1. << Fourier Transform of a Derivative | Fourier Transforms and Wave Packets |
Find the Fourier transform of the (simplified) Gaussian function \begin{equation} f(x)=e^{-x^2} \end{equation} You may want to use the value of the following integral \begin{equation} \int_{-\infty}^{\infty} e^{-x^2}\, dx = \sqrt{\pi} \end{equation}
group Small Group Activity
5 min.
assignment Homework
assignment Homework
assignment Homework
Find \(N\).
face Lecture
30 min.
assignment Homework
Consider the following wave functions (over all space - not the infinite square well!):
\(\psi_a(x) = A e^{-x^2/3}\)
\(\psi_b(x) = B \frac{1}{x^2+2} \)
\(\psi_c(x) = C \;\mbox{sech}\left(\frac{x}{5}\right)\) (“sech” is the hyperbolic secant function.)
In each case:
assignment Homework
Remember that the delta function is defined so that \[ \delta(x-a)= \begin{cases} 0, &x\ne a\\ \infty, & x=a \end{cases} \]
Also: \[\int_{-\infty}^{\infty} \delta(x-a)\, dx =1\].