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

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Let us consider a system of 2 identical particles, 1 and 2.

Let, ψa(1) is the amplitude of finding particle 1 at state a, and ψa(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

N.F * (ψa(1)ψa(2)+ψa(2)ψa(1)) = 2 * N.F * ψa(1)ψa(2)

While in case of fermions, that amplitude is

N.F * (ψa(1)ψa(2)−ψa(2)ψa(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 ψa(1)+ψa(2) in case of fermions? If not, What is the correct equation?

b) Is it simply ψa(1)−ψa(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.

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