As we have discussed in lecture, there is a close relationship between the measurement of position and momentum and the mathematical concept of the Fourier Transform. Many of the known mathematical properties of the Fourier Transform correspond directly to physical concepts which we have only yet been able to discuss qualitatively. To be able to discuss these physical concepts on a quantitative level, we must develop a working knowledge of the Fourier transform. The purpose of this problem is to develop just such knowledge. Fourier transforms play a key role in nearly all area of physics. It will be easier for you to remember their properties when you have the physical insights which this and the next problem develop.
The operation of the Fourier transform we shall write as
The inverse relationship is
For this problem you will consider a wave packet where
