Answer to Question #166027 in Quantum Mechanics for khalida

Question

Find the wave function of two systems of identical, noninteracting particles: first consists of two bosons

and the second of two spin-1/2 fermions.

Answer 1

Consider two identical, non interacting particles (1 and 2) which may exist in two different states a and b.

$\Psi_I = \Psi_a(1) \Psi_b(2)$

$\Psi_{II} = \Psi_a(2) \Psi_b(1)$

If the particles are indistinguishable, then we cannot tell whether the number is in state $\Psi_1$ or $\Psi_2$ and because both states are equally likely we write the system wave function as a linear combination of $\Psi_1 and \Psi_2$ .

If the particles are bosons, the system wave function is symmetric :

$\Psi_b = \dfrac{1}{\sqrt2}[\Psi_a(1)\Psi_b(2)+\Psi(2)\Psi(1)] = \Psi_s$

If the partices are fermions, the wave function is antisymmetric

$\Psi_F = \dfrac{1}{\sqrt2}[\Psi_a(1)\Psi_b(2)-\Psi_a(2)\Psi_b(1)] = \Psi_A$