7. Consider two noninteracting particles in a 1D simple harmonicoscillator (SHO) potential, which has 1-particle spatialwavefunctions ψ n ( x), where n = 0, 1, 2, … (we ignore spin byassuming both particles have the same spin quantum number or arespin 0). These wavefunctions are normalized to 1 and satisfy ψ n *( x)ψ m ( x)dx −∞ ∞ ∫ = 0 when n ≠m , i.e., they are orthogonal.The energies are !ω0 n + 1 2 ( ) . (a) (4 pts) Using the ψ n ( x),write the normalized wave function for two particles Ψ(1, 2) in thelowest energy state for the cases where they are distinguishable,bosons, or fermions. Be careful about symmetry. (b) (2 pts) What isthe total energy (in terms of !ω0 ) and degeneracy for each case?(c) (2 pts) Using the shorthand (nm) − (mn) notation forantisymmetric wavefunctions, write all possible combinations of twofermions to be in the 2nd excited state. (d) (2 pts) What is thetotal energy for (c)?
