This may seem somewhat unorthodox, but what I would strongly recommend is to start not by reading books on physics or math but read some books on history of physics (and math) first. This will give you some intangible basic knowledge, or a sense, of what scientific research is all about, so that many things that otherwise may end up being puzzling to you when you come to learn the "hard science", won't. One recommendation I can make is Rhodes' The Making of the Atomic Bomb.
As a mathematician, I disagree, but this is probably because I already understand the math pretty well. I'd much rather have a clear picture of the most useful mathematical models for the various areas of physics (and where they fail) than slog through the centuries of failed experiments and feuds. Every time I ask a physicist for a clear picture of what light is (mathematically, what the model says it is), I get a mix of "it's a wave but also a particle" and random things that dead physicists have said about light.
I honestly don't care if Einstein thought God doesn't play dice, or if someone way back when thought an atom was like plum pudding. I do want physical intuition, but I want it through clear and well-explained mathematics.
That is, when I'm in the mindset to actually learn physics. Hearing about the history is fine and dandy if I want entertaining stories.
There is not such a thing. In case we'd been handled a manual of the physical world, it would make sense to tell people to RTFM, but there is no FM, and the whole point of Physics is that we can't just make it up.
There are two kinds of people that fight actual Physics from within, those that say Math is not important: physical intuition "has to" be enough, and those that say that it's just Math, so they are equally happy with perfectly reasonable correct models and completely absurd correct models. They churn out both, usually more of the latter. Both schools of opinionated thought miss that wherever you go you have to learn the trade and be humble.
They should have told you that we don't know what things are, we just deal with how they do their thing.
It's a wave, it's a particle... it's a state, a linear combination of the eigenstates of the EM Hamiltonian, which span a Hilbert space. That's what we got and every physicist knows it, they are just being polite. If they told you that, your next question would be: why on Earth would you think that? why not just [random brainstorming here]? I don't think you're really interested in the really long answer, but it has to do with the boring fact that Physics is, above all, an experimental science.
> There is not such a thing. In case we'd been handled a manual of the physical world...
I never asked for this, but rather for a clear overview of the models that exist, and where they fail. I understand it's experimental, and what interests me personally is (a) how can I use the existing models to do other fun things like write simulations and (b) where do the models break down so I can be more informed about the open directions in physics, which are almost always mathematical (what mathematical model describes the observations we see? how can we design an experiment that confirms or refutes a given mathematical model? etc.).
I honestly think a sort of guide or text your asking for would be interesting. Absent that, next time you find yourself in the "learning physics mood" around your physics friends it might be beneficial to restrict your question to particular regimes, or even better, just ask about very specific physical phenomena. I'm guessing the reason you get these big picture, nebulous responses is the questions you might be asking are a little too general...I'm basing off the one data point I have from your post, "what is light?"
"What is light?" garners a very nebulous answer because there is a wide range of phenomena called "light" and work done in understanding what light is. "What is light in a fiber optic cable?" is more specified because it specifies a length (and thus wavelength) scale, an energy scale (not high intensity that makes you have to worry about plasma generation), and a time scale (steady state physics which allow talking about modes (ie., fourier analysis), unless you care about transients). The length scale and time scale rule out quantum mechanics, and probably will lead you to essentially to solving Helmholtz, which will be much more your speed. See, you might not know to specify all those scales, but by asking for a specific example, your physicist friend will restrict their universe of discourse down instinctively to a model you could use.
So light may be a bad example. The same could be said if you ask, "what is gravity" or "what are magnets?". Better questions are like, "how do we understand orbits in the solar system?" or "why do magnets stick to refrigerators?"
>'Better questions are like, "how do we understand orbits in the solar system?" or "why do magnets stick to refrigerators?"'
What about how exactly do people predict how likely it is the orbit of an asteroid/comet will intersect that of the earth? That is a very interesting problem. It is not at all solved as well as it could be.
You can divide physics into roughly these core subjects: classical mechanics, classical field theory, special relativity, quantum mechanics, quantum field theory, general relativity, thermodynamics. What you are asking for is basically what is already taught in a physics curriculum, so you could read physics textbooks on the subject that you are interested in.
I believe you want to start with the Michelson-Morley experiment, and the reasons why it failed.
Modern physics, so far as anyone has explained it to me, seems to be two partial responses to the results of that experiment (and the behavior of their test device, the interferometer) that we're in the process of synthesizing in to one result.
Wikipedia reasonably outlines the main features that modern physics has to account for, and links out to the two main bodies of work.
The problem is once you get away from those broad properties the model has to satisfy, there's several competing inplementations with somewhat different features/explanatory power.
I partly agree with you, but at the same time it seems that when teaching physics (intro level) they spend a bit too much time first teaching incorrect theories and even worse trying to pass it off as correct when there clearly is a better one.
