Answer to Question #178759 in Quantum Mechanics for samsan

Question

Fermions and Bosons There is a fundamental difference between fermions and bosons which is responsible for the construction eigenstates of atoms with several electrons. Here we study a very simple case. For a single particle with mass m in a one-dimensional box-potential a < x < a, the energy eigenvalues are En = E1 *n^2, n = 1, 2 ,3, with E1=hbar^2pi^2/(8ma^2) and corresponding eigenfunctions φn(x).

Consider now the ground state for three identical non-interacting spinless particles with coordinates x1, x2, and x3, respectively.

(a) What are the symmetry properties of the ground state wavefunctions φg for bosons or fermions? Express these properties for the three-particle wavefunction φg(x1,x2,x3).

(b) Write down the values of ground state energy Eg for bosons and fermions.

(c) Write down the ground state wavefunction φg(x1,x2,x3) for bosons and fermions.

Answer 1

(a) Fermions are usually associated with matter, whereas bosons are generally force carrier particles

Particles with spins that come in half-integer multiples (e.g., ±1/2, ±3/2, ±5/2, etc.) are known as fermions; particles with spins in integer multiples (e.g., 0, ±1, ±2, etc.) are bosons. There are no other types of particles, fundamental or composite.

(b) The ground state energy for bosons and fermions is-

For bosons- $E=\dfrac{\bar{h}^2\pi^2}{8ma^2}$

For Fermions- $E=\dfrac{\bar{h}^2\pi^2}{8ma^2}$

(c) The ground state wave function for bosons and fermions is-

$\Psi=Asin(\dfrac{\pi \lambda}{x})$