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Spin and intrinsic angular momentum

  1. Aug 22, 2013 #1
    Spin and "intrinsic angular momentum"

    After searching this forum for what exactly spin is my inference is that it is a misnomer for an electrons "intrinsic angular momentum".....so some doubts; may be I am being too blunt;
    What is the difference between +1/2 and -1/2 spin?
    Is the electron actually moving(it is has got to be either stationary or moving right?)
    How exactly was this angular momentum calculated?
    Since L=IW(OMEGA)
    and I=k m r^2 what value of k and r was taken to say that to provide for its angular momentum electron must be moving at a velocity greater than that of light...was all possible values of r explored?(sure it must have been,but what is the maximum possible value of r here)?

    And frankly there seems to be something wrong about saying that something processes an "intrinsic angular momentum"....
     
    Last edited: Aug 22, 2013
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  3. Aug 22, 2013 #2

    jtbell

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    The magnitude of a particle's intrinsic angular momentum ("spin") is fixed at [tex]S = \sqrt{s(s+1)} \hbar[/tex] where (lower case) ##s## has an integer or half-integer value that is fixed for a particular type of particle. For an electron, ##s = 1/2##, so ##S = \frac{\sqrt{3}}{2} \hbar##.

    Angular momentum is a vector quantity which also has a direction (orientation), and can be decomposed into vector components. (How much do you know about the mathematics of vectors?) The component of the spin along any particular direction (usually we call it the z-drection) has the value [tex]S_z = m_s \hbar[/tex] where ##m_s## can have values ranging from ##-s## to ##+s## in steps of 1. For an electron, this means either ##m_s = -1/2## or ##m_s = +1/2##. This is what people mean by "spin -1/2 or +1/2" or "spin down or up".

    (Maybe I'll address your other questions if no one else has done so after I finish dinner and other stuff tonight...)
     
  4. Aug 22, 2013 #3
    Physically, +1/2 and -1/2 spin correspond to the two possible outcomes of a measurement of a spin 1/2 particle's moment of angular momentum. Take a look at the Stern-Gerlach experiment: http://en.wikipedia.org/wiki/Stern-Gerlach_Experiment. Mathematically, they correspond to the two orthonormal basis vectors for the Hilbert space in which the spin state of a spin 1/2 particle lives.
    A stationary electron has a magnetic moment just like a moving one. An example of an experiment on a stationary spin-1/2 particle would be Electron Paramagnetic Resonance. http://en.wikipedia.org/wiki/Electron_paramagnetic_resonance . It is usually not helpful to think of the electron's spin magnetic moment as arising from any sort of rotational motion--it's just an intrinsic magnet similar to a bar magnet.
    At a molecular level, this classical calculation does not apply. For example, electrons in atoms have an orbital angular momentum, but if we were to assume this is due to classical orbital motion of the electron, we find that atoms are not stable (since they would emit dipole radiation, leeching off the kinetic energy of the orbiting electron, leading it to spiral into the nucleus in a very short time.)

    So instead of classical mechanics, we apply quantum mechanics. There are actually two ways to attack angular momentum: One is by calculating the wavefunctions of electrons in atoms (this is called the analytical method and yields spherical harmonic solutions), and another is the more abstract algebraic method which uses the properties of quantum mechanical angular momentum operators.

    If you read through the two methods I mentioned--the analytical method and the algebraic method--you will find a very enlightening point that indicates the fundamental difference between spin and orbital angular momentum. If we start out using the analytical method of solving the schrodinger equation for the electron's wavefunction as a function of space [this wavefunction would include everything there is to know about its position/velocity], we find that wavefunctions are only consistent (i.e. if we move around the nucleus by 2π radians, the wavefunction must return to the original form since we're back at the same point) if moment of angular momentum has integer values of spin: ..., -2, -1, 0, 1, 2, ...

    However if we are only looking at the properties of QM angular momentum operators, we are not necessarily dealing with a wavefunction in space. Instead we consider an abstract Hilbert space. Since we are in a more mathematically abstract space, we don't have any reason to say things return to the same value after moving around by 2π radians, so there is no periodicity requirement, and it turns out there are more solutions--the half integers. ..., -2, -3/2, -1, -1/2, 0, 1/2, 1, 3/2, 2, ... If an object has half-integer angular momentum, it must be due to spin, and particles observed with any spin are said to have more "degrees of freedom" which live in a "spin space" separate from the space the wavefunction lives in. So to describe a spin 1/2 particle completely you need two parts: the wavefunction [spatial degrees of freedom] and the spinor [intrinsic degrees of freedom.]

    This indicates that spin angular momentum in half-integer values cannot be due to motion through space [the wavefunction]. It is an intrinsic quality of the spinor.
     
