symmetric wavefunction
A symmetric wavefunction is a multiparticle state that comes back completely unchanged when you exchange two of the particles. Write down where particle one is and where particle two is, swap the two labels, and the whole expression reads exactly the same as before — same value, same sign. This is the exchange behaviour that defines bosons, and it is built by adding the two label arrangements together rather than subtracting them.
Constructed this way, the wavefunction treats both particles even-handedly: there is no first or second, only the shared pattern. A simple example with two particles in states labelled a and b combines 'particle one in a, particle two in b' with 'particle one in b, particle two in a' using a plus sign. The result has no memory of which particle started where, exactly as indistinguishability requires.
The plus sign carries a physical flavour all its own. Symmetric combinations actually enhance the chance of finding the two particles close together or in the very same state, compared with treating them as distinguishable. This subtle bunching is not a force in the usual sense, yet it shapes real phenomena — it is the seed of stimulated emission in lasers and of the dramatic crowding seen in Bose–Einstein condensates.
The plus sign makes the state identical under a swap — the signature of bosons.
When the two single-particle states are the same, the symmetric form is still perfectly allowed for bosons. It is only the antisymmetric form that collapses to zero in that case, which is why fermions, not bosons, obey exclusion.