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Baryon Predictions. Wave functions of Baryons. Baryon Magnetic Moments Baryon masses. Need to explain Parity and Charge Conjugation. Hadrons Magnetic moments. m q related to the intrinsic spin S of the quark. m = (q/mc) S and therefore for each spin-up quark:.

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Baryon Predictions

  • Wave functions of Baryons.

  • Baryon Magnetic Moments

  • Baryon masses.

  • Need to explain Parity and Charge Conjugation


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Hadrons Magnetic moments

  • mq related to the intrinsic spin S of the quark.

  • m=(q/mc)S

  • and therefore for each spin-up quark:

Spin down just changes the sign


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Hadron Magnetic moments

  • Need a particles which are long-lived and have some intrinsic spin. Proton!

Total Magnetic Moment should equal the vector sum of the

magnetic moments of the constituent quarks.

Reminder: The order of the spin arrows designates which quark has that spin.


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Hadron Magnetic moments

Doing the calculation for the first term:

So we expect mproton to be:


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Hadron Masses

  • Seems Simple enough

    • Just add up the masses of the quarks

      • Mp = Mu + Md = 2*Mu = 620 MeV/c2

      • Experimentally  Mp = 139 MeV/c2

        • What????

p+ is |u, d-bar>.

This is a particle made up of two like-sign charged quarks.

Why doesn’t it fly apart?

Strong Nuclear Force!


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Hadron Masses

Electromagnetic Force

Hyperfine splitting in hydrogen atom:

Caused by the spin of the electron interacting with the spin of the proton

Strong Nuclear Force!


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Hadron Masses

Strong Nuclear Force!

Masses are more equal, Force is much more powerful.

Fit to some meson masses and find

As = 160*(4pmu/h)2 MeV/c2

S1•S2 Meson Calculated Observed

p 140 138

r 780 776

K 484 496

K* 896 892


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Hadron Masses

Amazingly we can take the meson mass formula as the lead for

estimating baryon masses:

Fit to some baryon masses and find

As’ = 50*(4pmu/h)2 MeV/c2

Caution: There are tricks you need in order to calculate those spin

dot products. Example: if all masses are equal (proton, neutron):

Again see Griffiths, page 182.


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More Conserved Stuff

  • We need to cover some more conserved quantum numbers and explain some notation before moving on.

  • Parity and Charge Conjugation:

    • Parity Y(x,y,z)Y(-x,-y,-z) notreflection in a mirror!

    • Define the parity operator ‘P’ such that:

      • P | Y(x,y,z)> = | Y(-x,-y,-z)>

      • |> is an eigenstate of P if P|> = p|>

      • P2|> = p2|> = |> so p = 1

      • Parity is a simple group. Two elements only.


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Eigenstates of Parity

  • Suppose we have a force that only acts radially between two particles.

    • Then the wave function Y = y(r)yqyqbar

      • P | yq>  | yq> = -P| yqbar>

      • Parity is a Multiplicative quantum number, not additive.

        • Given q1 and q2

        • J = S1 + S2

        • P = P1*P2


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Eigenstates of Parity

  • For once, Baryons are easy!

    • For Mesons with no ang. Momtenum

      • P|Yb>|Ybbar> = -1 |Yb>|Ybbar>

    • DEFINE: P |Yb> 1 P |Ybbar> -1

    • So in general, for baryons with orbital angular momentum between the quarks:

      • P |Yb> = (-1)l |Yb>

  • Unfortunately, because baryon number is conserved anyway this relation is essentially useless.


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Eigenstates of Parity

  • y(r) can be separated into the angular part Ylm(,) and a purely radial part so:

    • y(r) = (r) Ylm(,) space-part of wave function

    • P Ylm(,) = (-1)l Ylm(,)

    • And P| Y > = (-1)l pq pqbar| Y > = (-1)l(1)(-1)| Y >

    • P| Y > =(-1)l+1 | Y >

      • For MESONS only (since pq=1, pqbar=-1)


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Charge Conjugation

  • Cis an operator which turns all particles into antiparticles:

    • C|q> = |q-bar>

      • changes sign of charge, baryon #, flavour quan. Num.

      • Leaves momentum, spin, position, Energy unchanged.

  • Most particles are NOT eigenstates of C

    • C|Y>  a|Y> (where a = number)

    • eg.


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Charge Conjugation

  • Neutral Mesons are eigenstates of C

|Y>=|Y(space)>|Y(spin)>|q,qbar>

If we apply C to the diagram on the left we change nothing but the ‘particleness’.

This doesn’t effect |q,qbar>

but has the same effect on |Y(space)> as if we’d used the parity operator.

C |Y(space)> = (-1)l+1 |Y(space)>


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Charge Conjugation

  • Neutral Mesons are eigenstates of C

|Y>=|Y(space)>|Y(spin)>|q,qbar>

If we apply C|Y(spin)> what do we get?

Lets try this on a S=1 or 0 meson |ms> = |0>

C |Y(spin)> = (-1)s+1 |Y(spin)> so

C |Y> = (-1)l+s |Y> neutral mesons only


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Conserved by Strong force:

  • Isospin, Quark Flavor

    • (I, I3, U, D, S, C, B, and T)

  • Parity

  • Charge Conjugation

  • Electric Charge

  • Energy/momentum

  • Angular Momentum / Spin


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