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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.

Thanks in advance.

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