Notices
Results 1 to 30 of 30
Like Tree12Likes
  • 4 Post By mvb
  • 1 Post By mvb
  • 1 Post By mvb
  • 1 Post By John Galt
  • 1 Post By mvb
  • 2 Post By mvb
  • 2 Post By mvb

Thread: Quarks, Leptons, and Bosons

  1. #1 Quarks, Leptons, and Bosons 
    Forum Freshman
    Join Date
    Feb 2013
    Location
    U.S. of A.
    Posts
    21
    I'm one of those ultra geeky teenagers who is trying to teach himself Quantum Mechanics. I think I've exhausted youtube, wikipedia, and any other site I trust, but I'm still a little hazy. I can grasp quarks (up, down, strange, charm, top, bottom) and the Leptons, but that's where my knowledge ends. I would love to know more about bosons, beyond just their respective force. If you can, also explain the tachyons, gravitons, and higgs, but they are less important because of their 'theoretical' label. I'm more interested in W and Z bosons, since they are the ones I know the least about. Explanations, away.


    Reply With Quote  
     

  2.  
     

  3. #2  
    Forum Professor
    Join Date
    Jul 2008
    Location
    New York State
    Posts
    1,069
    W and Z bosons are the carriers of the weak force.
    Weak interaction - Wikipedia, the free encyclopedia

    Start here.


    Reply With Quote  
     

  4. #3  
    mvb
    mvb is offline
    Thinker Emeritus
    Join Date
    Dec 2012
    Location
    Delaware, USA
    Posts
    195
    The W and Z are analogs of the photon, which is the carrier of the electromagnetic force. The Higgs boson is needed to provide mass for the W and Z without having weak-interaction calculations all be irretrievably infinite, and it has recently been detected at CERN.

    I could make a bit more sense of bosons vs. the other particles if I know how much you know about angular momentum, so I would know how much to say about spin. Could you tell us if you have picked up any of that?
    Reply With Quote  
     

  5. #4  
    Forum Freshman
    Join Date
    Feb 2013
    Location
    U.S. of A.
    Posts
    21
    I know all quarks have color, mass, charge, and spin, and their spin is measured in fractions, like their charges. And that, my friend, is the extent of my knowledge on spin.
    Reply With Quote  
     

  6. #5  
    mvb
    mvb is offline
    Thinker Emeritus
    Join Date
    Dec 2012
    Location
    Delaware, USA
    Posts
    195
    OK, let's see how much I can get into this. If I am lucky, I will suggest things you would like to know more about.

    As you apparently know, there are two overall classes of elementary particles. The particles that have spin = 1/2, 3/2, 5/2, ... are called Fermions, and particles that have spin = 0, 1, 2, ... are called Bosons. The Fermions include the quarks, the leptons, and particles made up of 3 quarks plus a variable number of quark-antiquark pairs. The Bosons include the force carriers: the photon, the W, the Z, the graviton if it exists, the Higgs, and for all we know some others. There are also Bosons that are combinations of a quark and an antiquark, plus maybe additional quark-antiquark pairs. The spin of two particles taken together obeys an odd set of rules:

    1/2 + 1/2 = 0 or 1
    0 + any number = that number [ 0 + n = n]
    1 + 1 = 0 or 1 or 2.

    In other words, the union of two Fermions gives a Boson, but the union of two Bosons gives another Boson. As you have probably run into, the proton and neutron are three-quark combinations with spin of 1/2. There are also Fermions related to the proton and neutron which are 3 quarks combined into a single particle of spin 1/2 or 3/2.

    Edit: Note that the full rule on combining Fermions is that combining an even number of Fermions makes a Boson, and combining an odd number of Fermions makes a Fermion, possibly of a new type.

    Now integer spin particles can disappear, converting their spin into (integer) orbital angular momentum. But spin 1/2 and 3/2 particles cannot just disappear. Two of them can combine in a way that gives integer spin, and then the combination may be able to disappear, but a single Fermion can at most turn into another Fermion. This means that the behavior of Fermions and Bosons is fundamentally different. In particular, Bosons can carry forces directly between other particles, and Fermions cannot.

    Now, what does it mean to "carry a force" in quantum mechanics. There is a macroscopic analogy which makes this a bit clearer. Suppose you and a friend are on a very, very slippery surface, and you are carrying a fairly heavy ball. If you throw it to your friend, when he catches it he will "recoil," moving away from you. Meantime, as you threw it you pushed yourself backwards. So the ball has exerted a force on you and your friend, simply by being thrown from one of you to the other.

