Consider two noninteracting particles in a 1d simple harmonic oscillator (sho) potential, which has 1-particle spatial wavefunctions ψ n ( x), where n = 0, 1, 2, … (we ignore spin by assuming both particles have the same spin quantum number or are spin 0). these wavefunctions are normalized to 1 and satisfy ψ n * ( x)ψ m ( x)dx −[infinity] [infinity] ∫ = 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 the lowest energy state for the cases where they are distinguishable, bosons, or fermions. be careful about symmetry.

(b) (2 pts) what is the total energy (in terms of ! ω0 ) and degeneracy for each case?

(c) (2 pts) using the shorthand (nm) − (mn) notation for antisymmetric wavefunctions, write all possible combinations of two fermions to be in the 2nd excited state.

(d) (2 pts) what is the total energy for (c)?