5 WKB Quantum States

We now assume that the system is in a quantum regime; we will perform its semiclassical quantization a la Bohr-Sommerfeld, i.e. considering the action as an integer multiple of⁶ $\hbar$. Remebering that all the computations of the previous section are in terms of the adimensional variables defined in (14) and (15), we have

$$S \left( Q , H \right) = H ^{2} \bar{S} \left( \Theta \right) = H ^{2} \bar{S} \left( \frac{Q}{H} \right) ,$$ (25)

so that we can rewrite the Bohr-Sommerfeld quantization condition as

$$S \left( Q , H \right) = l _{\mathrm{P}} ^{2} n = n \quad , \quad n = 0 , 1 , 2 , \dots .$$ (26)