Advanced Solid State Physics

3 Quantization

The first topic in this chapter is the appearance ofRabi oscillations. For understanding their origin
a few examples would be quite helpful.

Ammonia moleculeNH 3 :
The first example is the ammonia moleculeNH 3 , which is shown in fig. 1.

Figure 1: left: Ammonia molecule; right: Nitrogen atom can change the side

The nitrogen atom can either be on the "left" or on the "right" side of the plane, in which the hydrogen
atoms are. The potential energy for this configuration is a double well potential, which is shown in
fig. 2.

−3 −1.5 −1 −0.5 0 0.5 1 1.

−2.

−

−1.

−

−0.

0

0.

1

1.

2

x / a.u.

energy / a.u.

Figure 2: Double-Well Potential of ammonia molecule

The symmetric wavefunction for the ground state of the molecule has two maxima, where the potential
has it’s minima. The antisymmetric wavefunction of the first excited state of the molecule has only one
maximum in one of the minima of the potential, at the other potential minimum it has a minimum.
If the molecule is prepared in a way that the nitrogenN is on just one side of the hydrogen plane,
this is a superposition of the ground state and the first excited state of the molecule wavefunctions.
It’s imagineable that the resulting wavefunction, which is not an eigenstate, has a maximum on the
side where the nitrogen atom is prepared and nearly vanishes at the other side. But attention: The
energies of the ground state and the first excited state are slightly different. Therefore the resulting