    The same thing happens in quantum mechanics, but with an additional gimmick. There isn't generally a photon (or one of the other force carriers) accompanying another particle. But there is nothing to prevent a Boson from appearing from nowhere. It can then push off one particle, travel over to another, and push on the second particle while disappearing altogether. Hence the exchange of a Boson permits two other particles to exert forces on each other. In quantum mechanics, the exchange can produce attraction as well as repulsion, but I would rather duck that for the moment.

    On the other hand, a single Fermion cannot just appear or disappear because of its half-integer spin. So Fermions don't work (directly) as force carriers.

    An additional strange fact is that a photon can be produced near another particle, or disappear on hitting another particle, only if that particle is charged. Otherwise the photo goes on its way with no interaction with the (uncharged) particle. Thus uncharged particles don't feel electromagnetic forces.

    Now I haven't said anything about the Higgs Boson or the differences between force carriers and bosons made up of quarks. However, this is already a reasonable-sized post and I am reaching the point that my explanations might get even more mysterious. I'll continue later, but meantime are there any questions or anything you (or anyone else, of course) would like me to tackle? I don't have to ask about whether any corrections are needed. Those will be pointed out automatically by hordes of helpful people.
    Last edited by mvb; February 24th, 2013 at 08:09 PM.
    Reply With Quote  
     

  7. #6  
    Forum Freshman
    Join Date
    Feb 2013
    Location
    U.S. of A.
    Posts
    21
    This was great. I think I'm going to spend a couple days on this, but if you want to do more, please, by any and all means, go ahead. If you could explain the mysterious Higgs (although I understand the basics of that, the higgs field, and the excitation known as the Higgs Boson) and maybe go a little more advanced, like the Tachyon. Personally, I have no idea what the tachyon is, what force its supposed to carry, or how it achieves its 'extreme speeds'.
    Reply With Quote  
     

  8. #7  
    Brassica oleracea Strange's Avatar
    Join Date
    Oct 2011
    Location
    喫茶店
    Posts
    16,965
    Quote Originally Posted by TheOtherGuy View Post
    This was great. I think I'm going to spend a couple days on this, but if you want to do more, please, by any and all means, go ahead. If you could explain the mysterious Higgs (although I understand the basics of that, the higgs field, and the excitation known as the Higgs Boson)
    There are, as far as I can tell, two possible description of the Higgs mechanism (and associated field and boson).

    One is the "cocktail party analogy" (which I think Peter Higgs himself cam up with). This compares the Higgs field to a party full of guests (Higgs bosons). When a famous guest (a massive particle) enters the room all the other particles crowd round slowing him down giving the appearance of "mass".

    Personally, I think that is one of the least useful analogies ever.

    The other explanation requires understanding all of the mathematics of quantum theory, gauge theories, symmetry breaking, and everything else.

    There doesn't seem to be anything in between.

    and maybe go a little more advanced, like the Tachyon. Personally, I have no idea what the tachyon is, what force its supposed to carry, or how it achieves its 'extreme speeds'.
    A tachyon is just a name for a hypothetical particle that travels faster than light. There is no evidence they exist (and no reason to think they might). They would have some odd characteristics. For example you need to give them energy to slow them down. It would take an infinite amount of energy to slow them down to the speed of light. If you reduce their energy they go faster.

    They are just a possible solution to the math; I don't think anyone seriously expects them to be real. (People have done experiments to detect them, with no success).
    ei incumbit probatio qui dicit, non qui negat
    Reply With Quote  
     

  9. #8  
    mvb
    mvb is offline
    Thinker Emeritus
    Join Date
    Dec 2012
    Location
    Delaware, USA
    Posts
    195
    Quote Originally Posted by Strange View Post
    There are, as far as I can tell, two possible description of the Higgs mechanism (and associated field and boson).

    One is the "cocktail party analogy" (which I think Peter Higgs himself cam up with). This compares the Higgs field to a party full of guests (Higgs bosons). When a famous guest (a massive particle) enters the room all the other particles crowd round slowing him down giving the appearance of "mass".

    Personally, I think that is one of the least useful analogies ever.

    The other explanation requires understanding all of the mathematics of quantum theory, gauge theories, symmetry breaking, and everything else.

    There doesn't seem to be anything in between.
    Let me see if I can get somewhere in between those two. The middle ground is much longer than the "cocktail party analogy," but you can avoid actually doing the mathematics. There will surely be questions left by what I say, and I am not sure how quickly or deeply I can go in answering them. Now that the Higgs has been observed, I am reviewing some of the details of the theory, and I might have to go hunting for answers to more technical questions. I didn't like the idea of the Higgs when it was proposed, so I didn't study the details as much as I want to now.

    It is worth noting that essentially all correct theories in particle physics are beautiful. However, sometimes the beauty isn't apparent before the correctness is. The Higgs theory will now be generally acknowledged to be beautiful.

