4. In this question you should use the properties of the Gaussian probability distribution given in Problem 1 above. Consider a particle of mass m described by the wave function 111m 2 Nexp {g/2:12), (4} where a is a constant length. a} Use the properties of the Gaussian probability distribution to conrm that the expectation values of the position and square of the position are (12 {I} = (l and {IQ} = 3. [5} b] Show, without lengthy calculation, that the expectation values of the momentum and the square of the momentum are g (1)):0 and (p2) = . (63 c} Hence, show that the uncertainty in position, AI, and the uncertainty in momentum, A-p, for this particle obey the Heisenberg uncertainty relation 11 Amp = E. [7}