I doubt that they teach incorrect theories today. A thing to understand here is that all theories are considered correct, the only difference between them being the area which they describe the best (or more efficiently). They are all approximations and thus can be thought of as all being correct and incorrect at the same time. From the philosophical standpoint, however, you are correct - modern theories do replace the older ones; unfortunately, to apply the quantization procedure you need to be able to understand what it is that you are applying this procedure to - and that is something that is described by the old, "incorrect" theory.
I think context/scale/application is important. Classical Newtonian physics takes us pretty far pretty accurately until we get to more micro or macro scales. For example, if wanting to model a physics engine in software and the Newtonian models are considerably more performant, then could make sense to sacrifice the accuracy.
Well, it has to be said: a mathematician's outlook on physics is completely different from a physicist's. For a mathematician, physics is not much more than a bunch of more or less fixed mathematical models, each describing, approximately, some aspect of the physical world. For a physicist, on the other hand, not only are these models in a perpetual state of flux, but the physical world itself is seen as an indivisible whole, so that these "aspects" - the ones that are being modeled mathematically - are themselves just narrow views into the real physical world in which we live...
My brother was a physicist at Sandia. He said bluntly that mathematicians are not scientists. My point is not snark. It's that you may want to study physics from a physics/science POV rather than a mathematician's POV.
Also, I'd recommend Stewart over Thomas. I'd recommend Halliday Resnick and I'd recommend against Giordano.
I learned just enough vector calculus in college to know I didn't know enough to actually understand Maxwell's equations & electromagnetic fields, or most of the other fun mathematical techniques introduced in mid-level physics and engineering classes for modeling interesting "real-world" systems.
Later, when I realized what I was missing out on, I tried to teach myself the missing concepts. I failed, until I found H.M. Schey's "Div, Grad, Curl, and All That: An Informal Text on Vector Calculus." It's a pragmatic, friendly, slim little math book that reads more like lecture notes than a classic textbook, and I can fairly say it's taught me everything I know about those operators (which isn't much).
So if you are like me, and got to calc III, vector math, and/or liner algebra without learning div, grad, curl and partial differential equations... check out the book. it's great: https://www.amazon.com/Div-Grad-Curl-All-That/dp/0393925161
Math is about given the definition and understand the results through consistency. For example, given the set of Maxwell's equation and a set of boundary condition, figuring out or understanding the solution -- that is mathematics. So as a mathematician, you would like to have the definition (assumption) and equation, which provides the clear (mathematical) picture.
But physics is different. Physics is about the question whether those assumption (including many hidden assumption) can be wrong and how to interpret the mathematical equation even when the equation gives confirmed results. So Maxwell's equations is not a clear picture on what the light is. Maxwell's equation just describes one aspect of how light behave under one set of assumptions. There are other aspects of light and there are different set of maths that physicists tried to describe them. There are physics intuition on what light is, but that is subjectively dependent on which physicist you are asking. In physics, there is simply no mathematically equivalent of "clear picture" where you simply just accept what is given and go from there. In physics, it is "what is given" in question.
It depends on which subset of physics you restrict to. Often times. Math is often "bent" to match physical intuition and people drop some mathematical models over others because they match intuition and experiment better. As a result, there is no consistent narrative across the whole field and even across subfields in some cases.
Often times, there are no "failed theoretical frameworks", it's not like special relativity is "right" and newtonian mechanics is "wrong", they are right in different physical regimes. If they are down right wrong, then they wouldn't be known to us anyway.
If you stick to parts where theoretical physicists live, it's easier to find mathematical consistency and coherence, like QFT. Outside of that region, what things are and what approximations you use really come down to "what matches experiment best" at best and "what people will allow me to publish with" at worst.
> it's not like special relativity is "right" and newtonian mechanics is "wrong", they are right in different physical regimes
Hence I said, "where they fail" not "whether they fail." Is there even a single mathematical model of the physical world that does not fail in some way?
I was specifically referring to the part of that sentence regarding "centuries of failed experiments and feuds." Yes, you don't need to go through all the models, but many of what people with a lay knowledge of physics call failed models aren't really failed, they just apply in regimes that they could probe at the time in terms of theory and experiment. You might not have been referring to that.
And yes, there is no single mathematical model of the physical world that I am aware of. None.
One last thing that I should say that I probably didn't say clearly enough: from physics comes the mathematical models, not the other way around. Relativity didn't come out of newtonian mechanics, it came out of a change of the underlying physical model which then yield relativistic models that reduce to newtonian mechanics.
As someone who as a student was good at math and bad at physics, I'd rather disagree. Take for example the gravity formula F_{A->B} = G * M_A * M_B/d(A,B)^3. It's very easy to interpret that, but where does that come from? I can see why the gravity gets lower when the distance between A and B is higher, but why does it depend of the inverse of the cube of the distance? The formula is very easy to read, but since I didn't know where it came from, I would forget easily.