    Anyway, here goes:

    Perturbation Theory: Quantum Field Theory is too complicated to be solved exactly. There has been a lot of progress in numerical solutions of Quantum Chromodynamics using a network of points ["solving on a lattice"]. Otherwise, all that you can do is perturbation theory. All the descriptions of particle forces include a constant, called a coupling constant. With the exception of QCD that constant is small compared to one. So iyou can try to write your proposed solution as a power series in that constant. The terms multiplying a given power of the constants are integrations, and the integrations get more and more difficult as the power of the coupling constant goes up. But the coefficient in front of those integrals goes rapidly down [small2 < small], so you can hope to actually have to calculate only the simplest of the integrals. Of course, this will work only if the integrals don't get numerically bigger as they get more complicated. Unfortunately, in all cases in particle physics the more complicated integrals involve integrations of internal momenta running for 0 to infinity, and these integrals actually diverge [i.e. become infinite]. So the terms in your expansion certainly don't go down. You can't even get a usable number out of most of them. Now What?

    Renormalization: In electromagnetism, all these divergences can be absorbed into the parameters of the theory. For instance, take the mass of the electron to be m0. After including all the perturbation terms, you can prove that in all interesting expressions m0 appears multiplied by the same constant Z. Suppose you have a charged particle with a mass in the equations of m0. You can write your expressions for all physical quantities using a new ["physical"] mass m = m0*Z. All your measurable quantities are in terms of m, so the value you have for the measured mass of the particle is given by m, not by m0. So you pull a trick. You assume that m0 is zero, permitting m0Z to be finite [0*infinity is anything you want it to be]. Now you can calculate anything you want. There follows a spree of calculations of everything imaginable, all of which agree with experiment to very, very high accuracy. So you call this decision "renormalization" and use it for everything in electromagnetic theory. It works, so don't knock it.

    Gauge Symmetry: [Don't worry, we are getting to the Higgs eventually. It just takes a while.] You would like to know that renormalization really works no matter how far you go in the expansion. For electromagnetism, this can be proven. It is necessary for the success of the proof that the photon is massless. If the photon has nonzero mass, renormalization fails, and all your calculations give infinity. There is a symmetry in electromagnetism, in other words there is a modification of the wave functions of the various particles in the theory that doesn't change any of the basic equations. This symmetry, called a gauge symmetry, is only present because the mass of the photon is zero. So photon mass = 0 implies gauge symmetry which implies that renormalization is possible which permits use of perturbation theory which is the only way we can describe particle physics.

    Gauge Theories: [We're close enough to smell the Higgs now.] Now we move to the weak interactions. The analogue of the photon as the force carrier in electromagnetism is a set of three particles, the W+, W-, and Z0. Unfortunately, each of these particles has a nonzero mass, in fact a fairly substantial mass. As a result the direct theory of weak interactions does not satisfy gauge invariance and cannot be renormalized. Although the perturbation calculation to first order in the weak coupling constant g gives good agreement with experiment, the higher order calculations [proportional to gn with n>1] all give infinity. So the calculations are suspect to say the least. The theory looks very much like electromagnetism, but the nonzero mass destroys gauge invariance. What Peter Higgs [see, we have gotten there] and some others showed is that gauge transformations suggest adding a massive scalar to the weak interaction theory. This particle has since been named after Peter Higgs. If this scalar particle has the right gauge transformations itself, the combination of the W, Z theory and the behavior of the Higgs boson will restore gauge symmetry to weak interactions. The existence of the gauge symmetry allows you to renormalize the theory, and everything works. The gauge invariance of electromagnetism and of weak interactions now looks the same, and the two are referred to as Gauge Theories. As a side effect of the gauge theory, the Higgs particle becomes responsible for the mass of the W and Z particles and can be made responsible for the masses of all the quarks.

    Some Philosophy: The Higgs particle is important for, and was predicted because of the need for, the ability to calculate particle physics in perturbation theory instead of attempting an unbelievably difficult exact analysis. I for one consider it rather remarkable that the universe has a particle whose most important role is to make it possible for humans to calculate using expansions in powers of small numbers. One of the "lucky accidents" of various aspects of particle physics and cosmology, I guess.

    Questions are welcome and indeed expected.
    Markus Hanke likes this.
    Reply With Quote  
     

  10. #9  
    mvb
    mvb is offline
    Thinker Emeritus
    Join Date
    Dec 2012
    Location
    Delaware, USA
    Posts
    195
    Well, I guess that it is high time I posted something about tachyons. I will keep this pretty short, but I may come back later and expand it. The reason for all this will become obvious shortly.