    Last edited: Aug 22, 2013
  5. Aug 23, 2013 #4
    Thanks for the responses....
     
    Last edited: Aug 23, 2013
  6. Aug 24, 2013 #5
    Jolb the links provided were useful...but I have some doubts remaining.

    1.Was this value of "Intrinsic angular momentum" ie h/2 calculated experimentally-say from location of spots after Stern Gerlach expt etc or was it merely predicted by some formula?

    2.Is the electron in an atom actually rotating(though that need not necessarily be the cause of its magnetic moment) and may not be at the velocity to produce the effect seen.Has there been any evidence to the contrary ie electron is not rotating etc.

    3.What exactly is meant by half integer spin? From Stern-Geralch experiment I just inferred that it just indicates the number of possible values "the intrinsic angular momentum" can take but somewhere else I read that it implies that Fermions have to rotate 720^o to come back to the same state unlike Boson's which have to rotate just 360^o.Has this been experimentally verified (True it is difficult to conduct experiments on electrons without interfering with them, but has there been any indirect evidence that it has to rotate (or I don't know what other term to use) to come back to the same state

    And again I may be being silly, but I don't believe in any 4th spatial dimension.My understanding is that it is space that existed first and we introduced 3 dimensions to describe it.Had it required 4 dimensions we would have used 4D representation...Is my inference wrong? It seems ok to think of hidden variables but "dimension" is purely a human construct right?
     
  7. Aug 24, 2013 #6

    Bill_K

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    Historically, it was first derived from a study of atomic spectral lines, by Uhlenbeck and Goudsmit. See here for a nice summary.

    The electron, as well as all the other elementary particles, are point-like, meaning they are smaller than any size we have been able to observe. The idea that they have a finite size and/or rotate like a classical object might is false.

    Unfortunately you do often run across this statement. Physics is invariant under a 360-degree rotation, and that includes systems containing fermions as well as bosons. The reason for the misunderstanding is that a fermion wavefunction ψ(x, t) is double-valued. That is, there's an implicit ± sign in front. The fermion wavefunction requires a 720-degree rotation to come back to its original value, and under a 360-degree rotation changes sign. But this represents the same state.
     
    Last edited: Aug 24, 2013
  8. Aug 24, 2013 #7
    1. The S-G experiment demonstrates the effects of the spin magnetic moment, but is silent about the spin angular momentum.

    The idea that spin is connected to angular momentum - not just magnetism - is suggested by the relativistic analogue of the Schroedinger equation (the Dirac equation).

    The orbital angular momentum operator does not commute with the Dirac Hamiltonian, suggesting that it is not a constant of the motion in the relativistic theory. However, one can find an operator corresponding to the sum of orbital and half-integer spin angular momenta that does commute with the Hamiltonian. The implication is that spin is intimately connected with angular momentum (not just magnetic effects).
     
  9. Aug 24, 2013 #8

    Bill_K

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    And also by the nonrelativistic equation for spin-1/2 particles, the Pauli Equation, which came a year earlier.
     
  10. Aug 24, 2013 #9
    Via Heisenberg uncertainty, it always has some motion.

    The terminology might be unappealing or perhaps not entirely descriptive, but it is nevertheless an observed characteristic. 'Spin' is just one of many characteristics which are not fully understood. Why particles themselves even exist as observed and why they have certain discrete characteristics and not others, in fact even just what a particle actually IS, leaves a lot of room for discussion and disagreement.

    It might be worth noting that the Standard Model of particle physics contains a hodge podge of theory [like quantum electrodynamics, the strong force, and so forth] and a bunch of characteristics for which we have no fundamental theory[ like the charge and mass of the electron, and Planck's constant, h]. So all this is pasted together in a successful framework for defining observations, that is explaining what happens, but not so much why.

    So even though your questions about spin are entirely appropriate, it seems all characteristics of particles are 'strange', at least from a classical macroscopic perspective and likely well beyond.

    edit: In fact although we have theoretical models, just what the math means is a subject of 50 or more years of debate and discussion. Even Werner Heisenberg came to change his mind about exactly what his own uncertainty principle meant after discussion and review with his peers.

    Richard Feynman summed up nature this way:

     
    Last edited: Aug 24, 2013
  11. Aug 24, 2013 #10
    1.So is it the "intrinsic angular momentum" that gives rise to an electrons magnetic moment? I thought it had to do with the orientation of the orbital.

    2.If we perform Stern-Gerlach experiment with Bosons say with spin 1 where exactly are these particles likely to be found on the screen?

    Thanks Naty for that first answer of yours. So it is improbable to visuluaze the classical analogy of electron's rotation but is it impossible?

