# 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.

Thanks in advance.

asked Apr 25, 2019
edited Apr 25, 2019

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