Merely a small typo, but just to preemptively note it, you of course meant to write that bolded word as “momentum”.
Er, sorry, didn’t mean to say “um” there. 
And you’re right, that is what I meant to say.
Think of it this way. Momentum is equal to frequency times a constant. So to know the momentum exactly you’d have to have an infinitely high infinitely thin spike at a specific frequency (which is called a Dirac delta function).
But if you have a single frequency then the wave function for finding the position probability is a pure sine wave, which means that the particle could be anywhere at all.
And vice versa.
No, it’s about half right, and that’s exactly what I mean when I write that you need to have an understanding of the concept before you, in your own words, “go around telling people this now”. Chronos spoke of the precise position of a wave pulse (or wavelet), which is a single, non-periodic impulse. This is not the same thing as a “wave packet”, which is a collection or envelope of waves all traveling together, possibly with some time-varying characteristics (modulation). Although a wave packet doesn’t generally have a single wavelength (unless it is uniform, and no real wave packet will be completely uniform) it will have a characteristic distribution of waves (spectrum) which are readily definable. The wave packet will have an approximate interval over which it occurs but no specific position, and depending upon the spectrum and sensitivity of any observing medium that definition may be more or less precise, and cannot be more precise than a minimum threshold.
An explanation that “looks pretty good,” in physics is often just as false as one that is completely wrong, especially if one doesn’t understand explicitly what is meant by the terminology and how it relates to the concept at hand. This is especially true when applying analogy to describe a situation; it is often tempting to further reason from that analogy even though this will generally bite you in the ass, especially with quantum mechanics where no aspect of it can really be discovered by intuition.
I recall from previous posts that you’re a smart guy, but are trained in the discipline of philosophy and tend to reason from what appear to be first principles in a peripatetic fashion. This works to a greater or lesser extent in other fields, but doesn’t provide a reliable basis for understanding in non-classical physics.
The mention wasn’t any kind of attempt to project anything onto you; it was only an extension of the explanation and endorsement for this book, which unlike many books on quantum theory for the layman doesn’t engage in any pseudoscientific babble about the mysteries of entanglement and observer interactions and in fact presents these phenomena is a cogent and rationale fashion.
Stranger
Oh, come off it, Stranger, that’s a minor terminological quibble and you (ought to) know it. Frylock clearly understood what Chronos was saying, and just happened to use the phrase “wave packet” for what you would call “wave pulse”. Do you really think Frylock was laboring under some conceptual misunderstanding there?
I’m no physicist, but I am familiar with the uncertainty principle via harmonic analysis (this being precisely what Chronos was referring to), and Frylock’s explanation seemed perfectly satisfiable to me, particularly at the level of abstraction he phrased it at.
“Shut up and calculate!”
Sorry, I didn’t mean to be rude. That’s just what legions of physics instructors have directed students to do when they ask these kinds of questions. It is imperative to understand that virtually none of the parameters in quantum theory have any kind of physically-realizable analogues in the macro world. It may seem really wacky that people have come up with a working theory that is so mathematically rigorous but had no qualities that are directly measurable, but then you need to understand the history and development of quantum theory in order to appreciate why this is so.
Basically, some scientists (both physicists and chemists) started trying to explain radiative behavior in atoms like the photoelectric effect and some seemingly discrete behavior in what should have been continuous statistical distributions. From thence came progressively more useful models of the atom, which behaved in a clockwork fashion, and then things occurring within the atom or between atoms. As the theory progressed, it became clear that while the phenomena were discrete, the interactions had a stochastic basis; that is, the particles had discrete parameters like energy levels, quantum states, et cetera, but the interactions between them (including any kind of observation) occurred in a fashion only predictable by statistically distributions rather than explicit calculations, although it followed those distributions very precisely. From there, a theory pretty much just fell out of attempts to give the model a consistent set of rules, however arbitrary they may have seemed.
