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Homework Help: Which is pion(0) makeup?

  1. Mar 18, 2006 #1
    In my book (Greiner's Quantum Mechanics, vol2, symmetries) says after calculation with Clebsch-Gordan coefficients,

    [tex]|\pi^0\rangle = \frac 1 2 (u\uparrow \overline{u} \downarrow + d\uparrow \overline{d} \downarrow - u\downarrow \overline{u} \uparrow - d\downarrow \overline{d} \uparrow ),[/tex]

    And I confirmed it.

    Ignoring spin, It's [tex]|\pi^0\rangle = \frac 1 {\sqrt{2}} (u \overline{u} + d \overline{d})[/tex]

    However, some sites denote, it's the same: the sum (ex, this one, but others denote it's different; minus sign (ex. this one).

    I wonder which is correct. I konw Internet resource is sometimes incorrect. And the latter is wiki... Or they mean the same? because [tex]u \overline{u}[/tex] and [tex]d \overline{d}[/tex] are orthogonal.

    But if I do [tex]\hat{T_-}|\pi^+}\rangle = \hat{T_-}u\overline{d} = \frac 1 {\sqrt{2}} (u\overline{u} + d\overline{d}),[/tex] only plus is correct. (phase is ignored and each hand is normalized.)

    So will anyone give me any hint which is correct or both are correct? I think at least plus is correct.

    Thanks in advance!
    Last edited: Mar 19, 2006
  2. jcsd
  3. Mar 19, 2006 #2

    Meir Achuz

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    That sign is different in different books.
    Many times the sign doesn't matter in a particular calculation if yhou are consistent throughout.
    I think the minus sign is more common.
    That is related to the common usage that charge conservation turns the d into -dbar (if I remember it correctly).
    The Ispin doublet for the antiquarks is (ubar,-dbar).
    Just be consistent in whatever you do.
    I think the sign is only important in getting the G parity right.
    Last edited: Mar 19, 2006
  4. Mar 19, 2006 #3
    Thank you for the reply. I don't understand exactly what you mean right now (why ispin doublet for the antiquarks is [tex](\overline u,-\overline d)[/tex]. In my understanding, it's [tex](\overline u, \overline d)[/tex] and it must make difference), but I'll proceed keeping it in mind, because I don't have a book denoting minus explicitly...

    Last edited: Mar 19, 2006
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