Discusión:Quantum-classical

In the classical case, the transformation

$Q_{c}={\sqrt {\omega (t)}}q_{c}$

where the function $\omega (t)$ fulfills the differential equation

$\omega {\ddot {\omega }}-{\frac {3}{2}}{\dot {\omega }}^{2}+2\left[\omega ^{2}-\Omega ^{2}\right]\omega ^{2}=0$

remember that

$q=\rho \cos \left(\int {\frac {W}{\rho ^{2}}}dt\right)$

or equivalently

$q={\sqrt {\frac {W}{\omega }}}\cos \left(\int \omega dt\right)$

Thus, the transformation yields

$Q_{c}={\sqrt {W}}\cos \left(\int \omega dt\right)$

that corresponds to Hector eq (12) in the semiproofs with $W=1$.

linearly independent solution

The orthogonal solution from the Wronskian is then

$P_{c}={\sqrt {W}}\sin \left(\int \omega dt\right)$