Determine the function \psi(x).
(25 marks) Consider a wave packet defined by \(\psi(x)=\int_{-\infty}^{\infty} A(k) \cos k x d k,\) where \(A(k)\) is given by
$$ A(k)=\left\{\begin{array}{ll} 1 & \text { for } 0 \leq k \leq k_{0} \\ 0 & \text { otherwise } \end{array}\right. $$
The width of \(A(k)\) is thus equal to \(\Delta k=k_{0}\). (a) Determine the function \(\psi(x)\). (b) What is the value of \(\psi(0) .\) (c) Sketch the function \(\psi(x)\). (d) The function \(\psi(x)\) is sharply peaked at \(x=0\). Define the width of the function by \(\Delta x=2\left|x_{0}\right|,\) where \(x_{0}\) is the position where \(\psi(x)\) first vanishes. Calculate the product \(\Delta x \Delta k\). [Remark: This example illustrates the concept that the widths of a wave packet \(\psi(x)\) and that of its Fourier transform \(A(k)\) are directly related. \(]\)
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