6. The Hamiltonian of a 1D quantum harmonic oscillator is given by $H = \hbar\omega(a^\dagger a + \frac{1}{2})$, with $\omega$ the frequency of the oscillator and $a$ ($a^\dagger$) the lowering (rising) operator. Let us indicate with $|n\rangle$, where $n = 1, 2, \dots$, the number eigenstates forming an orthonormal basis for the system. Calculate the mean energy of the oscillator and the expectation values of the operators $Q = \frac{(a + a^\dagger)}{\sqrt{2}}$ and $P = \frac{(a - a^\dagger)}{i\sqrt{2}}$ when the state of the system is $|\psi\rangle = \frac{|2\rangle + |1\rangle}{\sqrt{2}}$.
