated with the "wave-like" states. In other words, how particles or matter can occupy those solutions. It turns out that electrons, which are the basic building blocks for how we (humans that is) experience chemistry and materials, have very specific rules for how they assume a given state. Electrons are indistinguishable particles, meaning they cannot be individual tracked or resolved,  but they have a unique property associated with the "wave-like" state that requires them to take on a specific intrinsic angular momentum which is called spin$^\star$. This spin characteristic gives rise to "wave-like" states such that only a single electron is allowed to acquire a give "wave-like" state of specified spin. In QM this is called the Pauli exclusion principal for fermions which was postulated by Wolfgang Pauli based on spin-statistics theorem. It wasn't till Paul Dirac's work with relativistic QM of an electron that "spin" was deduced to be a consequence of the inclusion of Einstein's special relativity, at least this is my understanding of the outcome of the Dirac Equation.

So now let me try and give analogy to re-explain this, not sure it will work out.

The Race Track (Quantum) Architect

We start our conceptual  understanding of a quantum system which describes electrons in an environment of atomic nuclei.

A quantum architect is told that they will need to design a race track. The race track has several constraints in how it can be built. The first piece of information the architect is given is where the race track will be built. This will dictate the shape, number of bends and slopes of the race track. The race track terrain is our analogy for the potential energy surface created by the combinatoric interactions among the atomic nuclei and electrons.

We then tell the quantum architect how many lanes/tracks are needed to accommodate the drivers, these are our analogy for the "wave-like" states an electron can be in. There are additional pieces of information we need to tell our architect about this race track, that is, the drivers of this race track have unique characteristics. The first is that they all look identical and drive the same car, so we cant tell them apart, therefore we never know who is really in each lane. The next unique characteristic is that each driver has a preferred direction (spin) of driving, forward or reverse, and preferred lane (spatial location). The third characteristic is that each driver dislikes all other drivers, so they try to avoid each other at all costs and never use the same lane, with one exception, that is when two drivers who drive in opposite directions use the same lane/track they both usually perform better — we will take this to mean lower energy — but no lane can ever accommodate drivers who are moving in the same direction.

So lets break down our analogy. The number of lanes corresponds to the number of "wave-like" state solutions we can have which depends on the type of system we are interested in; here the system is our race track configuration based on the number of drivers, terrain, etc. The fact that all the drivers look identical and drive the same car is a characteristic of electrons, they are indistinguishable particles. The preferred driving direction of each driver corresponds to a innate property of electrons (more broadly fermions) dubbed "spin", where its value corresponds to a positive (up) or negative (down) sign due to its behavior in magnetic fields. The fact that the drivers dislike each other but can have special configurations in the lanes corresponds to the Coulomb and exchange forces, where the latter is due to spin and the Pauli exclusion principal.

With all this information the architect is ready to build our race track, but, we through in an additional caveat, that we need him/her to build it in such a way that the number and shape of the lanes can change dynamically. At this point he/her is annoyed, but says their company is so skilled that they can dynamically adjust the state of the race-track to accommodate any number of drivers. So now drivers can be added or removed from the rack track simply by the architects ability to add or remove lanes.

Okay, so we have described the quantum state of electrons using an analogy.  I'm not sure how good this was but I will continually revisit it to improve. No quote for this blog given it is not final.

$^\dagger$ Please comment if you find any explanation of analogy incorrect. This blog post was a result in trying to explain to my wife, who is not trained in a STEM field, the physics that gives rise to the chemical and material world around us. What inspired this was my reading of books discussing self-learning approach dubbed the Feynman technique, pioneered by the famous physicist Richard Feynman. The idea is you take a topic or idea that your familiar with but not sure your level of understanding and work through the topic/idea (e.g. Quantum States) as if you were teaching it to someone else. The purpose is to help identify areas where you may lack understanding or capability (i.e. mathematics). Since I'm always trying to improve my level of knowledge in quantum mechanics, this blog post was my attempt to do so.

$^\mp$ I have chosen to use the quoted term "wave-like" instead of the more common and probably accurate term wavefunction. My reason for doing so is at this level of discussion I don't think any additional clarity is gained by using the word wavefunction, but "wave-like" in my opinion, conveys that the wavefunction will have a functional form that is reminiscent of a wave or wave packet. Anyhow, I will use the two interchangeably throughout this blog.

$^\ddagger$ My understanding is that the leading view in quantum (field) physics is that particles are a manifestation of excitations in quantum fields. In other words, it is quantum fields that exist not particles, and when we speak of particles, say an electron, we are just talking about quantized excitation in an electron field.

$^\star$ The origin of the term spin is unfortunate because it forces a physical picture of the electron which isn't necessarily true in the framework of QM. It comes from the initial view that an electron is not a structure-less point object in space, but that it has some diameter/volume  associated with it and spins about its own axis (as the earth does). To my knowledge the diameter of a electron has never been observed or measured. As mentioned in the main text, the spin is a result of special relativity which adds chirality (handedness) to the spatial part of the wavefunction, also called a spinor.

Reuse and Attribution