3.
(a) Evaluate
(i) [tex]\left[\frac{d^2}{dx^2}, x^4\right][/tex]
(ii) [tex]\left[P_x, x^n\right][/tex]

(b)
(i) In deep water, the phase velocity of water waves of wavelength [tex]\lambda[/tex] is given by [tex]V_p = \sqrt{\frac{g \lambda}{2 \pi}}[/tex]. Determine the group velocity of such waves.

(ii) Determine the energy of a system in a state [tex]\psi = \exp \left[i\left(\frac{px}{\hbar} - \omega t\right)\right][/tex].

(iii) The first excited state of a simple harmonic oscillator is described by the wave function [tex]\psi = x \exp \left(-\alpha x^2\right)[/tex]. Determine [tex]\Delta x[/tex], the uncertainty in its position in the state.