Question

We send in a particle of mass m from x=∞ towards x=0. This is described by the localized incident wave-packet Ψinc(x,t) given by
Ψinc(x,t)=∫∞0dkf(k)e−ikxe−iE(k)t/ℏ,for x>R,

where f(k) is a real function whose magnitude peaks sharply about k=k0>0. As usual, E=ℏ2k22m. This incoming wave encounters some kind of impenetrable barrier in the region x∈[0,R] and a reflected, outgoing packet is produced of the form
Ψout(x,t)=−∫∞0dkf(k)e2iδ(k)eikxe−iE(k)t/ℏ,for x>R,

where δ(k) is a calculable function of k called a phase shift (and is not the Dirac delta function).

Use the stationary phase approximation to find the relation between x and t that describes the motion of the “peak" of the incoming packet Ψinc(x,t).

Write your answer in terms of t, m, k_0 for k0 and hbar for ℏ as necessary.

x(t)=

Answer 1

Answer:

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