    "Tachyon" is the name applied to any particle that travels faster than the speed of light. It's content is rather empty, since no such particles have ever been observed. Given the mass-energy relation from special relativity, E = m0 c2 [ 1 - ( v2 / c2 )]1/2 , a tachyon's energy would be expected to be imaginary unless its rest-mass was imaginary. In quantum field theories, tachyon-like objects sometimes appear, and they have obnoxious properties and are generally taken as evidence that there is something wrong with that particular field theory. As a result, when CERN's OPERA experiment appeared to have determined that neutrinos were tachyons, the fast-moving neutrinos were sometimes referred to as phantoms. And indeed, in the long run, the Phantom of the Opera disappeared.

    However.

    I have recently become aware of a paper by James M. Hill and Barry J. Cox in the Proceedings of the Royal Society A, vol. 468 (2012), p. 4174, which details a perfectly respectable extension of Einstein's theory to velocities greater than c. This extension need have no imaginary rest-masses or any other obvious anomalies. [Warning - I am a bit biased in favor of this article since I got similar but partial results after the CERN experiment and before I found their paper.] They get two possible extended Lorentz transformations, one of which yields a velocity-addition formula for superluminal velocities in the same form as the verified one for velocities below c. So there now exists a way of describing tachyons with no anomalies.

    I have not completely digested this paper, but I have seen enough to feel that it is a very important paper, especially if tachyons are ever observed. It could possibly suggest places to look for them. Once I understand its details, I may come back and rewrite this blurb about tachyons.
    Reply With Quote  
     

  11. #10  
    Forum Freshman
    Join Date
    Feb 2013
    Location
    U.S. of A.
    Posts
    21
    Ok, here's a few questions in general, not necessarily relating to anything said here. Any answer you got I'll take.

    1) How do bosons 'disappear', and doesn't that break the law of conservation of matter?

    2)How do we determine integer/half integer spin, and why do particles work like that?

    3)How do fermions combine? And, if the combination of fermions make bosons, doesn't that mean nuetrons/protons are bosons? And, if bosons carry a force, what force would protons/nuetrons carry?

    4)Can protons disappear with orbital Angular Momentum?

    what is orbital Angular momentum?
    Reply With Quote  
     

  12. #11  
    mvb
    mvb is offline
    Thinker Emeritus
    Join Date
    Dec 2012
    Location
    Delaware, USA
    Posts
    195
    Quote Originally Posted by TheOtherGuy View Post
    Ok, here's a few questions in general, not necessarily relating to anything said here. Any answer you got I'll take.

    3)How do fermions combine? And, if the combination of fermions make bosons, doesn't that mean nuetrons/protons are bosons? And, if bosons carry a force, what force would protons/nuetrons carry?
    I'll try to tackle all of these questions fairly soon, but I want to do number 3 immediately since I see I left an incorrect impression. Fermions can combine if they feel attractive forces toward each other. In that case they may be able to go into states where they do not easily go away from each other, and therefore act much like a single particle. An example of this are an electron and a proton, both of which are fermions (see next paragraph for the proton). If a bare proton and an electron come close enough, they can give off some energy (maybe carried off by a photon) and join together into a hydrogen atom. All states of hydrogen have integer spin, so the hydrogen atom is in fact a boson.

    Now for the combination of fermions to make a boson: The rule is that 2 fermions make a boson, 3 fermions make a fermion, 4 fermions make a boson, etc. Combining any even number of fermions gives a boson, and combining any odd number makes a (possibly different) fermion. So protons and neutrons, which are essentially each made of 3 quarks, are fermions. Therefore they can disappear only in a reaction with their respective antiparticles and not spoken of as "carrying" a force. However, nothing is really simple. An electron can be electrically attracted to two different protons, which then "share" the electron between themselves. The attraction of the electron prevents the protons from easily flying away from each other, and it is often said that there is a force due to the "sharing" of the electron. However, this is a mechanism that is entirely different from the carrying of a force by a boson which disappears in the process.
    Reply With Quote  
     

  13. #12  
    Moderator Moderator Markus Hanke's Avatar
    Join Date
    Nov 2011
    Location
    Ireland
    Posts
    7,302
    I have not completely digested this paper, but I have seen enough to feel that it is a very important paper, especially if tachyons are ever observed. It could possibly suggest places to look for them. Once I understand its details, I may come back and rewrite this blurb about tachyons.
    Is there an online version for this ? I'd be interested in seeing how this was done - mind you, an extension to the Lorentz transformations would imply a possible amendment to the causal structure of space-time itself.
    Reply With Quote  
     

  14. #13  
    mvb
    mvb is offline
    Thinker Emeritus
    Join Date
    Dec 2012
    Location
    Delaware, USA
    Posts
    195
    Quote Originally Posted by Markus Hanke View Post
    Is there an online version for this ? I'd be interested in seeing how this was done - mind you, an extension to the Lorentz transformations would imply a possible amendment to the causal structure of space-time itself.
    You can find it starting from rspa.royalsocietypublishing.org/, but the download is not free. I got it from a university library.

