Graduate Energy Levels and Wave Functions of Identical Particle Systems

[Rafaelmado]

TL;DR

Consider a system of two noninteracting spin 1/2 identical particles moving in a common
external harmonic oscillator potential.

a) Find the energy levels of the ground state and the first excited state.
b) Find the wave functions (in the coordinate representation) of the ground state and the first excited state.
Hints: For a particle of mass m in a harmonic potential of angular frequency ω, the energy of the particle in the n = 0, 1, 2,... state is given by (n + 1/2)ħω; the wave functions for the ground state (n = 0) φ₀(x') and the first excited state (n = 1) φ₁(x') are given by φ₀(x') = 1/sqrt(sqrt(π)R) e^{−x'2/(2R²)} , φ₁(x') = sqrt(2/sqrt(π)R³)x'e^{−x'2/(2R²)}, with R = sqrt(ħ/(mω)). You can use a table of Clebsch-Gordan coefficients.

 

[PeterDonis]

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