## Fourier Transform of Cosine and Sine

• Periodic Systems 2022
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.