Quantum-Charge Ratio

Theorem

The ratio of Planck's constant to the elementary charge is given by:

\(\ds \dfrac h \E\)

\(\approx\)

\(\ds 4 \cdotp 13566 \, 7697 \times 10^{-15}\)

joule seconds per coulombs

\(\quad\) This sequence is A343571 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

\(\ds \)

\(\approx\)

\(\ds 4 \cdotp 13566 \, 7697 \times 10^{-7}\)

erg seconds per abcoulomb

\(\ds \)

\(\approx\)

\(\ds 1 \cdotp 37951 \, 0386 \times 10^{-17}\)

erg seconds per statcoulomb

Proof

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

\(\ds \E\)

\(=\)

\(\ds 1 \cdotp 60217 \, 6634 \times 10^{−19} \, \mathrm C\)

that is: coulombs

\(\quad\) Definition of Elementary Charge

\(\ds \leadsto \ \ \)

\(\ds \dfrac h \E\)

\(\approx\)

\(\ds 4 \cdotp 13566 \, 7697 \times 10^{-15} \, \mathrm {J \, s / C}\)

\(\quad\) by calculation

\(\ds h\)

\(=\)

\(\ds 6 \cdotp 62607 \, 015 \times 10^{-27} \, \mathrm {erg \, s}\)

that is: erg seconds

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

\(\ds \E\)

\(=\)

\(\ds 1 \cdotp 60217 \, 6634 \times 10^{−20} \, \mathrm {abC}\)

that is: abcoulombs

\(\quad\) Definition of Elementary Charge

\(\ds \leadsto \ \ \)

\(\ds \dfrac h \E\)

\(\approx\)

\(\ds 4 \cdotp 13566 \, 7697 \times 10^{-7} \, \mathrm {erg \, s / abC}\)

by calculation

\(\ds h\)

\(=\)

\(\ds 6 \cdotp 62607 \, 015 \times 10^{-27} \, \mathrm {erg \, s}\)

that is: erg seconds

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

\(\ds \E\)

\(=\)

\(\ds 4 \cdotp 80320 \, 42510 \times 10^{-10} \, \mathrm {statC}\)

that is: statcoulombs

\(\quad\) Definition of Elementary Charge

\(\ds \leadsto \ \ \)

\(\ds \dfrac h \E\)

\(\approx\)

\(\ds 1 \cdotp 37951 \, 0386 \times 10^{-17} \, \mathrm {erg \, s / statC}\)

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 mantissas of these figures as:

$4 \cdotp 135 \, 708$ with an uncertainty of $\pm 14$ corresponding to the $2$ least significant figures

$1 \cdotp 379 \, 523 \, 4$ with an uncertainty of $\pm 46$ corresponding to the $2$ least significant figures