    Edit: See next post, by John Galt, for a free source.
    Last edited by mvb; February 25th, 2013 at 12:14 PM.
    Markus Hanke likes this.
    Reply With Quote  
     

  15. #14  
    Universal Mind John Galt's Avatar
    Join Date
    Jul 2005
    Posts
    14,168
    The full paper is available here, at no charge.
    http://www.cnd.mcgill.ca/~ivan/Faste....2012.0340.pdf

    Perhaps that is the university library mvb refers to. (Google Scholar is definitely your friend.)
    mvb likes this.
    Reply With Quote  
     

  16. #15  
    mvb
    mvb is offline
    Thinker Emeritus
    Join Date
    Dec 2012
    Location
    Delaware, USA
    Posts
    195
    Quote Originally Posted by John Galt View Post
    The full paper is available here, at no charge.
    http://www.cnd.mcgill.ca/~ivan/Faste....2012.0340.pdf

    Perhaps that is the university library mvb refers to. (Google Scholar is definitely your friend.)
    Actually, I missed that one, probably because I could get the paper from the library at the university I retired from. Having that relatively simple access I didn't search very hard for another one. I'm glad that others now have simple access.
    Reply With Quote  
     

  17. #16  
    Moderator Moderator Markus Hanke's Avatar
    Join Date
    Nov 2011
    Location
    Ireland
    Posts
    7,302
    Excellent, thank you guys. I will examine this carefully when I have time, since this is a topic I am personally interested in
    Reply With Quote  
     

  18. #17  
    mvb
    mvb is offline
    Thinker Emeritus
    Join Date
    Dec 2012
    Location
    Delaware, USA
    Posts
    195
    Quote Originally Posted by TheOtherGuy View Post
    Ok, here's a few questions in general, not necessarily relating to anything said here. Any answer you got I'll take.

    1) How do bosons 'disappear', and doesn't that break the law of conservation of matter?
    Basically, particles can disappear if that disappearance doesn't violate any conservation laws [statements that the amount or size of some quantity or other never changes]. In particular, this means that something must carry away the particle's energy and momentum. So generally, a particle cannot disappear without some kind of interaction with another particle. In general, if a reaction does not violate any conservation laws and involves a force that at least two nearby particles can feel, the reaction is likely to happen. There may be exceptions, but not many.

    The conservation of matter is an important law, but it isn't simple. There are several laws: One is that the total charge cannot be changed in a reaction. Another is that the total number of baryons minus the total number of antibaryons coming into a reaction does not change during the reaction. Without this law it might be difficult for us to exist. Another is that the total number of leptons including the corresponding neutrinos minus the number of their antiparticles cannot change. These latter two laws provide the conservation of "matter" in the narrow sense. As a word, "matter" is often used rather loosely, so in discussing the permanence of matter in the universe it is better to talk of the "baryonic matter" conservation law.

    Anyway, since mesons don't appear in these conservation laws, there is nothing stopping them from being created at one spot and destroyed at another, thus "carrying" a force from one particle to another.

    It is also thought that here may be a very tiny probability of violating baryon conservation. If there is, it would explain why the observable universe has more baryons than antibaryons.
    nnunn likes this.
    Reply With Quote  
     

  19. #18  
    mvb
    mvb is offline
    Thinker Emeritus
    Join Date
    Dec 2012
    Location
    Delaware, USA
    Posts
    195
    Quote Originally Posted by TheOtherGuy View Post
    Ok, here's a few questions in general, not necessarily relating to anything said here. Any answer you got I'll take.

    2)How do we determine integer/half integer spin, and why do particles work like that?

    what is orbital Angular momentum?
    OK, it took me a while, but I'm going to tackle angular momentum. At that point I will have answered all the questions I feel that I can answer. This answer will be a bit more abstract than the other answers, although for a strange reason. Angular momentum is actually simpler than the other issues, which tempts me to go a bit further into the formalism. [Maybe too far?]

    Classical Mechanics: [Review, basically] Angular momentum describes the extent that a body is moving around an axis. If the axis goes through the body's center of mass, it is called spin, but spin is just a special case of angular momentum. Angular Momentum is usually denoted L and is defined by L = sum (ri X pi). In this formula, ri is the distance of a given part of the body from the axis, with ri perpendicular to the axis, pi is the momentum of that part of the body, X denotes a vector (cross) product, and the sum is over each little bit of the body. The sum is needed since different parts of the body are different distances from the axis. The total angular momentum of an isolated body is conserved, i.e. it does not change with time.

