Problem: Consider a free-particle wave packet Ψ(x) = ∫A(k)ekxdk with momentum amplitude A(k) = Cexp[- |k - k0|/Λ] where C, k0 and Λ are constants. In the following assume that t = 0.
1) Determine the wave function Ψ(x) and the value of C so that Ψ is normalized. Hint: let p = k - k0 and consider the integral over regions were p is positive and negative.
2) Determine the expectation values
, , , , and and the uncertainty product ΔpΔx.
3) What is the probability that the particle has momentum in the range p and p + dp? Provide a sketch of the appropriate momentum probability distribution.
4) What is the probability that the particle is in the region between x and x + dx? Provide a sketch of this distribution. How is the width of this distribution related to that of part (c)? Is this relation between the widths of a general nature?
5) How must A(k) be modified so that = x0?
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