Problem 1:

Determine the energies and the degeneracies of the two lowest levels of a system composed of three particles with equal masses m, where the particles are
(a)  distinguishable non-interacting spinless particles in a 3-dimensional simple harmonic potential with spring constants k_x = k_y = k_z = k.
(b)  distinguishable non-interacting spinless particles in a 3-dimensional Coulomb potential
V(x,y,z) = -Ze²/r, where r = (x² + y² + z²)^½.
(c)  indistinguishable non-interacting spin ½ particles in a 3-dimensional cubic box (with impenetrable walls) of dimensions L_x × L_y × L_z, where L_x = L_y = L_z = L.

Problem 2:

Submit this problem on Canvas as Assignment 9.  If you used an AI as a Socratic tutor, submit a copy of your session leading to your solution and reflect on your session.  If you did not need any help or worked with another student, explain your reasoning, do not just write down formulas. 

Consider a helium atom where both electrons are replaced by identical charged particles of spin quantum number s = 1.  Ignoring the motion of the nucleus and the spin-orbit interaction, the Hamiltonian is given by
H = P₁²/(2m) + P₂²/(2m) - 2e²/r₁ - 2e²/r₂ + e²/|r₁ - r₂|.
Construct an energy level diagram (qualitatively) for this "atom", when
(a) both particles are in the n = 1 state, and when
(b) one particle is in the n = 1 state and the other is in the state (n l m) = (2 0 0). 
Do this by treating the e²/|r₁ - r₂| term in the Hamiltonian as a perturbation.  Write out the space and spin wave functions for each level in terms of the single particle hydrogenic wave functions ψ_nlm and spin wave functions χ_s,ms.  Show the splitting qualitatively, and state the degeneracy of each level.  Don't forget to include the effect of the e²/|r₁ - r₂| term in your qualitative discussion.

Problem 3:

Consider an excited Nitrogen atom with a an electronic configuration (1s)², (2s)², (2p)², (3s).
Find the spectroscopic terms ^2S+1L_J that characterize the system.

Problem 4: