# Probability of finding a particle in a two/three particle system

Let us consider a system of 2 identical particles, 1 and 2. Let, $$ψ_a(1)$$ is the amplitude of finding particle 1 at state $$a$$, and $$ψ_a(2)$$ is the amplitude of finding particle 2 at state $$a$$. Let N.F is an arbitrary normalization factor. Now, if 1 and 2 are bosons, we know that, finding both 1 AND 2 at a has an amplitude

\eqalign{&(N.F)(ψ_a(1)ψ_a(2) + ψ_a(2)ψ_a(1)) \cr &= 2(N.F)ψ_a(1)ψ_a(2)\cr}

While in case of fermions, that amplitude is

$$(N.F)(ψ_a(1)ψ_a(2) - ψ_a(2)ψ_a(1)) = 0$$

[Pauli exclusion Principle]

Now,

1. What is the amplitude that 1 OR 2 will be state a? (Here OR means inclusive OR).

a) Is it simply $$ψ_a(1) + ψ_a(2)$$ in case of fermions? If not, What is the correct equation?

b) Is it simply $$ψ_a(1) - ψ_a(2)$$ in case of bosons? If not, What is the correct equation?

2. What would be the amplitude in case of 3 particle system? Again, we are assuming the OR case, not the AND case. Please reply

a) for fermions, and

b) for bosons.