(Solved): Write down the Hamiltonian of a 1-D simple harmonic oscillator. Consider linear transformation of ...

Write down the Hamiltonian of a 1-D simple harmonic oscillator. Consider linear transformation of position and momentum: \[ \sqrt{\omega} Q=\alpha p+\beta \sqrt{\omega} q ; \quad P=\gamma p+\delta \sqrt{\omega} q \] where \( \alpha, \beta, \gamma \), and \( \delta \) are constants (which can be complex) and \( \omega \) the angular frequency of the oscillator. Which canonical transformation (i) would leave the Hamiltonian unchanged? Find a geometrical picture for this transformation.