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Spinors and spatial coordinates

  1. Mar 23, 2009 #1
    I was reading a web page (http://electron6.phys.utk.edu/qm1/modules/m12/spinor.htm) that claims that the state vector of a spin-1/2 particle is completely specified by a two-component spinor, just as the state vector of a spinless particle is completely specified by its components in position space.

    This confused me greatly - it seems to me that the complete description of a spin-1/2 particle would be a state vector with components in the tensor product of both the "position space" and the "spin space" (the latter being two-dimensional). How do you get a "complete description" by collapsing all this information into a spinor with two complex components, and how do you extract the spatial coordinates of the particle from the spinor? By "spatial coordinates," I mean the components of the particle's state vector in that tensor product space I mentioned.

    I'm trying to understand this as a step toward understanding the Dirac and Pauli equations. It seems very straightforward that the Schrodinger equation generates a time evolution of the position-dependent wavefunction for a spinless particle, but it's not at all clear to me how these former two equations generate such a time evolution for a particle with spin when they seem to concern only the spin components of the wavefunction, and not the spatial components.
     
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  3. Mar 23, 2009 #2

    George Jones

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    I don't see where the web page says this.
    The web page says this, too.
     
  4. Mar 23, 2009 #3
    It says that in the "notation" section, right after the sentence with the green words "two component spinor."

    So I'm still wondering about my original questions.
     
  5. Mar 23, 2009 #4

    George Jones

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    The author has just been a bit careless; the page should read

    "For a spin ½ particle [itex]| \psi >[/itex] is completely specified by the position-dependent two component spinor, just as for a spinless particle [itex]| \psi >[/itex] is specified by the wave function [itex]\psi (r) [/itex]."

    From what is written above this, this is clearly what the author means. Note that in the first paragraph, state space is defined as a tensor product.
     
  6. Mar 23, 2009 #5
    I see. Are the components of the spinor just complex numbers, or are they themselves vectors? (the latter would make more sense to me, given the state space)
     
  7. Mar 23, 2009 #6

    Ben Niehoff

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    The components of the spinor are functions.

    Essentially, the wavefunction is a spinor-valued function of position.
     
  8. Mar 23, 2009 #7
    Ah, OK - thanks, Ben and George.
     
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