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if i have two particles in states: |1> and |2> respectavly , and they are not identical then the combined state is |1>|2> , but if they are identical then the labels 1 or 2 are arbitrary and could be swapped so supposedly we get two options:

  • bosons = |1>|2> + |2>|1>
  • fermions = |1>|2> - |2>|1>

Why there are only these two options ? can't you have any arbitrary superposition of the two states ? (eg: a|1>|2> + b|2>|1> , a*a + b*b = 1)

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Wikipedia states:

It appears to be a fact of nature that identical particles do not occupy states of a mixed symmetry.

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