On the other hand, it would have been intellectually pleasing(at least for me) to know how Physicists came with it, what's the motivation behind 1/d(A,B)^3, how did people come up with G, how it was measured, etc.
You find that by studying physics, not history of physics.
> it would have been intellectually pleasing(at least for me) to know how Physicists came with it, what's the motivation behind 1/d(A,B)^3, how did people come up with G, how it was measured, etc.
Yes, understanding a specific change in the state of physics at time T is most certainly helped by understanding it at time T-1. But make no mistake here, this is still physics and not history of physics (whatever that means).
Thanks for correcting me. The fact that I remembered the formula wrong proves that I didn't understand that formula well enough to remember it. It was been about 13 years since I didn't do any physics, and I forgot most of it, however, I remember well the math I learnt because I understood them much better.
it's r^2 in the denominator. one idea is to think of the geometry: the surface area of a sphere is proportional to r^2. therefore if you think of a constant "influence" at a given distance, you might divide it by the surface area (to get r^2 in the denominator)
Yes, it's trivial to deduce Newton's law of universal gravitation from the divergence theorem. But to deduce that the divergence theorem applies, you need the lagrangian and the least action principle.
Conceptually the least action principle is the fundamental concept in classical mechanics, everything is derived from it, plus symmetry.
Historically this has all been done backwards. Once you have Newton's law of universal gravitation, it's trivial to prove that the gravitational flux must be constant for every enclosing surface. It's easy, in terms of mathematics required, to come up with a least action principle from that (albeit not as easy as the inverse deduction). However, if you think of Newton's laws as fundamental, it is not a very natural thing to do. Why would you do it? Lagrange was a very deep thinker to see why the least action principle is the truly fundamental thing. He was doing this way before Noether's theorem. He was doing this before the concept of energy was formalized! They knew that a quantity with what we now know as units of energy was conserved, and they knew momentum was conserved, but they didn't know what these things were, especially as no other form of energy except kinetic and potential energy was known back then. Truly a visionary thinker.
Once you understand the maths clearly, it becomes much easier to understand the failures of the past. For example, reading Newton and Leibniz on calculus makes it clear that it was epsilon/delta waiting to get out.
I guess my point is that, since most people find the math hard, writers put history in as what appears to be filler, which I find as mostly an obstacle.
Ha, this immediately calls to mind Scott Aaronson's approach to teaching Quantum Mechanics in one of his lectures:
> There are two ways to teach quantum mechanics. The first way -- which for most physicists today is still the only way -- follows the historical order in which the ideas were discovered. So, you start with classical mechanics and electrodynamics, solving lots of grueling differential equations at every step. Then you learn about the "blackbody paradox" and various strange experimental results, and the great crisis these things posed for physics. Next you learn a complicated patchwork of ideas that physicists invented between 1900 and 1926 to try to make the crisis go away. Then, if you're lucky, after years of study you finally get around to the central conceptual point: that nature is described not by probabilities (which are always nonnegative), but by numbers called amplitudes that can be positive, negative, or even complex.
> Today, in the quantum information age, the fact that all the physicists had to learn quantum this way seems increasingly humorous. For example, I've had experts in quantum field theory -- people who've spent years calculating path integrals of mind-boggling complexity -- ask me to explain the Bell inequality to them. That's like Andrew Wiles asking me to explain the Pythagorean Theorem.
> As a direct result of this "QWERTY" approach to explaining quantum mechanics - which you can see reflected in almost every popular book and article, down to the present -- the subject acquired an undeserved reputation for being hard. Educated people memorized the slogans -- "light is both a wave and a particle," "the cat is neither dead nor alive until you look," "you can ask about the position or the momentum, but not both," "one particle instantly learns the spin of the other through spooky action-at-a-distance," etc. -- and also learned that they shouldn't even try to understand such things without years of painstaking work.
> The second way to teach quantum mechanics leaves a blow-by-blow account of its discovery to the historians, and instead starts directly from the conceptual core -- namely, a certain generalization of probability theory to allow minus signs. Once you know what the theory is actually about, you can then sprinkle in physics to taste, and calculate the spectrum of whatever atom you want. This second approach is the one I'll be following here.
Here's the full lecture.[0] The approach was interesting enough that I bought his full book[1], but unfortunately it was a little over my head.
I think you can start either at a complex simulation paper and read up on the theory backwards from there (if you wish to simulate something) or then from basics and move up from there.
Physics is like exploring properties of a set where you are given a relation between individual elements of the set. Physical intuition us the knowledge of general relations that can be expected to hold for all elements.
If you don't want the physical intuition, then you don't want the physics at all; You should stop claiming to want to learn physics and never stray further from the one true path than the applied mathematics building.
And as a physicist, I disagree with you. Not specifically about your view of history (which I still think is important), but the way you think about Nature in general.