Since then, quantum theory, and in particular quantum field theories (which introduce mechanics for the interactions between different particles) have been organized into the Standard Model of particle physics, which is an attempt to tie two gauge fields (special unitary groups that describe how different facets of the particles interact with each other based upon a set of fundamental rules) together to obtain a single, consistent set of behaviors that can then be used to predict the measurable qualities (mass, electric charge, decay period) of known and undiscovered particles. Experimental particle physics has tried to produce and measure those particles in order to give specific values for unknown parameters in the theory.
So, if quantum theory appears to be all math and no “physics” in the Keplerian sense of observing planetary motion and then coming up with consistent empirical equations that later show a theoretical basis, well, that’s pretty much true. It is all math, and most of it is completely unverifiable except that it is internally consistent and allows us to make predictions that are very accurate about the few qualities we can measure, like the strength of the electric charge or the Lamb shift. That’s just the way it is.
Stranger
Except it isn’t a “minor terminological quibble”; calling it a “wave packet” and not understanding why that is different from a wave pulse is a significant conceptual error. That is the problem with repeating explanations without really understanding the phenomena and explicit meaning of the terminology behind them; you make what seem to be simple mistakes that are actually gross errors, which are then repeated ad nauseam. This is why we have high school physics teachers informing their students that “centrifugal force doesn’t exist,” (despite what you obviously experience on the Tilt-A-Whirl); they hear “fictitious force” and don’t understand that it means “inertial force” rather than “not a real force,” giving no end of headaches to students continuing on in their studies of physics.
Stranger
I’m sorry, I think maybe I made the nature of my confusion unclear. The shut up and calculate philosophy is perfectly fine; it’s not that I want an interpretation of “What does it all really mean, really?”. I don’t care about that; tell me the laws, and I’ll be happy.
Except I don’t actually know what the laws are! I only know them halfway. I know the mathematical framework in which they are formulated, but not how the mathematical structures are to be lined up with empirical claims. So I could calculate fine, but I have no idea how to set up the calculations or what the results tell me…
That is, I want an interpretation of “Ok, I want to get to calculating to solve some physical problem now. Uh, wait, what do I calculate? I know I’m supposed to start with some Hilbert space somehow…”. Does that make sense? It’s as if someone opaquely told me “To understand the behavior of electrical circuits, remember that I = V/R” and I could calculate I from V and R, and V from I and R, and all the rest of it to my heart’s content, but I had no idea what those letters stood for and how to turn these calculations into predictions about the results of experiments involving bits of copper wire and batteries and what have you.
An introductory QM textbook. Alastair Rae’s Quantum Mechanics is pretty popular over here in the UK and it does devote a chapter or so justifying the postulates of QM.
Try asking your question here: http://www.physicsforums.com/
I think a state space in quantum mechanics is just an abstract way of keeping track of all the information in that system. So basically when the postulates of QM say that the state of a system is completely described by it’s state space, that is the definition of a quantum state space. A Hilbert space is choosen for the state space and Hermitian operators for observables due to their desirable properties (e.g. if the possible values of a physical measurement are goign to be represented by the eigenvalues of an operator, you better makes sure those eigenvalues are real, whence Hermitian operators in QM)
One illustartion of how concepts like state spaces and so on are abstract rather than having ontological value in their own right is that quantum mechanics (the Schrodinger picture) and matrix mechanics (the heisenberg picture) are physically the same theory. In the Schordinger picture the information about the time evolution of the system is encoded in the state and it’s the state that evolves in time. In the Hesienberg picture the time evoltuion information is encoded in the linear operators representing observables and it’s these that evolve in time.
Holy crap! How did I miss that site? It looks like exactly what I need for help with my self-study. Thanks a million.
“Wave packet” is a phrase I got from the Wikipedia article on the uncertainty principle. I guess they got that wrong then, which happens. But I understood the phrase to be able to designate what you’re calling a “wave pulse.” In fact from the wiki on wave packets, as far as I can tell, a wave pulse just is a special case of a wave packet? But again, it’s wikipedia so that could easily be wrong.
I can’t find anything explaining what a “wave pulse” is.
As for the idea that philosophy involves a peripatetic process of reasoning from first princples, almost no philosopher would endorse such a method, and it certainly has nothing to do with anything I’ve said in this thread.
