The * exchange operator* acts on a function resulting in the swapping of labels of any two identical particles, i.e.

where the label refers to the -th particle.

If we apply twice on the function,

Therefore, , where is the identity operator.

###### Question

Show that the eigenvalues of are .

###### Answer

If is an eigenfunction of , then . Since , we have . As an eigenfunction must be non-zero, .

Experiment data reveals that the wavefunction of a system of two identical fermions (particles with spin ) is ** antisymmetric** with respect to label exchange (i.e. the eigenvalue is -1 when the exchange operator acts on the wavefunction), while the wavefunction of a system of two identical bosons () is

**with respect to label exchange (eigenvalue of +1). The antisymmetric property of fermion wavefunctions and the symmetric property of boson wavefunctions can be regarded as postulates of quantum mechanics.**

*symmetric*For identical fermions, . Since the way we label identical particles cannot affect the state of the system, the commutation relation between and the Hamiltonian is

This implies that, for a system of identical fermions, we can select a common complete set of eigenfunctions for and , with the eigenfunctions being antisymmetric under label exchange.

###### Question

Show that commutes with , , , , , , , , , , and .

###### Answer

Similarly, . For and , we have and respectively. So and . Since , we have . Similarly, commutes with , , and .

Next, . Expanding this equation using eq76 and eq179, and noting that and , where and , we have . It follows that and that .