# Δκ of a superposition of waves

• tjkubo
In summary, the conversation discusses properties of matter waves in Eisberg & Resnick's Quantum Physics. The author defines the extent of a superposition as the maximum amplitude to half-maximum amplitude width and estimates it to be about 1/12. They also define the range of reciprocal wavelengths of the components and estimate it to be about 1. However, there seems to be some confusion about the definition and how it was estimated based on the figure. The uncertainty relation between the deviation of a variable and its spectrum is also mentioned.
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?

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.

## 1. What is Δκ of a superposition of waves?

The Δκ of a superposition of waves refers to the difference in wavenumbers between two or more waves that combine to form a superposition. It is a measure of the spatial frequency of the waves and is typically represented by the Greek letter delta (Δ) followed by the wavenumber symbol (κ).

## 2. How is Δκ calculated?

Δκ can be calculated by subtracting the individual wavenumbers of the waves from each other. For example, if there are two waves with wavenumbers κ1 and κ2, the Δκ would be equal to κ1 - κ2.

## 3. What does a large Δκ value indicate?

A large Δκ value indicates that the superposition of waves is made up of waves with significantly different spatial frequencies. This can result in a more complex interference pattern and can affect the overall behavior of the wave.

## 4. How does Δκ affect the amplitude of the resulting wave?

The amplitude of the resulting wave in a superposition will depend on the relative phases and amplitudes of the individual waves. However, in general, a larger Δκ value can result in a larger amplitude for the resulting wave.

## 5. Can Δκ be negative?

Yes, Δκ can be negative. This occurs when the wavenumber of one wave is larger than the other, resulting in a negative difference. This can also affect the overall behavior and interference pattern of the superposition of waves.

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