Quandary: Is QM Spin Angular Momentum Conserved?

Discussion in 'Physics & Math' started by Schneibster, May 1, 2016.

  1. Schneibster Registered Member

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    The story of the neutrino comes immediately to mind.

    It is often stated that neutron decay cannot result in a proton and an electron, because both have SAM 1/2, and the neutron they come from also has SAM 1/2. Therefore, there must be a neutrino to provide the extra 1/2 that allows this interaction.

    However, it is also stated that SAM is not conserved because it can only take discrete values of + and -, and discrete symmetries are not correspondent to conservation laws under Noether's Theorem, only continuous ones (i.e., differentiable, i.e. definable by a Lie group).

    Please resolve this apparent paradox for me.
     
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  3. Fednis48 Registered Senior Member

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    Good question! The key distinction here is that while the symmetry of a system must be differentiable for Noether's Theorem to apply, the conserved variable itself need not be. In the case of classical angular momentum, Noether's Theorem applies because rotation is a differentiable process; as far as I know, the differentiability of each body's angular momentum never comes into play. For spin angular momentum, the rotation symmetry is a little more abstract, but it's still there. The simplest example is for a spin-1/2 particle: we can map the Hilbert space of such a particle's state vectors to the surface of a sphere, such that zenith angle determines the probability to be in spin up vs. down and azimuthal angle determines the relative phase between the two spin states. All possible (unitary) transformations of the spin state are then described by rotations of the sphere, corresponding with the SU(2) Lie group. Higher spins aren't so easy to interpret geometrically, but I'm pretty sure any spin-s system transforms under the SU(2s+1) Lie group.
     
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  5. Schneibster Registered Member

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    Wow, thanks Fednis48, I'll have to think about that for a bit. But I do have a couple immediate questions that I think might help me understand:
    1. How does the relative phase work in this manner? And how does that link up with joint conservation of J (total momentum) and with the fact that L (OAM) + S (SAM) add up to J?
    2. Does this actually mean that spin symmetry on its own is a continuous symmetry? Or is that only in combination with the relative phase? And how does this interact with my first question above?
    3. What quantity is conserved under Noether's Theorem dual to the rotation symmetry of spin?
     
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  7. Fednis48 Registered Senior Member

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    Hmm. Those are good followups, and I'm now second-guessing my understanding a little bit. The way I explained it above would seem to imply that the rotation of the Bloch sphere (i.e. the sphere onto which we map a spin-1/2 state space) is indeed its own symmetry. But that can't be right, because spin-orbit coupling happens in atoms all the time, so L and S are not independently conserved quantities. Let me try and talk this through.

    1. My understanding of Noether's theorem can be summed up as: "If the laws of physics are invariant under a given (differentiable) transformation, then the generator of that transformation is conserved."
    2. In generalized coordinates, p is the generator of q iff \(e^{ipt}\) is a displacement in q. This means S is the generator of Bloch sphere rotations.
    3. The behavior of a spin can depend on the relative phase between spin states. For example, the states \(|\uparrow\rangle+|\downarrow\rangle\) and \(|\uparrow\rangle-|\downarrow\rangle\) in the z basis are \(|\uparrow\rangle\) and \(|\downarrow\rangle\) in the x basis, respectively. This means they will experience opposite forces when exposed to a magnetic field gradient along the x-axis.
    4. Since the "direction" of a state vector on the Bloch sphere interacts with the physical directions of magnetic fields as described in point 3, physics changes when the Bloch sphere and physical space are rotated independently.
    5. On the other hand, I strongly suspect that physics stays the same when the Bloch sphere and physical space are rotated by the same amount, although I wouldn't know how to go about showing that.
    6. If my increasingly hand-wavey beliefs are correct, then we're looking at symmetry under simultaneous rotation of the Bloch sphere and physical space. Since L generates physical space rotations while S generates Bloch sphere rotation, their sum J is what generates simultaneous rotation, so J is the conserved quantity.

    This is stretching my understanding of the connection between spin and orbital angular momentum, so I'm not sure I'll be able to give a more rigorous argument. Maybe rpenner will show up and blow my explanation out of the water.

