assignment Homework

Fourier Transform of Cosine and Sine

1. Find the Fourier transforms of \(f(x)=\cos kx\) and \(g(x)=\sin kx\).
2. Find the Fourier transform of \(g(x)\) using the formula for the Fourier transform of a derivative and your result for the Fourier transform of \(f(x)\). Compare with your previous answer.
3. In quantum mechanics, the Fourier transform is the set of coefficients in the expansion of a quantum state in terms of plane waves, i.e. the function \(\tilde{f}(k)\) is a continuous histogram of how much each functions \(e^{ikx}\) contributes to the quantum state. What does the Fourier transform of the function \(\cos kx\) tell you about which plane waves make up this quantum state? Write a sentence or two about how this makes sense.

assignment Homework

Spring Force Constant

assignment Homework

Sum Shift

In each of the following sums, shift the index \(n\rightarrow n+2\). Don't forget to shift the limits of the sum as well. Then write out all of the terms in the sum (if the sum has a finite number of terms) or the first five terms in the sum (if the sum has an infinite number of terms) and convince yourself that the two different expressions for each sum are the same:

1. \begin{equation} \sum_{n=0}^3 n \end{equation}
2. \begin{equation} \sum_{n=1}^5 e^{in\phi} \end{equation}
3. \begin{equation} \sum_{n=0}^{\infty} a_n n(n-1)z^{n-2} \end{equation}