    Quantum Mechanics: [Still reviewing, mostly] In quantum mechanics, the average value of a property of a particle, in this case angular momentum, is given by



    Since the operator for pi is -i hbar (d/dri), [watch out, the subscripts are suddenly vector components] the operator corresponding to L is



    where the last factor is the gradient operator: it means the derivative with respect to the proper component of r . I now intend for the r operator to refer to the position of the center of mass of the object, in which case the angular momentum is referred to as the "orbital angular momentum." Angular momentum due to the rotation of the internal structure of the object also exists, and is usually called "spin," denoted S, to be specific about its source. [Important:] The quantity that is most likely to be constant in time is the combination L + S, as it is in classical mechanics.

    Recalling that in quantum the "commutator" or a and b is given by the expression [a,b] F = abF - baF, we can easily see that
    [x, d/dx]F = x dF/dx - (d/dx)(xF) = i hbar F . This means that when the operator for momentum, d/dr, is involved, it matters whether you have ripj or pjri, unlike when you are using ordinary numbers. In particular, [Li op , Lj op] is not zero. The terminology for this expression is that the commutator of two different components of L is nonzero. This fact produces most of what I want to mention about angular momentum in particle physics.

    One more piece of terminology from quantum mechanics is needed. If an object has a definite value of a given property, such as Lz, we have
    the formula Lz op (r) = lz (r), where the lower case lz means an ordinary number and psi is the wave function of the property. The number lz is referred to as the eigenvalue of the operator, and the wave function is an eigenvector of the operator.

    Particle Physics: [Definitely not reviewing, and I will switch to talking about what happens without displaying the mathematical details.]
    [Now we have finally gotten to the fun part] In either relativistic or nonrelativistic quantum mechanics, you can solve a differential equation to determine that the possible eigenvalues of L2 are hbar2 l(l+1), where l is any integer, and that the eigenvalues of any one Li are hbar m, where m is an integer which may have any value between -l and +l. However, only one component at a time of L can have definite values, because different components of L do not commute and different operators that do not commute cannot both have definite values at the same time. For convenience, that component with definite values is usually taken to be the z component, unless there is a definite reason to worry about more than one component in the same experiment.

    Rather amazingly, you can get exactly the same results using only the commutators of the three Li , except that you cannot specifically determine that the numbers l,m must be integers. In fact, it is possible for m to take values that are integer steps apart and range from
    -l to +l for both integers and half-integers [Try it for l = 1/2 and l = 3/2]. If you also use the commutators [L, , rj ], you can restrict l to be an integer, but you can't from the angular momentum commutators alone. So orbital angular momenta must have integer eigenvalues, but you can't prove it from the properties of angular momentum alone.

    Even more amazing, when you experiment with particles such as electrons, protons, and the rest of the menagerie, you find that their angular momentum is not conserved unless you include a contribution from an internal angular momentum, even though for leptons it appears that there is no internal structure. Moreover, that internal angular momentum can have half-integer eigenvalues. So the quantity conserved in nature contains an orbital part with integer components and an intrinsic component which may have integer or half-integer eigenvalues, depending on which particle you are dealing with. To emphasize the difference, the intrinsic part is called spin and denoted by S. The orbital part is denoted by L, and the conserved combination L+S is denoted J. The orbital part always involves r X p. The intrinsic part cannot involve r x p even using the internal r, because the commutator with r would forbid half-integer eigenvalues.

    [Aside to finish the questions] The energy of a particle in a constant magnetic field involves a term mu jz . By measuring this energy for states of the particle with various m's [the Zeeman effect] and counting the number of values, you can determine the values of j and m . If the number of states is even, then j is half-integral. If the number of states is even, j is integral.

    OK, I think that is it. Questions are certainly welcome as always. I'll restrain myself from philosophical comments about the conservation of the two types of angular momentum only in combination. At least for the moment.

    Edits: Inserting some tex display; no substantive changes
    Last edited by mvb; March 6th, 2013 at 08:40 PM.
    Markus Hanke and nnunn like this.
    Reply With Quote  
     

  20. #19  
    Moderator Moderator Markus Hanke's Avatar
    Join Date
    Nov 2011
    Location
    Ireland
    Posts
    7,302
    Nice one

    It would be easier to follow though if you were to use LaTeX to produce a well formatted maths output. You can get LaTeX code from the online editor

    Online LaTeX Equation Editor - create, integrate and download

    without ever having to learn how to code ( mostly, anyway ). Just put the code between "tex" and "/tex", each of which enclosed in angle brackets.
    Reply With Quote  
     

  21. #20  
    Forum Freshman
    Join Date
    Feb 2013
    Location
    U.S. of A.
    Posts
    21
    Ok, I'm sort of understanding... Care to shed any light on my other questions?
    Reply With Quote  
     