> The physicist needs a facility in looking at problems from several points of view. The exact analysis of real physical problems is usually quite complicated, and any particular physical situation may be too complicated to analyze directly by solving the differential equation. But one can still get a very good idea of the behavior of a system if one has some feel for the character of the solution in different circumstances. Ideas such as the field lines, capacitance, resistance, and inductance are, for such purposes, very useful. So we will spend much of our time analyzing them. In this way we will get a feel as to what should happen in different electromagnetic situations. On the other hand, none of the heuristic models, such as field lines, is really adequate and accurate for all situations. There is only one precise way of presenting the laws, and that is by means of differential equations. They have the advantage of being fundamental and, so far as we know, precise. If you have learned the differential equations you can always go back to them. There is nothing to unlearn.
> It will take you some time to understand what should happen in different circumstances. You will have to solve the equations. Each time you solve the equations, you will learn something about the character of the solutions. To keep these solutions in mind, it will be useful also to study their meaning in terms of field lines and of other concepts. This is the way you will really “understand” the equations. That is the difference between mathematics and physics. Mathematicians, or people who have very mathematical minds, are often led astray when “studying” physics because they lose sight of the physics. They say: “Look, these differential equations—the Maxwell equations—are all there is to electrodynamics; it is admitted by the physicists that there is nothing which is not contained in the equations. The equations are complicated, but after all they are only mathematical equations and if I understand them mathematically inside out, I will understand the physics inside out.” Only it doesn’t work that way. Mathematicians who study physics with that point of view—and there have been many of them—usually make little contribution to physics and, in fact, little to mathematics. They fail because the actual physical situations in the real world are so complicated that it is necessary to have a much broader understanding of the equations.
> What it means really to understand an equation—that is, in more than a strictly mathematical sense—was described by Dirac. He said: “I understand what an equation means if I have a way of figuring out the characteristics of its solution without actually solving it.” So if we have a way of knowing what should happen in given circumstances without actually solving the equations, then we “understand” the equations, as applied to these circumstances. A physical understanding is a completely unmathematical, imprecise, and inexact thing, but absolutely necessary for a physicist.
From personal experience, there are such mathematics-oriented people in physics too, not just mathematicans. And some of them push a lot of papers, typically by applying the same method over and over again in different subfields, resulting in infinitesimal incremental "advances".
Related Suggestion: Start with experiments, then read theories. This is very true for QM (or any other modern science). Start with older texts. Read the thought process of past giants.
Avoid all "textbooks". Especially the one with >1ed/decade. These will only teach you how to pass in exams and useless "tricks". These will result in worse understanding of nature. Remember go to the source.
lol. "Avoiding textbooks" is probably the worse advice imaginable. Now, I'm not saying go out and buy all of Pearson's catalog, but textbooks are the only books designed from the ground up to impart knowledge to a reader that meets the minimum standards described in the first few pages. Plus, the good ones describe the experiments used to develop whatever theory they're trying to impart to the reader.
I will agree there are lots of bad textbooks, but the best-in-class textbooks are better than any self-learning guide you could make up yourself (within categories).
As a physicist I completely agree. The history is the best part. And it's fun seeing how much scientists had a love/hate relationship with each other. But you also learn why they invented/discovered what they did. Which is important in advanced physics where sometimes the reason or purpose gets lost.
> but read some books on history of physics (and math) first.
That's great advice. Do you have any more recommendations?
As a working scientist myself, I find that a historical context really helps. This is especially true when a lot of quantum concepts are taught from a quasi-classical approach first, which arguably makes things harder for everyone in the long run, but feels more natural when you consider the historical thinking that led there (electron spin, I'm looking at you).
Edit: Some recommendations of my own... A math teacher in school very kindly leant me a biography of James Clerk Maxwell (might have been 'The Man Who Changed Everything' but I'm not sure) and Marcus du Sautoy's 'Music of the Primes.' They were both great introductions, and definitely changed my thinking.
I recommend Connections or The Day the Universe Changed (out of print, I think?), both by James Burke. These aren't comprehensive histories of science, but they show how science, technology, and society are intimately connected. They are incredibly well written with lots of carefully selected images. In addition, there accompanying videos were also made (with very high production value) which are also a great deal of fun to watch.
In addition to Connections and tDtUC, I also recommend this[1] interview, in which Burke discusses current problems related to privacy and technology, the balkanization of traditional social structures, speculation about using technology to replace important decision-making with a continuous Delphi-technique-style referendum, and social responsibility in a post-scarcity future.
Another kind of source of inspiration and a subtle form of knowledge that may be hard to gain from elsewhere is older physics books, for example Richtmyer and Kennard's Introduction to Modern Physics (4th ed., 1947). Often written in an impeccable style, these books contain information that is now considered of historical value only and yet may be invaluable in providing the exactly those - historical - details that are omitted or glossed over in more modern texts.
"The End of Physics" is a good one. You may also re-think your desire to learn physics. If you get through the book and still want to learn, then you will do it completely without illusions.
