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Homework Help: Proving that an alpha particle is a boson

  1. May 9, 2014 #1
    1. Is there any way to prove that the alpha particle is a boson (its total wave function is symmetric), given that it's made up of two protons (fermions) and two neutrons (fermions)?

    2. Relevant equations
    The total wave function for two identical particles that are
    (bosons) ψ_tot = 1/√2 * (ψ_a (particle 1) ψ_b (particle 2) + ψ_b (particle 1) ψ_a ( particle 2))
    (fermions) ψ_tot = 1/√2 * (ψ_a (particle 1) ψ_b (particle 2) - ψ_b (particle 1) ψ_a ( particle 2)),
    where "particle 1" and "particle 2" designate the coordinates of each particle, and
    a,b designate states of each of the particles.

    3. The attempt at a solution

    For a single alpha particle,
    ψ = P*N, where P is the wave function for the two protons and N is the wave function for two fermions.
    P=1/√2 * (P_a(Proton 1)P_b(Proton 2) - P_b (Proton 1) P_a (Proton 2))
    N= 1/√2 * (N_c(Neutron 1)N_d(Neutron 2) - N_d (Neutron 1) N_c(Neutron 2)),
    where a,b describe the states for each of the two protons
    and c,d does the same for each of the two neutrons.

    For a pair of alpha particles,
    ψ_tot = 1/√2 * (ψ_{abcd} (Alpha particle 1) * ψ_{efgh} (Alpha particle 2) \pm ψ_{efgh} (Alpha particle 1) * ψ_{abcd} (Alpha particle 2)},
    and our goal is to know whether we should use the plus sign (if the alpha particle is a boson) or the minus sign (if the alpha particle is a fermion).

    At this point I'm stuck. How can we find out which sign to use? I appreciate any help!
    Last edited: May 9, 2014
  2. jcsd
  3. May 9, 2014 #2
    Do it step by step. Show that combining two fermions you get a boson and combining two bosons you get another boson and combining a boson with a fermion produces a fermion. Than show by iteration that the combination of multiple particles will be a fermion if and only if (iff for mathematicians) it contains an odd number of fermions inside. Finally count the total number of fermions inside an alpha particle to figure out whether or not it is a fermion.
  4. May 9, 2014 #3
    So take two fermions. Together, they can form a particle described by the wave function, ψ:
    What I need to do is show that this composite particle of two fermions is a boson.

    I see how this is a simplified version of my original problem, but I don't see how I can use this new wave function to show that the composite particle is a boson.
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