Exercises

1. Demonstrate that the particle interchange operator, $P_{12}$ , in a system of two identical particles is Hermitian.

2. Consider two identical spin-$1/2$ particles of mass $m$ confined in a cubic box of dimension $L$ . Find the possible energies and wavefunctions of this system in the case of no interaction between the particles.

3. Consider a system of two spin-$1$ particles with no orbital angular momentum (i.e., both particles are in $s$ -states). What are the possible eigenvalues of the total spin angular momentum of the system, as well as its projection along the $z$ -direction, in the cases in which the particles are non-identical and identical?