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    As an aside, I think this question belies a subtle misunderstanding of Noether's theorem. For both orbital and spin angular momentum, the system dynamics are what's symmetric under rotation. Spin (or S plus L) is the conserved quantity associated with this symmetry.
     
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  8. Schneibster Registered Member

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    This is great stuff, Fednis48, and I will try to respond when I have more attention to give to the matter.

    Just briefly, I think that characterizing spin as S+L is incorrect; spin is only S. S+L is total momentum, or J. J is conserved, according to my understanding; however neither S nor L is since momentum can be freely interchanged between them according to the frame of reference of the observer. Please correct me if I am wrong on this point.
     
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  9. Schneibster Registered Member

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    Thanks!

    Well, you've come to the same conclusion I essentially did, so I don't think your analysis is incorrect here. But I do take issue with the statement that J is spin as I previously posted.

    Now, this can be explained if the "spin" that needs to be conserved in the neutron interaction is actually J, not S; and this would at least make some sense of the argument that one of the key quantities that dictates that neutrinos (actually electron anti-neutrinos, to preserve lepton number conservation) must be formed from the interaction to preserve conservation of total angular momentum.

    So my next question is, could it be J rather than S that is actually conserved in the neutron decay interaction?

    And could I just be looking here at a case where some popular science writer has ignored the difference between J and S?
     
  10. Fednis48 Registered Senior Member

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    Once again, a really good question! I hadn't thought about it before, but it does seem odd that the argument for neutrinos' existence seems to assume conservation of S, rather than conservation of J. Naively, it seems reasonable that an electron and a proton could combine to generate some motionally excited neutron, such that half the system's original spin angular momentum has been converted into orbital angular momentum. I have no idea why that process is never considered; maybe the energy it takes to excite a nucleus like that is just too high to get significant combination cross-sections? If another forum member has any insight, I'd be very interested.
     
  11. Schneibster Registered Member

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    Hmmm. I dimly recall something called "K capture" google google google...

    Yep. In this case, an electron is captured from the innermost electron shell by a proton and an electron neutrino is emitted, resulting in a neutron. This is followed by emission of a gamma ray, as the new nucleus returns to ground state. A number of proton-rich nuclides undergo this interaction. This was first discovered by Alvarez in 1937; the theory was worked out in 1934 by Gian-Carlo Wick, according to the Wikipedia article. Alvarez went on to win a Nobel in Physics for, among other things, inventing the bubble chamber.

    One must also remember that there is an interaction of a proton with an electron antineutrino, in which a positron is emitted and the proton converts to a neutron. This interaction is rare, but has been detected, first at Savannah River in 1956 by Reines and Cowan. Wolfgang Pauli had predicted it in 1930, and when the news was telegraphed to him in '56 he and some friends reportedly consumed a case of champagne.

    Back to our discussion of spin, it must also be remembered that protons and neutrons have L, due to the orbits of quarks inside them. Thus the nucleus can have L as well.

    Our only outstanding problems, then, are
    1. to show how L is involved in the decay of a naked neutron,
    2. what L is for the ejected electron and antineutrino in the original interaction, and
    3. what L is for the incoming antineutrino in the second type of inverse beta decay;
    we can assume without likely being in error that these other interactions all involve L in the nucleus and electron shell. Also I think that solving any one of these three will give the answer to the other two, and my intuition tells me that there will be L=0 in the three making J=S but this is only a conjecture. Can you think of anything nice and neat that might tie this all up with a bow?
     
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  12. Farsight

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    IMHO you need to backtrack a bit Schneibster. Start with the electron:

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    Remember that in atomic orbitals, electrons "exist as standing waves". So what do you think they exist as outside of an orbital? What with magnetic moment and the Einstein de Haas effect and the Poynting vector, and the wave nature of matter and the wave in the box and spin ½ spinors, you ought to be able to visualize its spin angular momentum:

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    If this electron was surrounding a proton, you'd be looking at a hydrogen atom. But this hydrogen atom is not the same as a neutron. It's different, as we know from free neutron decay. We call that difference an antineutrino. It travels at c and it has a helicity.
     
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