Math solution De-Broglie wavelength

Problem

Calculate the de Broglie wavelength of an electron moving with a velocity of $1\times 10^6\text{m/s}$. The mass of an electron $𝑚$ is $9.11\times 10^{-31}\text{kg}$.

Solution

1. Identify the given values:

   • Velocity of the electron, $𝑣=1\times 10^6\text{m/s}$
   • Mass of the electron, $𝑚=9.11\times 10^{-31}\text{kg}$
   • Planck's constant, $ℎ=6.626\times 10^{-34}\text{Js}$

2. Calculate the momentum $𝑝$:

   $$𝑝=𝑚𝑣=(9.11\times 10^{-31}\text{kg})\times (1\times 10^6\text{m/s})$$
   $$𝑝=9.11\times 10^{-25}\text{kg}\cdot \text{m/s}$$

3. Substitute the values into the de Broglie wavelength formula:

   $$𝜆=\frac{ℎ}{𝑝}=\frac{6.626\times 10^{-34}\text{Js}}{9.11\times 10^{-25}\text{kg}\cdot \text{m/s}}$$

4. Perform the calculation:

   $$𝜆\approx 7.27\times 10^{-10}\text{m}$$

So, the de Broglie wavelength of the electron moving with a velocity of $1\times 10^6\text{m/s}$ is approximately $7.27\times 10^{-10}\text{meters}$, or 0.727 nanometers.

Conclusion

By following these steps, you can use the de Broglie wavelength formula to find the wavelength associated with any particle given its mass and velocity.