I'm curious whether any physicists here have read this book and found its claims justified. I'm just a layman who enjoys reading about physics from time to time, but it seems to me that there are actually a large number of unsolved problems in physics that are experimentally accessible to us right now, and there's plenty of space for new theories to yield testable predictions. People were making claims similar to the ones in this book 100 years ago; they were unjustified then, and I expect them to turn out to be unjustified now.
I'd suggest that in addition you should read at least some parts of modern translations of early scientific works starting with Aristotle. I don't mean read everything of course but try to get a sense of perspective, not everything was invented in the last hundred years.
Also take a look at Carlo Rovelli's monograph: "Aristotle's Physics: a Physicist's Look" at https://arxiv.org/abs/1312.4057. The kind of thinking that Rovelli describes is essential to success in science
That book is amazing, speaking as a non-physicist. (Fine, I did study maths, so I suppose I'm almost qualified as a theoretical physicist, except I'm not good enough at maths ;)
Really recommend it to anybody who's interested in that part of history. Amazing writing, and an incredible story. (You know it's gripping when you read about some of the experiments and feverishly hope they don't blow themselves up, nevermind knowing the outcome)
I have to agree with this. My physics education was sort of being dumped into late 20th century physics and could not understood how we got to quantum and relativity. It wasn't until recently that I went back and kind of walked through the history of physics from the natural philosophy days to the cavendish in the late 1800s that modern physics started to make sense.
Generally speaking, I find that the more expensive a rare academic tome is, the more likely there are legal/etc. versions online (PDFs, HTML, DjVu, etc.), indexed by Google. That seems to be much more true for STEM texts than those of the humanities, though.
Highly agree. I found that learning in sync with looking up its history, on why something was invented or why something is done in a particular way, helped we understand it better as I got more interested in the thing am studying. It may even help find a way to do it better. It gives various perspectives involved in decision making and scientific process. That kind of learning helped me to reach to solutions faster.
This is the right context for Newton, Einstein and quantum physics, it's written by an astrophysicist and is up to date, with beautiful illustrations and easy to read.
I would definitely agree, partially. Intuitively classical physics makes sense but it was definitely helpful to me to understand the historical perspective for quantum mechanics.
Or at least is where I think the undergrad Griffiths QM book gets it wrong. If I remember correct it just starts by basically stating the Schrodinger equation and going from there. One of the most helpful lectures I had in undergrad QM was on matrix mechanics formulation. It really helped to solidify my understanding.
I don't know if this makes sense for someone looking for graduate-level knowledge, but it I'm sure it would make school a lot more interesting for children. It certainly helps to understand things if you know what people's motivations were and what tools they had when they made discoveries.
I think this is right. But the good news is that many childrens books on science are fairly historical -- at least the ones I had as a kid. One had, or example an illustrated page with Gallileo dropping weights from the leaning tower, and then another one with Newton and his apple.
The good lower-division textbooks will intersperse some history with the concepts. Even the Modern Physics textbook that we used had historical information in it.
For sure, it's relevant, but you'll get a helping of it in every decent textbook.
I can confirm. I just sniped a book about "pressure", the historical pathway with steps and non famous scientists gave a completely different feel than any physics textbooks with models and equations.
> The Theoretical Minimum is a series of Stanford Continuing Studies courses taught by world renowned physicist Leonard Susskind. These courses collectively teach everything required to gain a basic understanding of each area of modern physics including all of the fundamental mathematics.
That's a great list. The only thing that's missing would be a linear algebra course. The OP mentions it in passing, but a good understanding of LA goes a long way. I did UGRAD in engineering, and when I switched to physics everything was over my head, but my knowledge of LA still managed to keep me afloat. Also, matrix quantum mechanics is essentially straight up linear algebra (vectors, unitaries, projections, etc.)
Now switching to a shameless plug mode, I'll mention my math+mech+calc book, which would be a good addition to the section 1. Introduction to Mechanics. Chapter2 of the book (on topic) is part of the preview: https://minireference.com/static/excerpts/noBSguide_v5_previ...
Thanks for plugging your book. Math has always been a struggle for me and considering I got my ass kicked through undergrad barely passing my math courses, I constantly I'd enjoy math as a programmer if I had just had the proper schooling in highschool.
I was always amazed that more physics curricula do not require linear algebra. You don't even need to get to quantum for LA to be amazingly helpful. Our classical mechanics text from undergrad was swarming with LA...
I was amazed that my EE degree didn't required it. We had a rushed treatment as part of a required ODEs course, but I didn't really get it until I took a theory heavy course dedicated to the subject. It and probability both made my life a lot easier and are essential parts of my daily work.
I'll second that Lay was the textbook I used throughout my Engineering Degree. It is sitting on my shelf at work right between "The C++ Programming Language" and "Numerical Recipes"
I've always loved physics but I don't think it loves me back.
