Question

Here, we consider integrals of the form
Ψ(x)=∫∞−∞dkΦ(k)eikx,

where Φ(k) is a function that is sharply localized around k=k0.

In each of the following cases, use the stationary phase argument to predict the location of the peak of |Ψ(x)|. Then compute the integral exactly to find Ψ(x) and |Ψ(x)|, and to confirm your prediction.

Useful Integral: Valid for complex constants a and b, with real part of a positive:
∫∞−∞e−ax2+bxdx=πa−−√exp(b24a),when Re(a)>0.


PART A

Φ(k)=Ce−L2(k−k0)2

Here L is a constant with units of length, and C is a constant that provides the needed units for Φ(k) and is required for proper normalization. We do not ask you to compute C in this problem.

Estimate the location of the maximum of Ψ(x) using the stationary phase approximation. To confirm your result, compute Ψ(x) and examine |Ψ(x)| to find the position of the peak value.

Write your answers in terms of x, L and k_0 for k0.

Ψ(x)=C _________________________________


Peak value occurs at x= _________________________

Answer 1

Explanation:

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