Question

Prove that even number of fermions produces boson using wave function.

Answer 1

All fundamental particles in nature can be divided into one of two categories, Fermions or Bosons

Any object which is comprised of an even number of fermions is a boson.

A boson is a particle whose wave function remains the same under rotation by 360°. A fermion is a particle whose wave function changes by a factor of -1 under 360° rotation. The combination of an even number of them changes by -1 to an even power, i.e. 1, so that's a boson.

Let us assume that for bosons:

• wave-function is given as, :   
• After 360 degree rotation : ?

Let us assume that for fermion:

Let fermion particle 1, wave function given as,

• wave-function is given as :
• After 360 degree rotation : ?

Let fermion particle 2, wave function is given as,

• without any rotation :
• After rotation of 360 degree :

let as take 2 (even number of ) particle's wave function combination:

  

Here after rotation of 360 degree if two fermion particle rotate system will not change its parity.

So, if we assume that,

Wave-function of fermionic system is :

After rotation of 360 wave function of fermionic system is :

Then,

Which is same as seen for bosonic system.

Hence it is proved that even number of fermions produces boson .