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

Exercice:

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

Exercice:

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

Exercice:

Propose a basis for the study of molecule ${\displaystyle BeH_{2}}$ 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 ${\displaystyle a}$ (potential zero for ${\displaystyle -a/2 and ${\displaystyle -a/2, potential infinite elsewhere). What happens if to this potential is added a perturbation of value ${\displaystyle \epsilon }$ on a quarter of the box (${\displaystyle 0 and ${\displaystyle 0)? Calculate by perturbation the new energies and eigenvectors. Would symmetry considerations have permitted to know in advance the eigenvectors?