  22. #21  
    mvb
    mvb is offline
    Thinker Emeritus
    Join Date
    Dec 2012
    Location
    Delaware, USA
    Posts
    195
    Quote Originally Posted by TheOtherGuy View Post
    Ok, I'm sort of understanding... Care to shed any light on my other questions?
    Actually, I thought I had hit all of them in one way or another. Perhaps you could tell me what I didn't give you at least a start on, or ask specific questions about things that weren't clear or that you want to know more about?
    Reply With Quote  
     

  23. #22  
    Forum Freshman
    Join Date
    Feb 2013
    Location
    U.S. of A.
    Posts
    21
    Well, my friend introduced me to Richard Feynman's biography, and part of that deals with his Diagrams. I get how photons can 'disappear' because of that, but I'm still confused about orbital angular momentum. If there's anything you can do concerning that, I'd be happy.
    Reply With Quote  
     

  24. #23  
    mvb
    mvb is offline
    Thinker Emeritus
    Join Date
    Dec 2012
    Location
    Delaware, USA
    Posts
    195
    Quote Originally Posted by TheOtherGuy View Post
    Well, my friend introduced me to Richard Feynman's biography, and part of that deals with his Diagrams. I get how photons can 'disappear' because of that, but I'm still confused about orbital angular momentum. If there's anything you can do concerning that, I'd be happy.
    Well I did a first attempt on that in post #18, in particular trying to distinguish orbital from spin angular momentum. I'm not sure how I would change that, so maybe the best thing would for you to post where you lost the thread and hopefully a question or two about what lost you or what details you might like to see. I'll grant you that #18 is a bit complex, but that is somewhat unavoidable with angular momentum, even in classical mechanics.
    Reply With Quote  
     

  25. #24  
    New Member
    Join Date
    May 2013
    Posts
    2
    I have a quick question. When a quark transformers into a different kind of quark, is the W boson created by the transformation, and then can become a lepton or a quark (depending on which flavor the quark was and transformed into) because they are both fermions? Or am I completely wrong and stupid.
    Reply With Quote  
     

  26. #25  
    mvb
    mvb is offline
    Thinker Emeritus
    Join Date
    Dec 2012
    Location
    Delaware, USA
    Posts
    195
    Quote Originally Posted by Curious person View Post
    I have a quick question. When a quark transformers into a different kind of quark, is the W boson created by the transformation, and then can become a lepton or a quark (depending on which flavor the quark was and transformed into) because they are both fermions? Or am I completely wrong and stupid.
    First: you can't be stupid, because you know you are uncertain. Stupid people are always sure they understand the things they actually have wrong.

    OK, now W bosons. There is a process in which, all at the same time, a quark changes type and a W boson appears. This process only occurs when the quantities which are (always) conserved are the same after the transformation as they were before. One of those conservation laws is that the number of Fermions must not change. Then since there was one Fermion (a quark) before the transformation and one (the new quark) after it, the W boson cannot be a Fermion1, and of course it isn't. It is a boson. However, if the quarks are not coming out of an additional interaction or headed into another one, when you calculate the energy and momentum before and after you find that those cannot be conserved if the W boson has the mass-value that it has as a free particle. However, the product of the uncertainty of the energy of the W and the time it is in existence as a W is of the order h-bar, so the boson can be temporarily "off the mass shell." However, some other process must make the W disappear soon enough to allow the escape hatch to work. So there will be another interaction, say the disappearance of the W and the appearance or change of type of some other particle(s). The conservation laws must be obeyed here also, so either there is another particle in the vicinity to absorb the W, changing its type if it is a quark or changing it between charged lepton and the corresponding neutrino if it is not a quark. Barring the existence of this second particle, the W can disappear leaving behind a quark-antiquark pair or a lepton-antineutrino pair or an antilepton-neutrino pair. You can verify that under these conditions energy and momentum are properly conserved in the overall process. So the relationships between the Fermions at either end of the interaction are a bit complicated.

    1. You could also argue that because you had one Fermion before the process and the W is a boson, the initial Fermion can only change into another type of Fermion.
    Markus Hanke and nnunn like this.
    Reply With Quote  
     

  27. #26  
    New Member
    Join Date
    May 2013
    Posts
    2
    Thank you for your help, I was curious because in my science class my teacher had told us about beta decay, only telling us that during beta decay a neutron was changing into a proton and releasing an electron or a proton into a neutron and releasing a positron, which left me wondering about what was in electrons and protons that could change to cause this. I did a little searching and found this site.
    Reply With Quote  
     

  28. #27  
    AI's Have More Fun Bad Robot's Avatar
    Join Date
    Apr 2013
    Location
    Washington state
    Posts
    6,114
    Wow! What a topic, kind of like reading a text book that's easy on the non math majors. Thanks mvb