    I am sorry if these questions seem too silly, but I come from engineering background and am just acquainting with the basics of physics..
     
    Last edited: Aug 24, 2013
  12. Aug 24, 2013 #11

    Bill_K

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    The electron's magnetic moment has contributions from both the spin angular momentum and the orbital angular momentum.

    Repeat after me... "The electron does not rotate." Despite any classical model you may imagine that pretends otherwise!
     
  13. Aug 24, 2013 #12
    I just made a thoughtless remark and is editing it.....What prevents electrons from rotating? what exactly happens at that level?
    Bill, these days I often come across this remark ie "some thing cannot be picturised in terms of classical mechanics" Thats fine but, it would have been ok if the quantum mechanical picture gets clear. It is quite uncomfortable with some vague notion of something and not being able to know more.
     
    Last edited: Aug 24, 2013
  14. Aug 24, 2013 #13
    Well Bill_K is right in saying that Ulenbeck and Gaudschmidt first predicted it via some formula, but if you do the Stern-Gerlach experiment, you can easily look at the magnitude of the deflection and experimentally verify it.

    Well there always is some uncertainty in position and momentum, so yes technically it is moving in the atom. But if we calculate an electron's orbital angular momentum in QM (using a somewhat different mathematical definition from classical mechanics), there are actually orbitals which have zero angular momentum. A hydrogen atom's ground state, for example, has quantum numbers n=0 (s orbital), l=0 (total orbital angular momentum is zero), ml=0 (no moment of orbital angular momentum.)

    This touches on the spin-statistics connection, which is a proof that if a particle has spin 1/2, 3/2, 5/2, etc., then it must be a fermion obeying the pauli exclusion principle [and similar conclusions for bosons]. One way to see that this leads to the "electrons must rotate 720° to turn all the way around" idea is as follows: This principle gives rise to an "exchange phase" which means that if two electrons swap positions [all other things held equal], the wavefunction gets multiplied by -1.

    If you're creative, you can convince yourself that the exchange phase must be equal to the phase acquired when a stationary object rotates by 360°. Here's an interesting reference by John Preskill that discusses this fact with in the context of anyons, which are [new and exciting] particles with an arbitrary exchange phase. See section 9.3: http://www.theory.caltech.edu/~preskill/ph219/topological.pdf
    You might be surprised to learn that even though QM only deals with a "real" space of 3 dimensions, the quantum state which determines the wavefunction actually must live in a Hilbert space with infinite dimension. Spinors live in finite dimensional hilbert spaces, so we already need way more than just 3 dimensions mathematically. But the actual universe these things describe is 3D (or whatever dimension you want).

    There is actually an analogous sort of hidden dimension in classical electromagnetism. The electric (and magnetic) fields are defined by giving their 3-dimensional vector at each position in space. Now if we tried to imagine that these vectors actually live in the real 3-dimensions of our universe, then I argue that according to special relativity, they should transform under lorentz transformations just like length vectors. But in reality they obey a different transformation law from length vectors. So these vectors are pointing in some 3-D space which obeys different rules from our 3-D space.

    Spin is the origin of intrinsic angular momentum. The motion of the particle through space determines its orbital angular momentum, which is not intrinsic.

    If we did it with a massive boson of spin 1, then we would see three spots on the screen instead of 2, one in the middle, corresponding to spin 0, and two to either side, one each for +1 and -1. That is what you'd normally expect from Quantum Mechanics. [However, things get changed around when we deal with relativistic particles, like the photon, which is a spin-1 boson.]
     
    Last edited: Aug 24, 2013
  15. Aug 24, 2013 #14

    Bill_K

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    :confused: When I say the electron's spin angular momentum does not arise due to its rotation, I mean like the Earth rotating on its axis. This does not happen.

    Orbital angular momentum, on the other hand, comes from the electron's motion, such as in a hydrogen atom, and this does happen. :smile: Although you can't describe this motion as a precise particle orbit, many of the states have a net "probability current" around the nucleus, and this leads to orbital angular momentum and an orbital magnetic moment.
     
  16. Aug 24, 2013 #15
    Now would it be right to imagine a changing shape but similar to earth rotating clockwise or anticlockwise or would the plane of rotation itself be inclined at different angles...ie moving hither and thither but yet a kind of rotation.
    (say like if the equatorial plane would orient itself at different angle each time wrt to some central plane).

    Also I read somewhere that electrons exhibit spin to exclude too many electrons nearby- that seems quite sensible. But that does not explain this notion of angular momentum being intrinsic, ie even when other electrons are not nearby...
    so is the orientation of orbital (ml) also driven by the desire to get rid of other electrons?
     
