Introduction to Mathematical Physics/N body problem in quantum mechanics/Exercises

Exercice:

Give the name of the symmetry group molecule $NH_{3}$ belongs to.

Exercice:

Parametrize the vibrations of molecule $NH_{3}$ and expand it in a sum of irreducible representations.

Exercice:

Propose a basis for the study of molecule $BeH_{2}$ different from those presented at section secnnne.

Exercice:

Find the eigenstates as well as their energies of a system constituted by an electron in a square box of side $a$ (potential zero for $-a/2<x<a/2$ and $-a/2<y<+a/2$, potential infinite elsewhere). What happens if to this potential is added a perturbation of value $\epsilon$ on a quarter of the box ($0<y<a/2$ and $0<y<a/2$)? Calculate by perturbation the new energies and eigenvectors. Would symmetry considerations have permitted to know in advance the eigenvectors?