Δκ of a superposition of waves

[tjkubo]

I am trying read through a chapter on properties of matter waves in Eisberg & Resnick's Quantum Physics. In section 3-4, a superposition Ψ of 7 sinusoidal waves, each with a different reciprical wavelength and amplitude, is shown along with all the component waves(fig. 3-9). He defines the extent of the group Δx as the maximum amplitude to half-maximum amplitude width of Ψ and estimates that it is about 1/12, which I understand from looking at the figure. However, we then defines Δκ as "the range of reciprical wavelengths of the components of Ψ from maximum amplitude to half-maximum amplitude" and estimates that it is about 1. I don't quite understand this definition, or how he estimated Δκ from the figure. Can someone carefully explain what he's doing?

 

[marcusl]

I don't have your book so I can make only general comments. It is established from the theory of Fourier transforms that an uncertainty relation exists between the deviation of a variable and that of its spectrum,
$$\Delta x \Delta k \geq \frac{1}{2}$$
If you define dx as the ratio of FWHM to peak amplitude (the inverse of what you said), then dk should be defined the same way and not as you wrote. For dx=1/12, we'd then expect dk=6.