Consider the Hamiltonian function for the harmonic
oscillator:
$$H = \frac{p^2}{2m} + \frac{kq^2}{2}$$
Prove that the transformation
$$Q = p + iaq$$
$$P = \frac{p - iaq}{2ia}$$
It is canonical. What is the form of the Hamiltonian function
of the harmonic oscillator when we carry out this
transformation? Is this transformation more convenient or
less convenient than the following transformation for the
harmonic oscillator?
$$q = \frac{P}{m\omega} sinQ$$
$$p = P cosQ$$