Compton Wavelength of Proton

Theorem

The Compton wavelength of the proton is:

\(\ds \lambda_{\mathrm p}\)

\(\approx\)

\(\ds 1 \cdotp 32140 \, 98553 \, 9(40) \times 10^{-15} \, \mathrm m\)

\(\ds \)

\(\approx\)

\(\ds 1 \cdotp 32140 \, 98553 \, 9(40) \times 10^{-13} \, \mathrm {cm}\)

Symbol

$\lambda_{\mathrm p}$

The symbol for the Compton wavelength of the proton is $\lambda_{\mathrm p}$.

The $\LaTeX$ code for \(\lambda_{\mathrm p}\) is \lambda_{\mathrm p} .

Proof

By definition, the Compton wavelength $\lambda_{\mathrm p}$ of a proton is given as:

$\lambda_{\mathrm p} = \dfrac h {m_{\mathrm p} c}$

where:

$m_{\mathrm p}$ denotes the mass of the proton

$h$ denotes Planck's constant

$c$ denotes the speed of light.

Then we have:

\(\ds h\)

\(=\)

\(\ds 6 \cdotp 62607 \, 015 \times 10^{-34} \, \mathrm {J \, s}\)

that is: joule seconds

\(\quad\) Definition of Planck's Constant

\(\ds m_{\mathrm p}\)

\(\approx\)

\(\ds 1 \cdotp 67262 \, 19236 \, 9(51) \times 10^{-27} \, \mathrm {kg}\)

that is: kilograms

\(\quad\) Mass of Proton

\(\ds c\)

\(=\)

\(\ds 299 \, 792 \, 458 \, \mathrm {m \, s^{-1} }\)

that is: metres per second

\(\quad\) Definition of Speed of Light

\(\ds \leadsto \ \ \)

\(\ds \lambda_{\mathrm p}\)

\(\approx\)

\(\ds \dfrac {6 \cdotp 62607 \, 015 \times 10^{-34} } {1 \cdotp 67262 \, 19236 \, 9(51) \times 10^{-27} \times 299 \, 792 \, 458} \, \mathrm m\)

\(\ds \)

\(\approx\)

\(\ds 1 \cdotp 32140 \, 98553 \, 9(40) \times 10^{-15} \, \mathrm m\)

\(\quad\) by calculation

\(\ds \)

\(\approx\)

\(\ds 1 \cdotp 32140 \, 98553 \, 9(40) \times 10^{-13} \, \mathrm {cm}\)

\(\quad\) by calculation

$\blacksquare$

Sources

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of Mathematical Functions ... (previous) ... (next): $2$. Physical Constants and Conversion Factors: Table $2.3$ Adjusted Values of Constants

which gives the mantissa as $1 \cdotp 321 \, 440 \, 9$ with an uncertainty of $\pm 90$ corresponding to the $2$ least significant figures