Stranger, are you aware that your approach in threads like these serves to actively quash inquiry? I’m not saying that as a criticism–I won’t be surprised if you in fact not only know that, but are doing it intentionally. That’s fine–I can see why someone might think that’s a good thing to do about certain topics.
First of all, I think what you’re attempting is a great thing to do. Sure, you’ve got a long road ahead of you, but what marvellous scenery it leads through! And don’t fret about getting to the end of it too much – even PhD physicists still learn all the time, that’s the beauty of it: the universe is big enough to not run out of new things to fascinate you with quickly. 
There’s already been some excellent suggestions made so far, but, for my money, I think the first thing I’d do is to acquire a roadmap: Roger Penrose’s aptly titled Road to Reality. Most of the things you’ll need to learn are at least mentioned there, though, being a single (though quite hefty) book, some ‘details’ have to be left out. But it covers, more or less, everything, in a ‘first pass’ kind of way: from basic algebra to string theory, from complex numbers to twistors. None of this, of course, is discussed in great depth, but there are some nice, intuitive expositions, aided by Penrose’s near unique capability to visualize complicated concepts (and draw pretty pictures of them). It’s got problems to do, too, but honestly for some of the more complicated ones, I’m not sure how well the book prepares you to do them.
That’s not to dissuade you from seeking out other, more specialized and detailed works, going through them in the proper order of depth and difficulty, to really get the full picture – to the contrary! --, but there’s no harm in sneaking a glance at what lies ahead – and with Penrose being among the most acclaimed contemporary scientists, you can be sure you won’t learn anything you’ll have to ‘unlearn’ later. That way, you can hear a bit of the music without having to become a composer first!
And I was going to link to physicsforums, as well – there’s generally someone there to help you with any problems you might have, plus, as you probably noticed, there’s a ‘Math & Science Learning Materials’ sub-forum, where you can discover such useful resources as, for instance, the Perimeter Institute Recorded Seminar Archive (PIRSA), which I check on a regular basis for new videos. They have one-shot lectures, but also whole courses – here’s a course on quantum theory, for instance. (Just check under 4. collections -> courses -> All for a full list.)
Perhaps you’d benefit from a more historical approach – how experimental discoveries forced the introduction of ‘quantization conditions’ into the physics, to separate ‘allowed trajectories’ from disallowed ones, and how that semiclassical theory ultimately lead to the quantum mechanics we know and… struggle with today. For instance, in the so-called ‘old quantum theory’, you basically asked of any system, in order to yield results that agreed with experiment, to fulfil a certain relation between momentum and position (given on the wiki page). So the whole thing was experiment-driven, and does not (like special or even general relativity) derive from any concrete physical principle (the search for which is much of what is subsumed under the problem of the interpretation of quantum mechanics). From there (since agreement with experiment was still lacking), everything just kind of evolved into yet more abstract realms.
Perhaps, if you get hung up about Hilbert spaces, let’s do away with them for the moment, and consider the more familiar phase space of classical mechanics, in which, loosely speaking, the points represent configurations of the system under consideration, represented through position and momentum variables. This is the natural home of Hamiltonian mechanics, which is just a (somewhat more mathematically sophisticated) reformulation of Newtonian mechanics, where everything nicely follows classical equations of motion. Those equations of motion are dictated by the Hamiltonian, which is, in the simplest sense, just the sum of potential and kinetic energy (and thus, the total energy) (I’m sorry, by the way, if that’s all old hat to you, but I usually feel it’s better to spell out things in as much detail as feasible than to get tangled up in unnecessary confusions). For a bit more on this, see the wiki.
On this phase space, you can do statistical mechanics – you can, for instance, for an ensemble of particles, ask, ‘what is the probability of finding a certain (classical) particle at a certain point in phase space (i.e. at a certain position with a certain momentum)?’ This probability is given by the Liouville density, which evolves in time according to Liouville’s equation. Up to this point, I hope everything seems sufficiently sensible from a physical point of view. Now here’s where the weirdness enters: I’ve noted above the ‘old quantum condition’, which essentially says that if you’re a quantum particle, there’s some rule that your position and momentum have to obey. In a sense, this amounts to the fact that a quantum particle can’t be ‘localized’ arbitrarily well in phase space; roughly, there’s a minimum area (of dimension [momentumposition] = [energytime] = Js, which is not coincidentally the dimension of Planck’s constant) it must ‘occupy’. This ties to the uncertainty relation.
