Indistinguishable Particles

When making a Measurement, we obtain one of the associated eigenvalues. Since particles are indistinguishable, we cannot tell apart $|ab⟩$ from $|ba⟩$. Describes as a Wave function, we have that $\psi (a,b)$ and $\psi (b,a)$ represent the same physical environment, meaning that

$$\begin{array}{rll}\beta |ab⟩+\gamma |ba⟩& =\psi (a,b)& \text{by indistinguishability}\\ & =\alpha \psi (b,a)& \text{where ∣α∣=1}\\ & =\alpha (\beta |ba⟩+\gamma |ab⟩)& \\ & =\alpha \beta |ba⟩+\alpha \gamma |ab⟩& \end{array}$$

From matching coefficients we get $\beta =\alpha \gamma$ and $\gamma =\alpha \beta$, which implies that $\alpha =\pm 1$. This property defines Bosons and Fermions:

$$\{\begin{array}{cccc}\alpha & =& 1& \text{Bosons}\\ \alpha & =& -1& \text{Fermions}\end{array}$$