Question

We observe 3 particles and 3 different one-particle states ψ_a(x), ψ_b(x) i ψ_c(x). How many different states that describe a system of three particles can we make:

(a) if the particles can be distinguished;

(b) if the particles are identical bosons;

(c) if the particles are identical fermions?

Note that, unlike the exercise task, there is a possibility that the particles may be in the same quantum state.

For example, ψ(x₁, x₂, x₃) = ψ_a(x₁)ψ_a(x₂)ψ_c(x₃)

Answer 1

Solution:

3 particles and 3 different one-particle states

So here ni =3 , gi =3

(i)Hence ,if the particles can be distinguished then different states that describes a system of three particles we can make is (gi)^ni=3^3=27

(ii)Hence, if the particles are identical bosons , then different states that describe a system of three particles we can make is W= (ni+gi–1)!/{ni!*(gi!–1)} = (3+3–1)!/(3!×2!) =10

(iii) Hence, if the particles are identical fermions , then different states that describes a system of three particles we can make is W = gCn =3C3 = 1.