Apply the Heisenberg uncertainty principle to estimate the zero point energy for the particle in the box. 

a. First justify the assumption that Δ x ≤ a and that, as a result, Δp ≥ h/2a. Justify the statement that, if Δp ≥ 0, we cannot know that E = p² /2m is identically zero.

b. Make this application more quantitative. Assume that Δx = 0.35a and Δp = 0.35 p, where p is the momentum in the lowest energy state. Calculate the total energy of this state based on these assumptions and compare your result with the ground-state energy for the particle in the box.

c. Compare your estimates for Δp and Δx with the more rigorously derived uncertainties σ _p and σ _x of Equation (17.13).