This of course means a complication for finding out how likely it is that a certain particle exists at a certain point in phase space, as the notion of ‘a certain point’ is no longer meaningful. I’ll have to mumble through the details a bit (so much for spelling it all out), but what you do in order to keep track of this mathematically amounts to a deformation of the algebra of functions on phase space (i.e. functions of where a particle is and how fast it’s going somewhere else, bluntly). This is known as phase-space or Weyl quantization. You end up with a (unique up to isomorphism) structure that is in a sense a generalization of classical (Hamiltonian) mechanics on phase space: if you let Planck’s constant (which is the deformation parameter, coming from the ‘quantum condition’/the ‘discretization’ of phase space) tend to 0, you recover the classical situation completely. In particular, all that really changes is that ordinary multiplication goes over to a (noncommutative, as it must be to do quantum mechanics) star product, (the Poisson bracket is replaced by the Moyal bracket, but I’ve not talked about that, so I only mention it in some vain attempt at an illusion of completeness), and the above-mentioned Liouville density gets replaced by the proper analogue, the somewhat unwieldily named Wigner quasi-probability distribution (sometimes Wigner density function), whose evolution is given by Moyal’s evolution equation. Everything else essentially retains its physical interpretation – allowing for some necessary generalizations, you can do something very much like Hamiltonian (statistical) mechanics on your new, noncommutative, deformed phase space.
And, as it turns out, this (to me, much more intuitive) picture of quantum mechanics as deformed statistical mechanics on phase space is completely equivalent to standard Hilbert space quantum mechanics! All phase space functions can be mapped onto Hilbert space operators using the Weyl map (and back using the Wigner map) (especially, the Hamilton function then yields the Hamilton operator, which perhaps gives some motivation for the idea that the eigenvalues of the latter have something to do with energy). In particular, the Wigner density function gets mapped to the quantum mechanical density matrix, which is a description of the quantum state of a system equivalent to the perhaps more familiar wave function (the line, or ray, in Hilbert space giving the state of the system). It evolves via the von Neumann equation (an equivalent to the Schrödinger equation), which is the Weyl transform of the Moyal evolution equation, which was the deformed version of Liouville’s equation. (Also, the Moyal bracket gets mapped to the commutator of operators in Hilbert space.)
So, you can view the Hilbert space formalism simply as an isomorphic mathematical description of deformed statistical phase-space mechanics. Liouville’s equation is what describes the time evolution of the phase space distribution function, which describes the probability of finding the system in a certain phase space volume – i.e., in the case of a single particle, of the particle having a certain momentum and position. The physical content is that at a certain time t, there is a certain probability that the particle (thought taken from an ensemble of particles) has position x and momentum p. That physical content stays (up to some awkwardness related to the occurrence of ‘negative probabilities’ in the Wigner density function) basically the same with the Moyal equation describing the time evolution of the Wigner function obtained after deforming the phase space algebra. The Hilbert space formalism, in which density matrices evolve according to the von Neumann equation (which is equivalent to wave functions evolving according to the Schrödinger equation), is then just a bit of mathematical re-writing.
Another formalism for quantum mechanics that’s just as equivalent is the path integral formulation, which at least in the beginning is pretty intuitive – however, it’s not recommended for mathematicians, as you guys tend to fret over whether or not functional integrals actually exist in any well-defined way and other insignificant details like that.