That is, I've always found it fascinating since high school but once you need calculus to understand some of the more advanced stuff I feel that I get lost in the math (which, admittedly, I suck at) and lose the intuition for what's really going on. Then it just becomes a giant math problem that prevents me from seeing the bigger picture.
It's just this problem I've had that I always sweat the small things and sometimes miss the bigger picture or the main concept when I get frustrated that I can't understand the details.
There is a point early in your education as a physicist (Quantum Mechanics) where it becomes impossible to have an intuitive understanding. For Quantum Mechanics you can just do the calculation, there is no way you can get an intuitive understanding, other than by becoming comfortable with the mathematics.
I used the feel the same as you, but then I gave up. One day I was like "screw it, I'm just going to take it for granted". I stopped caring about getting a feel for why things happen, just that they do and I know how to calculate them. When that changed I was suddenly free, I didn't have to worry about why things made sense or not anymore.
You still should understand what the problem is though. Physics isn't just about understanding the mathematics, it's also about understanding the physical arguments that goes along with it. Like how can we come up with the problem to solve in the first place?
One problem I have often had is that in a long derivation my brain will be so fried on the mathematics that I eventually forget what the terms in the equations actually represent. I'm like "what is q again? oh yeah it's a generalised coordinate".
It's cliche but the important part is not giving up. My favourite lecturer, who is a theorist, says that the main issue that students have is fluency. They can do the mathematics, but they aren't fluent at it. They aren't quick, they're slow, it takes them time to work it out, etc. That is what makes you forget, but eventually after seeing the mathematics so many times it will becomes ingrained in your brain, and it will just feel obvious, and you don't have to think about it. Eventually you will become fluent in the mathematics and it will disintegrate as a boundary, and all you'll have to think about is the physics. It's just practise.
> I used the feel the same as you, but then I gave up. One day I was like "screw it, I'm just going to take it for granted". I stopped caring about getting a feel for why things happen, just that they do and I know how to calculate them. When that changed I was suddenly free, I didn't have to worry about why things made sense or not anymore.
A very important development in my understanding of mathematics was developing an intuition of when something is worth visualizing. Sometimes visualization is extremely helpful. Sometimes it just makes understanding the problem more difficult (looking at you quaternions).
> One problem I have often had is that in a long derivation my brain will be so fried on the mathematics that I eventually forget what the terms in the equations actually represent. I'm like "what is q again? oh yeah it's a generalised coordinate".
I had a professor who said something along the lines of "a good notation liberates the mind while a poor one clutters it." I suspect he was quoting a famous mathematician (as he was wont to do), but I cannot remember who (Whitehead?).
Thanks for the last paragraph. I'm currently studying convex optimization and feeling pretty dumb despite having done well in some reasonably challenging math classes before. Gotta remember that it will get easier as these new topics trickle into the realm of intuition.
This is something that everyone who studies physics has to work hard to get past.
Understanding the math as a description of the physics rather than just a bunch of symbols takes time and hard work.
Someplace to start: find a differential equation you are trying to understand and walk through what it means. Look at each term and try to suss out the physics that is going on by thinking about the differentials as descriptive. It takes time, but it is very powerful (even when thinking about non-physics related equations)
You're not alone. I was lucky enough to go to undergrad with Nima Arkani-Hamed [1]. I later had lunch with him while he was at Harvard, and he mentioned that a lot of what he did in the intervening years was to get away from the very formal mathematical, proof-based thinking of our math courses back then.
Physics is made much harder than it needs to be by the fact that physics pedagogy is generally terrible. Physics texts start by just throwing equations at you, telling you "This is how it is" with no background or foundation about how we know that this is the way it is, or what it means that this is the way it is. There are some very good popularizations out there (like David Mermin's "Boojums all the way through") but very little that bridges the gap between these and "real" physics books. One of the things on my to-do list is to write a book to try to fill this void, at least for quantum mechanics.
I would on the whole agree that physics pedagogy is generally terrible. But there are a handful of textbooks that do an excellent job on explaining the WHY before the math. Griffith, as mentioned 3 times in the article is famous for his excellent books.
I'm not sure if you have read the Griffith's textbook on Quantum, but I would agree it does a reasonable job of introducing the topics before going too math heavy. The first 2 chapters are devoted to introducing concepts before the "boojums" of chapter 3.
But I wholly disagree with your assertion that QM should be taught _how we know_ before what it means. QM tends to need to get across 3 things to the introductory student, broadly, it's what the tools are (e.g. Schroedinger's, uncertainty principle), how they depart from the classical understanding, and what the mathematical foundations are (e.g. commutators and linear algebra). I think that just teaching the tools, then the math, then the departure is by far the best means of teaching QM. It's just too weird to contrast to classical. Contrasting to classical at all would lend the student to an understanding of QM in terms of classical, that is absolutely the wrong mindset to be.
> But I wholly disagree with your assertion that QM should be taught _how we know_ before what it means.