    Last edited: Aug 24, 2013
  17. Aug 24, 2013 #16
    Sharon25.... sounds like maybe you are thinking of the electron as in the old/outdated/inaccurate BOHR atomic model....??

    Not the best picture....see 'ORBITALS' in the next link
    http://en.wikipedia.org/wiki/Bohr_Atomic_Model



    I think of this several ways..it is not really a 'particle".....it is a wave...as in a standing [resonant wave] around a nucleus......and/or if you think of it as a point particle for interactions [like measurements] it is a POINT particle.

    See the illustrations here:

    http://en.wikipedia.org/wiki/Atomic_orbital#Orbitals_table
     
  18. Aug 24, 2013 #17
    No Naty, that was an error which I corrected after posting within a minute or two..see above
    I just mixed up "spin" and position of a particle for a while...

    Saying that electron is a wavicle is no substitute for explaining the logic behind something.
    Those links are disappointing... Please don't paste such silly links and divert the course of discussion, which in turn will prevent any sensible answers to come by

    (Here the concern is not the orbital but how the electron is spinning; if the usage rotation seems misleading please do read as spin)

    Sorry,I feel somewhat frustrated, not finding a straightforward answer for what seemed like a silly question a week ago...or maybe answers are not coming by because I have made some error here.
     
    Last edited: Aug 24, 2013
  19. Aug 24, 2013 #18
    Don't worry, if the links are "silly" everyone will ignore them; I suggest you NOT ignore them.

    They need to be read carefully....and likely some of the surrounding sections as well....The 'orbitals' illustrations show the wave like, not particle like, electron characteristics. A general example is the Schrodinger wave equation. The time evolution of quantum particles is best understood via wavelike descriptions; the interactions of particles seem best described as particle like interactions as in the Standard Model of particle physics.


    I believe the answers have been posted.

    What could be more straightforward than Bill_K's insight:

    I'm guessing right now you may still be locked into classical perspectives...like a finite size particle rotating about an axis.

    Perhaps if you reiterate your question, each followed each with several posted answers to summarize what has been provided, then you can express what you need in the way of further clarifications?? Remember, we don't know your experience nor mathematical training nor facial responses to posts.

    Maybe an insight from an eminent physicist will provide some perspective:

    Carlo Rovelli:
    What it means: Spin,for example, is a quantum state and apparently nobody knows how to clearly define it within a gravitational theory; This is why the standard model of particle physics only include special relativity, not general relativity, not gravity.

    So 'spin' is NOT a simple concept in quantum theory.

    A couple years ago I posted a question: "What is a particle?" If I recall it went on 13 pages...
    without a 'simple' answer. Here are two of my favorite replies in that discussion:

    Tom Stoer:
    What it means to me: 'particles' normally exist in a superposition of wavelike states....reread my comments on 'orbitals' earlier in this post. This means trying to visualize electron spin via classical analogies has many pitfalls.

    A related description from another forum expert:

    Marcus:
    What it means to me: So we can't observe such a wave, but an atomic nucleus with orbiting electrons knows exactly the spin characteristic of every particle...and force carriers as well!! That is mind boggling!!
     
  20. Aug 25, 2013 #19

    strangerep

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    It helps to understand the distinction between orbital and intrinsic angular momenta from a purely classical perspective. I wrote a bit about this in an older thread: Spin 4-vector
    (See post #51.)

    In particular, try to see how intrinsic angular momentum relates to a limit as the point at which the energy-momentum tensor is measured approaches the particle's world line. This is connected with Bill_K's emphasis about how "the electron does not rotate...".

    In a nonrelativistic context, it's sensible to consider the two concepts (orbital and intrinsic angular momenta) separately, but in a fully relativistic context they no longer remain invariantly separate concepts. Rather, they get mixed together under Lorentz transformations in general. In that case, it's better to speak only of "angular momentum".

    If you were to provide more information about your educational background and which textbooks you have consulted, we might be able to pitch the answers at the right level ('cause I'm guessing my pitching level above is still wrong).
     
  21. Aug 25, 2013 #20
    Naty, I am quite familiar with the idea presented in reply #18. I asked the question keeping that in mind. But thanks for that long reply
    Suppose we are observing an electron in a atom, ie now its a particle...so now would it be spinning about in different planes.?(Here again I am not assuming that it has a definite shape or size).I don't think I can put it more clearly.Now do point out if there is something wrong about this visualization..

    I am just repeating something which I asked before-

    I read somewhere that electrons exhibit spin to get rid of other electrons nearby. That seems quite sensible;But that doesnt explain "intrinsic angular momentum" ie when no other electron is nearby..?

    Is the orientation of the orbital also driven by the desire to exclude other electrons?
     
    Last edited: Aug 25, 2013
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