    I get the impression you are a retired physics professor that still gets excited by this subject. I see you are relatively new to this forum and very much hope it will not burn you out, with the repetitive topics and questions that keep making the rounds. I look forward to following most of your posts in the future.
    Reply With Quote  
     

  29. #28  
    mvb
    mvb is offline
    Thinker Emeritus
    Join Date
    Dec 2012
    Location
    Delaware, USA
    Posts
    195
    Quote Originally Posted by Bad Robot View Post
    Wow! What a topic, kind of like reading a text book that's easy on the non math majors. Thanks mvb

    I get the impression you are a retired physics professor that still gets excited by this subject. I see you are relatively new to this forum and very much hope it will not burn you out, with the repetitive topics and questions that keep making the rounds. I look forward to following most of your posts in the future.
    You are right about where I am coming from. I was a researcher, but I really enjoyed teaching more. I did a lot
    of experimental-methods teaching of physics, and I tried to get students to talk with me when they were puzzled. I think and hope it worked.

    After a career of teaching I don't expect to burn out on repetitive topics and questions. You might be surprised how different the same question from two people can be if you are listening for what the problem is, not just what the question is. The main problem here is the people who aren't interested in listening to standard science but only in pushing rather silly ideas. The good questions can get lost in the noise.

    If you are not watching the Physics Forum, you might want to look in there also. There is a link from the contents page for the Physics Section here. Both fora are fun, but there is a bit more signal to noise in that one.
    Reply With Quote  
     

  30. #29  
    AI's Have More Fun Bad Robot's Avatar
    Join Date
    Apr 2013
    Location
    Washington state
    Posts
    6,114
    I believe quarks are considered elementary particles, but are they really? It seems to me that matter, atoms & leptons both are mostly empty space and as our instruments improve and increase our awareness of the very small, even the so called elementary particles break down into even smaller components.

    Will we ever know the real nature of matter or will we reach our limits before we are able to find out?
    Reply With Quote  
     

  31. #30  
    mvb
    mvb is offline
    Thinker Emeritus
    Join Date
    Dec 2012
    Location
    Delaware, USA
    Posts
    195
    Quote Originally Posted by Bad Robot View Post
    I believe quarks are considered elementary particles, but are they really? It seems to me that matter, atoms & leptons both are mostly empty space and as our instruments improve and increase our awareness of the very small, even the so called elementary particles break down into even smaller components.

    Will we ever know the real nature of matter or will we reach our limits before we are able to find out?
    Quibble: atoms are indeed largely empty space, although the space is occupied by electron wave-functions. Leptons, however, are too small for their size to be detected in current experiments.


    "Reply": Leptons and quarks are generally considered, at least for the moment, to be elementary particles. The bosonic force carriers are also considered to be elementary. None of these have internal structure large enough for current experiments to detect.

    Any or all of these may eventually turn out not to be elementary, if experimental resolution continues to improve. The number of different quarks, and the parallel spectroscopy of quarks and leptons invites guesses that neither is elementary. There was an early theory of their potential make-up as combinations of "preons," but nothing useful resulted, at least not immediately. The current attempt at giving a structure to the two is string theory, which hypothesizes that both are excitations of a string in a space which has some additional dimensions beyond the 3+1 that we observe. The extra dimensions have all collapsed to very small sizes. Membrane ("M-theory") is a descendent of this original idea. The theories make predictions which are far from observable for the forseeable future, and the past history of theories with no immediate experimental implications is not encouraging for string theory. [I hope I made this comment soft enough to satisfy string theorists.]

    String theory proposes that the internal structure of "elementary" particles is small enough that general relativity is relevant to its description. If this is true, either by string theory or some other idea, determining the internal structure of particles will also tackle the current inconsistency of quantum mechanics and general relativity. When and if this occurs (with experimental consequencies), it will be a major development.
    Reply With Quote  
     

Similar Threads

  1. Strings or quarks?
    By wallplaster in forum Physics
    Replies: 4
    Last Post: August 30th, 2012, 11:32 AM
  2. Replies: 7
    Last Post: April 4th, 2012, 06:57 PM
  3. Leaping Leptons!
    By wraithedge in forum Physics
    Replies: 6
    Last Post: March 29th, 2012, 05:18 AM
  4. Top Quarks
    By Quantime in forum Physics
    Replies: 1
    Last Post: July 3rd, 2010, 10:27 PM
  5. Fermions and Bosons
    By EV33 in forum Physics
    Replies: 8
    Last Post: December 8th, 2007, 01:51 PM
Bookmarks
Bookmarks
Posting Permissions
  • You may not post new threads
  • You may not post replies
  • You may not post attachments
  • You may not edit your posts
  •