1. Solve the quantum harmonic oscillator using the analytical method.
2. Show that $[\hat{x}, \hat{p}] = i\hbar$.
3. Show that $\hat{H} = \hbar\omega (a_+ a_- + \frac{1}{2})$.
4. Show that $[a_+, a_-] = 1$.
5. Show that the action of the lowering operator is such that $\hat{H} [a_- \phi_n] = (E_n - \hbar\omega) [a_- \phi_n]$.
6. Determine the first and second normalized excited states of the quantum harmonic oscillator.
7. In what way can we correlate the solution of the Schrödinger equation for the free particle with Heisenberg's uncertainty principle?