I didn't assert that, so you can't disagree with it :-) I think the how-we-know and the what-it-means should both precede the math, but I don't have a strong opinion on which of those should come first.
I have not read Griffiths, but I took a quick look at:
and I was not impressed. It seems like a completely traditional presentation, and like all traditional presentations it completely misses the absolutely central role that entanglement plays in the conceptual foundations of QM. (In fact, the word "entanglement" does not even appear in the table of contents! Alas, the on-line text I found at the above link is not searchable so I can't tell you if he doesn't address it at all.)
[EDIT] I've now read more of Griffiths and I would like to revise and extend my above remarks :-) My original criticism still stands, but aside from that the book is actually quite good.
I've been thinking about reading The Feynman Lectures on Physics recently, but I always thought they were essentially textbooks; I was surprised to see them described as 'popular' here. I remember reading something about their origin, that some universities tried adopting them with the result being that students found them too difficult (and many professors considered the material to be a sort of fresh take on classical subjects).
Check out some of the first chapters. You expect to see some kind of Newtonian prelude with force, mass, acceleration, motion of projectiles, etc. It's not really there.
The FLP are really rewarding to browse around in -- I would hate to have to read it from cover to cover -- but they are not good as a textbook. It's not a systematic treatment. It's an idiosyncratic look at how one person enters a problem space and walks through it.
Friends who took classes at Caltech (in the 1980s, I think) using FLP found it to be a bad fit for people who did not have an immediate grasp of problem essentials. In other words, if you did not already have some mastery of the basics, it might not help. I can't imagine the confusion possible when the class was first taught in the 1960s, and there was only the verbal lecture, with no companion text.
I share your surprise. The Feynman Lectures on Physics - which by the way are available online at http://feynmanlectures.caltech.edu - are clearly a popular set of books as in "liked or admired by many people or by a particular person or group," but hardly popular as in "intended for or suited to the taste, understanding, or means of the general public rather than specialists or intellectuals."
How on earth can they be regarded as too difficult? The Feynman lectures were set books at Exeter Uni. for my applied physics degree '74-'77, I don't remember anyone complaining that they were too difficult, challenging of course.
The title is 'So you want to learn physics'. If you aren't trying anything challenging you aren't really learning physics, you are just getting a few of the highlights.
No. The site clearly places those lectures under the category "popular".
Also, "do you want to learn physics" might as well refer to high school level physics.
I know plenty of + 30 who don't have a "high school" level understanding of physics (despite their high school diploma) who might see that headline and think "hmm, maybe I ought too" and they would be barking up the wrong tree.
The level of math and physics required to follow a first semester college course in physics is beyond most highschoolers.
I can't recommend the book, 'Prime Obsession' by John Derbyshire enough. 'Gravity' by Hartle is invaluable and quite accessible. If you have a strong background in calculus, you can also check out 'Gravitation' by Misner, Thorne, and Wheeler.
I think this is a great list, and I'm so glad Susan took the time to lay all of this out!
I have to add these two books to the list. I was surprised to see they didn't make it, even though she nailed some of the other "bibles". The following were my favorite books as an undergraduate physics major:
- Introduction to Mechanics by Kleppner and Kolenkow
- Electricity and Magnetism by Edward Purcell
No true physics education would be complete without reading and going through the problems in those books. I knew physics was my passion before, but these books helped me fall in love with physics even more.
Learning physics requires more than simply reading text books. A significant portion of actually understanding the concepts laid out in the book is performing demonstrations and experiments in the lab. In college, we had a 3 hour lab each week to go with 3 1-hour classes and each was critical to learning. I certainly admire anyone who wants to learn physics on their own, especially without already having a strong mathematical education, but to really grasp the meaning of the words in a book requires practical exposure in a lab.
Depends what you are trying to achieve. If you are trying to gain an appreciation for it (as a layperson) then I think you can forgo many of the complicated experiments. On the other hand, if you are trying to prime yourself for a career in it then you definitely must perform the experiments.
I don't disagree that a lab isn't necessary if all you want is a cursory overview of the various topics but I'm assuming that the person wanting to learn physics wants a deeper understanding of the concepts. Really know what those equations mean. That takes practical experiences in my opinion.
Wondering if anyone here has run into the same issue as me: I've wanted to learn Physics properly (post college) but can't find time to balance it with a job/having a social life. I've had a few false starts where I try for a week or two and give up because the whole enterprise feels insurmountable. Anyone else experience this/figure out a good strategy?
There is a classical mechanics missing in the grad school section. One of the primary books used is by Goldstein.
I also recommend Classical Dynamics of Particles and Systems by Marion and Thornton.
As was mentioned in another post Linear Algebra is a must, and I think David Lay's book is a great one to start with.
As the author mentions, to learn physics you MUST DO PROBLEMS.
