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Question - 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.
a) 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.
b) Determine the expectation values <p>, <p2>, <E>, <x>, and <x2> and the uncertainty product ΔpΔx.
c) 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.
d) 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?
e) How must A(k) be modified so that <x> = x0?
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