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

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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?