# Compton Wavelength Calculator

Created by Krishna Nelaturu
Last updated: Sep 20, 2022

Our Compton wavelength calculator can help you determine a particle's Compton wavelength. It is our intent in this article to examine Compton wavelength and its implications briefly. In particular, let's discuss:

• Compton wavelength equation.
• Compton wavelengths of electrons and protons.
• Limitations on measurement.

You will benefit from learning about the dual nature of matter, so we recommend our De Broglie wavelength calculator first.

## Compton wavelength and its equation

Compton wavelength is a quantum characteristic of a particle. It is equal to the wavelength of a photon with the same energy as the particle's rest energy.

The rest energy of a particle is given by:

$E = mc^2$

Where:

• $E$ - Rest energy of the particle;
• $m$ - Particle's mass; and
• $c$ - Speed of light.

On the other hand, the energy of a photon is given by:

$E = h \nu = \frac{hc}{\lambda}$

Where:

• $E$ - Energy of the photon;
• $h$ - Planck's constant, equal to 6.62607 × 10-34 J/s;
• $\nu$ - Frequency of the photon; and
• $\lambda$ - Wavelength of the photon.

To obtain the Compton wavelength equation, let's equate these two energies:

\begin{align*} mc^2 &= \frac{hc}{\lambda}\\[1em] \implies \lambda &= \frac{hc}{mc^2} = \frac{h}{mc} \end{align*}

Thus, we obtain the formula for Compton wavelength of any particle with mass $m$ as:

$\lambda = \frac{h}{mc}$

Notice the inverse proportionality between a particle's mass $m$ and its Compton wavelength $\lambda$. Since particles often travel at speed close to $c$, the relativistic effects begin to play an essential role. Explore this topic more, for example, in our length contraction calculator!

## Compton wavelength of protons and electrons

Let's use the Compton wavelength formula on sub-atomic particles.

Compton wavelength of protons whose mass is 1.6726 × 10-27 kg would be:

\small \begin{align*} \lambda &= \frac{h}{mc}\\[1em] &= \frac{6.62607 × 10^{-34} \text{ J/s}}{1.6726 × 10^{-27} \text{ kg} \cdot 299792458 \text{ m/s}}\\[1em] \lambda &= 0.0013214 \times 10^{-12} \text{ m} \end{align*}

Similarly, the Compton wavelength of an electron with a mass of 9.11 × 10-31 kg is:

\small \begin{align*} \lambda &= \frac{h}{mc}\\[1em] &= \frac{6.62607 × 10^{-34} \text{ J/s}}{9.11 × 10^{-31} \text{ kg} \cdot 299792458 \text{ m/s}}\\[1em] \lambda &= 2.4263 \times 10^{-12} \text{ m} \end{align*}

## Limitation on measurement

Measuring the position of a particle requires bouncing light on the particle. We need to use the light of short wavelengths to adequately capture the subatomic particles (refer to the section to see how small their Compton wavelengths are). However, the shorter the wavelength, the higher the photon's energy. This could be a problem if the photon's energy exceeds the particle's energy $(mc^2)$, as the photon-particle collision would form a new particle, effectively throwing aside the point of "measuring" the particle.

Hence the Compton wavelength indicates how feasible the task of measuring a particle is. The shorter the Compton wavelength, the more challenging this task. At this cutoff, one would need to consider quantum field theory instead of quantum theory.

## Using this Compton wavelength calculator

Finding the Compton wavelength of a particle with our Compton wavelength calculator requires only one input from you - the mass of the particle.

You can also use this Compton wavelength calculator in reverse - enter a Compton wavelength and see how massive the particle must be.

If you're in the mood for more exciting interactions between photons, electrons, and the dual nature of matter, make sure to visit our photoelectric effect calculator!

Krishna Nelaturu
Mass
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Compton wavelength
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