 #1
Rafaelmado
 2
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 TL;DR Summary

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) φ_{0}(x') and the first excited state (n = 1) φ_{1}(x') are given by φ_{0}(x') = 1/sqrt(sqrt(π)R) e^{−x'2/(2R²)} , φ_{1}(x') = sqrt(2/sqrt(π)R³)x'e^{−x'2/(2R²)}, with R = sqrt(ħ/(mω)). You can use a table of ClebschGordan coefficients.
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) φ_{0}(x') and the first excited state (n = 1) φ_{1}(x') are given by φ_{0}(x') = 1/sqrt(sqrt(π)R) e^{−x'2/(2R²)} , φ_{1}(x') = sqrt(2/sqrt(π)R³)x'e^{−x'2/(2R²)}, with R = sqrt(ħ/(mω)). You can use a table of ClebschGordan coefficients.