Question 40.9: In a laboratory experiment, an electron has a speed of 3.00 ......
An Electron’s de Broglie Wavelength
In a laboratory experiment, an electron has a speed of 3.00 × 10^6 m/s. What is its de Broglie wavelength?
INTERPRET and ANTICIPATE
As in the previous example, find the electron’s de Broglie wavelength from its momentum. The electron’s speed is 0.01c—much less than the speed of light. Once again, we don’t need special relativity.
SOLVE
Find the magnitude of the momentum using Equation 10.1.
Substitute this momentum into Equation 40.18 to find the electron’s de Broglie wavelength.
\begin{aligned}\lambda & =\frac{h}{p} \quad \quad (40.18) \\\lambda & =\frac{6.626 \times 10^{-34} J \cdot s }{2.73 \times 10^{-24} kg \cdot m / s } \\ \lambda & =2.42 \times 10^{-10} m =0.242 nm\end{aligned}CHECK AND THINK
The electron’s de Broglie wavelength is about the size of an atom or of the space between atoms in a crystal. So when a beam of electrons is aimed at a crystal, the electrons form a diffraction pattern such as the one shown in Figure 40.16. In this case, the electrons behave like a wave, and we must use the wave model. In order to understand the structure of atoms (Chapter 42), we must model their electrons as matter waves.