Wow, that was a lot shorter in my head…
A wave pulse or wavelet is exactly what Chronos described, “a single sharp pulse” that has a defined peak amplitude at a specified time, but no wavelength (peak to peak distance). A wave packet can only be considered “like” a wave pulse in the sense that the envelope has a maximum amplitude and a group velocity, but unlike the wave pulse, which doesn’t change with time, the wave packet may change significantly with properties of a phase velocity and modulation. In other words, wave packets are a bunch of wave pulses combined together, or alternatively, a bunch of wave trains of differing wavelength and amplitude that are superimposed on one another. Now, wave packets are often used to represent a single discrete wave pulse, especially in digital signal processing where making a single squared off pulse shape isn’t physically possible with a single impulse; instead, you use a wave pulse and a band pass filter to create an approximately squared off pulse of a defined interval but in reality, it is a complex set of waveforms that is an approximation of a discrete signal. Understanding this gives a greater understanding of indeterminacy as a principle; the edges of the wave packet (and therefore its size and locational properties) are definable and measurable only to a specified precision.
I realize this may seem pedantic and overly nitpicky, but it is important to conceptually understand the phenomena as a gestalt and what the terminology describing them actually mean, lest you say something that is sort-of right, but actually very misleading or incomplete. As I said previously, in physics “half-right” is nearly as bad as completely wrong. By reading up on classical wave theory, you can understand what Chronos means about quantum theory being fundamentally a wave theory, and why the “wave/particle duality” as a paradox is a misconception.
I have no intention of quashing inquiry, and in fact I think asking questions, struggling with answers that are of a length and depth appropriate for a message board, and walking away with questions that have to be researched personally in order to understand the concepts behind them, is ultimately the only process by which one can learn anything. If you simply accept pat answers and then repeat them to a level of your entering comprehension, then you haven’t actually assimilated that bit of knowledge into a greater understanding of the phenomenon and how it fits in with the understanding of the natural world. But using the explicit language incorrectly or imprecisely means that one has not correctly fitted that piece into the jigsaw puzzle of knowledge,any more than learning a few legal phrases in Latin doesn’t offer an understanding of legal reasoning, which means your next attempt to learn something based on this knowledge will also be flawed.
I write this from experience, some of it actually coming from inaccurate, incomplete, or imprecise explanations offered by textbooks and supposed experts. While being insistently precise can be seen as (and in many contexts often is) excessive pedantry, in physics, and especially a field of study in which nearly our entire scope of knowledge is based on a formal and more-or-less rigorous mathematical system, is critical to understanding why the system is considered valid and complete despite our inability to directly observe or measure most of the parameters within it.
Stranger
Bolding mine. I thought you were correcting me by telling me that I should be speaking in terms of wave pulses, not in terms of wave packets. But in this last sentence, you are speaking in terms of wave packets. Can you clarify for this me?
No, the explanation of the distinction is appreciated. What quashes inquiry is insisting on valuing only utterances which are completely right, refusing to acknowledge the possibility of there being degrees of understanding which can have value in themselves. If I can only be either completely wrong or completely right then I’d better not inquire into the subject unless I’m prepared to make it my life’s work. Otherwise, I’d be wasting my time.
If it’s any comfort to you, when I “go around telling people” about this, I haven’t had any intention of using the term “wave packet” or “wave pulse” Rather, I was going to speak more directly in terms of what Chronos described–what happens when you push up and down a single time on your end of a rope, the other end of which is tied to a tree.
The confusion contained in what I wrote was purely terminological, not conceptual. There is definitely such a thing as a terminological-but-not-conceptual confusion. Telling the difference can sometimes be non-trivial, but there are definite clear and exclusive cases of each out there. Someone can understand perfectly well exactly what the differences between a cat, dog and fox are–yet believe for all that that the term “cat” can standardly be used to refer to foxes, as well as the term “fox”. This would be a terminological confusion, not a conceptual one.
Think of it this way: If I’d simply said “wave pulse” instead of “wave packet,” you would have found what I said incomplete but nevertheless correct, right?
Then just pretend that’s what I said, because it’s exactly what I meant, and it’s what I would communicate to anyone who asked me about the uncertainty principle.
BTW you put the word “like” in quotes above–I’m afraid you think you were quoting me. You weren’t. I’ve got no fixation on “analogical reasoning” though it seems you may think I do for some reason.