On another note I can't seem to find anyone that has mnemonic techniques for learning equations. So if anyone comes across a good method I'd like to hear it. And I'm not just talking about something like "low d high minus high d low, square the bottom and away we go". But to more complex equations, like memorize "memorize Einstein's field equation." A method that could potentially work for any arbitrary equation.
I think the only way to consistently memorize an equation is to deeply understand it. Not just how to apply it, but essentially how to derive it. What motivates it. Why it must take the form that it does.
With fundamental physical equations you have to be a little hand-wavy, because they are, well, fundamental, so they can't be derived from anything else. But you can motivate the form of the Schrodinger equation from the classical wave equation. (I should note that Feynman disagrees with this claim, but you can find a nice motivation for the form in Penrose's Road to Reality.) Regardless, going through these motions will serve to embed the form of the equation in your mind.
It also helps to write the form of the equation in a number of different ways. Some may be easier to remember than others. Maxwell's equations, for instance, are pretty easy to remember if you write them in terms of the EM stress-energy tensor.
This is always how I have memorized them, really through repetition (back to doing problems). The thing is that except for the basic ones and ones I use excessively, I forget; I think this is common. In comparison I can remember mnemonic lists of random items created years ago. I can't help but think there is a way to store math information similarly. Because that would suggest that it is a problem with the storage of the information.
I've always looked for more effective ways to study. And to counter your suggestion you can definitely fully understand the principles of an equation without being able to remember the equation itself. A simple example of this would be the Laplacian in spherical coordinates. It is easy to understand what is being done and the Cartesian form is trivial (most people have this memorized even) and the spherical can be derived from it simply. Problem is this takes way too much time. I don't think an example like this you could argue that there isn't an understanding of what is going on, just a familiarity issue. And thus how the information was stored. I'd argue that most of those that know this off the top of their head do so from repetition and not because of a better understanding than their Cartesian only counterparts.
Often if you understand the equations you can "half re-derive" them from the given information. That is, you can start to work them out from first principles to jog your memory, then write the rest from memory.
Other times, e.g. in probability theory, there are just a lot of equations, so even if you do a lot of practice problems, you will not have memorised all the equations. In these cases, usually a formula sheet is permitted on the final exam. Otherwise, you have to study math the same way you do biology - summarising and constant revision.
I'm out of school. But I can't help but think there is a better way. I always get the answer "When you understand it enough you'll have it memorized." I hate this answer because it is easily provable as false. Like my example in the other comment, the Laplacian is an easy concept, but remember it in spherical coordinates isn't trivial like it is in Cartesian. If I can memorize a list of 20 random items for years with minimal repetition I don't see why I can't do this with equations. It isn't about memorizing everything, it is about memorizing what is useful so I don't have to look it up on google every few weeks when I need it.
As for testing, when I was in school most classes didn't allow me to have a formula sheet. And anyone that does the physics GRE knows how much has to be remembered. But to me it is more about quick and easy access. Motivation isn't about some test.
These lists basically exist, though not conveniently in one place.
Just look at the required and elective courses at a few decent undergraduate programs for your field of interest, form a DAG from the list of prerequisites (which is sometimes conveniently shown as a flowchart by the program), and track down the required textbooks.
If your field of interest exists as a well-studied undergraduate major, then this list can be compiled in an hour or two.
For GR, I really liked Bernard Schutz's "A First Course in General Relativity" -- I read it from cover to cover.
Extremely lucid explanations of some very complex topics, and reading it for the first time blew my mind.
This book "teaches" you well (compared to other books where I feel like I really am putting in a ton of mental effort just to learn what the book is trying to say, much like reading mathematics articles on Wikipedia), and it still manages to move fast.
I understand the appeal of self learning physics for the sake of knowledge, but is there a market for self taught physicists in the same way that there is for self taught web devs?
I feel like you'd have to go through the university system. If not, what would that pathway look like?
> If you work through the all of the textbooks in the Undergraduate Physics list of this post, and master each of the topics, you'll have gained the knowledge equivalent of a Bachelor's Degree in Physics (and will be able to score well on the Physics GRE).
I am not so confident: the physics GRE is a notoriously difficult test, and is a significant barrier to acceptance to any Ph.d. program.
Each fall I teach a mini-course for our physics majors who want to take the Physics GRE. The trouble is that you can't do well on the test if you just go in and try to solve as many questions as you can: it's 100 questions in 170 minutes, and many (most?) of the questions look like full fledged homework questions (that would take even a pretty good student 5-10 minutes to solve, or sometimes much longer). Instead, you need to learn all of the common tricks and shortcuts that the test writers expect you to know and use (including things like checking the units on all of the multiple choice answers to eliminate some choices, or testing that the choices have the right limits like time->0 or mass->infinity).
It's a pretty obnoxious test, to be honest. Once you've learned the types of strategies to use, the GRE is testing something meaningful about physics knowledge and intuition, but I don't know how well that something correlates with either "successful in classes" or "successful in research".
