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Proving that an alpha particle is a boson
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[QUOTE="quantum_smile, post: 4742614, member: 503227"] [b]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)? [/b] [h2]Homework Equations[/h2] 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)) and (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. [h2]The Attempt at a Solution[/h2] 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! [/QUOTE]
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Proving that an alpha particle is a boson
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