A wave packet is something intermediate between a uniform plane wave and a wave pulse. It has a position that can be somewhat (but not entirely) defined, and a wavelength that can also be somewhat (but not entirely) defined. Of course, there are many other examples of things intermediate between a uniform wave and a pulse, too.
I have some experience with the topics at hand and the math involved. Since you say that you do have some math background, why don’t you just go ahead and see how deep you can get into the subjects? When you bump into some concept you don’t understand go back and dig into that some more, then try again. Go as far as you can in understanding your two subjects. That’s the fun. Who cares if you don’t understand it all when you’re done. You would have still learned an awful lot and that’s a better state than you’re in today. Go do it!
Well, that shows how much you know, because I spent several hours on physicsforums yesterday, and I didn’t see anything about PIRSA, which looks like another wonderful resource.
So thank you very much.
Also just got back from the library, where I checked out the Penrose book. It looks very interesting, and has the added bonus that if I can lift it up and down while I’m reading it, I won’t need to go to the gym.
Yeah, some GR books definitely try to illustrate their subject by curving space around them as much as possible. Misner, Thorne, and Wheeler is another example: I always have to be careful taking it off the shelf, lest I cause an avalanche.
I found volume 1 of Landau in my library, and all I can say is you must have been thinking of something else.
There is no comparison between his level and Feynman’s. Landau is already talking about Lagrangians by page 3. I would expect I’m at least two years away from being ready for that book. By comparison, I can read Feynman with great pleasure and understanding.
A wave pulse has a definite position insofar as having a single definite peak, but no wavelength (peak to peak) distance. A waveform (uniform wave) has a definite wavelength, but no real “position” as it exists across the entire interval. A wave packet, as Chronos notes, is kind of in-between; it has a sort-of maximum amplitude (although it is possible for that amplitude, as drawn as a continuous curve, to be actually greater than any individual amplitude of the individual pulses that make up the packet) and a soft of averaged position, though the actual location of the wave is distributed across an interval of time. Imagine this; if you were looking at a piece of tape on a jump rope as a wave packet transits it, you’d see it go up-and-down, up-and-down, up-and-down. After the entire pulse passes you’d be able to plot the maximum amplitude(s) and the overall shape of the pulse, but you’d be hard pressed to say precisely where the pulse started and where it ended. The more complex or longer the wave packet, the more difficult it is to say anything about its position or maximum amplitude with a particular degree of precision.
The issue is that there are some very discrete concepts in quantum mechanics that aren’t amenable to graduations of accuracy. The difference between a wave pulse and a wave packet is very significant in terms of understanding and explaining the basis of the theory. This isn’t a matter of making it one’s life work to understand, but at least understanding the complete set of tools (classical wave theory) before presenting it as the basis for another conceptual framework. Wave theory isn’t really very difficult to understand in concept, but there is a complete set of rules that needs to be digested before it is really useful, just as one has to understand all the rules in Eulerian algebra before you can solve equations.
I’ve been trying to look for a good reference about wave mechanics online but can’t find something that is unambiguous without being tied to a specific application. As I recall from my coursework in physics, wave theory was actually presented in a fairly piecemeal, “as you need it” fashion without any comprehensive introduction (other than a brief interlude in A-bomb about Fourier transforms), and in fact I think it was only when I took a signal processing class that wave theory was presented in any formal way, and then only incidentally as a necessary tool for discretizing signals. If I recall Feynman goes into length about waveforms in his Lectures, but I don’t remember if he actually does it in a very cohesive fashion. Wave theory as a whole is actually a quite simple extension of basic calculus once you understand the method and notation, but that is often only presented in more advanced coursework.
That may be; I was introduced to Landau in response to grousing about how poorly our E&M text presented the material (not Griffiths, which is the standard text). Landau (of which I’ve only read three or four books in the series) presented in a way that made far more sense, but that was after I’d read the material in the course textbooks and was looking for more guidance on actually understanding the problems conceptually rather than just crunching through problem sets. I haven’t looked at Landau in about twenty years, though I remember the volumes on classical fields and statistical mechanics being particularly insightful.
Stranger