The reduced Compton wavelength (\(\={\lambda }\)) of a particle represents its mass scale in quantum mechanics and is defined as:
\(\={\lambda }=\frac{\hbar }{mc}\)
where \(\hbar \) is the reduced Planck constant, \(m\) is the particle’s rest mass, and \(c\) is the speed of light. Because mass sits in the denominator, a heavier particle yields a smaller reduced Compton wavelength. [1, 2]
The latest CODATA recommended values for the proton and neutron are: [1, 2]
| Particle | Constant | Value (SI) |
|---|---|---|
| Proton | Reduced Proton Compton Wavelength (\(\={\lambda }_{p}\)) | \(2.10308910051 \times 10^{-16} \text{ m}\) |
| Neutron | Reduced Neutron Compton Wavelength (\(\={\lambda }_{n}\)) | \(2.1001941520 \times 10^{-16} \text{ m}\) |
Key Takeaways
- Neutrons are heavier, so their wavelength is shorter: Because the rest mass of a neutron is slightly greater than that of a proton (\(m_n > m_p\)), its reduced Compton wavelength is correspondingly smaller (\(\bar{\lambda}_n < \bar{\lambda}_p\)). [1, 2, 3]
- Scale context: Both wavelengths are on the order of \(0.21\text{ femtometers (fm)}\) (or \(10^{-15}\text{ m}\)). This is roughly \(4\) to \(5\) times smaller than the actual physical charge radius of a nucleon (\(\sim 0.84\text{–}0.88\text{ fm}\)).
Look at these values in a specific quantum mechanics equation (like the Dirac or Klein-Gordon equations) and calculate a Compton scattering wavelength shift. [1, 2]