Quantum Mechanics: Concepts and Applications · Problem 8.15

Approximation Methods for Stationary States — Nouredine Zettili

Physics — Quantum Mechanics · two-fermions-harmonic-oscillator-given-wavefunction

Problem

Consider a system of two noninteracting identical spin 1/21/2 particles (with mass mm) that are confined to move in a common one-dimensional harmonic oscillator potential. Assume that the particles are in a state with the wave function Ψ(x1,x2)=2πx02(x2−x1)exp⁡(−x12+x222x02)χ(s1,s2)\Psi(x_1, x_2) = \frac{\sqrt{2}}{\sqrt{\pi}x_0^2}(x_2 - x_1)\exp\left(-\frac{x_1^2 + x_2^2}{2x_0^2}\right)\chi(s_1, s_2) where x1x_1 and x2x_2 are the positions of particles 1 and 2, respectively, and χ(s1,s2)\chi(s_1, s_2) is the spin state of the two particles. (a) Is χ(s1,s2)\chi(s_1, s_2) going to be a singlet or triplet state? (b) Find the energy of this system.

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