Quantum Mechanics: Concepts and Applications · Problem 8.14

Approximation Methods for Stationary States — Nouredine Zettili

Physics — Quantum Mechanics · two-fermions-infinite-well-given-wavefunction

Problem

Consider a system of two noninteracting identical spin

1/2

particles (with mass

m

) that are confined to move in a one-dimensional infinite potential well of length

L

V(x) = 0

for

0 < x < L

and

V(x) = \infty

for other values of

x

. Assume that the particles are in a state with the wave function

\Psi(x_1, x_2) = \frac{\sqrt{2}}{L}\left[\sin\left(\frac{2\pi x_1}{L}\right)\sin\left(\frac{5\pi x_2}{L}\right) + \sin\left(\frac{5\pi x_1}{L}\right)\sin\left(\frac{2\pi x_2}{L}\right)\right]\chi(s_1, s_2)

where

x_1

and

x_2

are the positions of particles 1 and 2, respectively, and

\chi(s_1, s_2)

is the spin state of the two particles. (a) Is

\chi(s_1, s_2)

going to be a singlet or triplet state? (b) Find the energy of this system.