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Quantum Mechanics

Constants

This table gives frequently used constants for quantum mechanics. It is a study and calculation lookup table, not a metrology quotation sheet. Exact SI defining constants are marked exact; measured constants are rounded from CODATA 2022.

Convention

Section titled “Convention”

Values are in SI unless stated otherwise. The elementary charge ee is the positive charge magnitude. The electron charge is qe=−eq_e=-e.

Defining Constants

Section titled “Defining Constants”
ConstantSymbolValueStatus
Speed of lightcc299 792 458 m s−1299\,792\,458\ \mathrm{m\,s^{-1}}exact
Planck constanthh6.62607015×10−34 J s6.62607015\times10^{-34}\ \mathrm{J\,s}exact
Reduced Planck constantℏ=h/(2π)\hbar=h/(2\pi)1.054571817…×10−34 J s1.054571817\ldots\times10^{-34}\ \mathrm{J\,s}derived from exact hh
Elementary charge magnitudeee1.602176634×10−19 C1.602176634\times10^{-19}\ \mathrm Cexact
Boltzmann constantkBk_B1.380649×10−23 J K−11.380649\times10^{-23}\ \mathrm{J\,K^{-1}}exact
Avogadro constantNAN_A6.02214076×1023 mol−16.02214076\times10^{23}\ \mathrm{mol^{-1}}exact
Cesium hyperfine frequencyΔνCs\Delta\nu_{\mathrm{Cs}}9 192 631 770 Hz9\,192\,631\,770\ \mathrm{Hz}exact

Common Measured Constants

Section titled “Common Measured Constants”
ConstantSymbolRounded ValueTypical Use
Electron massmem_e9.1093837139×10−31 kg9.1093837139\times10^{-31}\ \mathrm{kg}atomic, magnetic, and nonrelativistic electron scales
Proton massmpm_p1.67262192595×10−27 kg1.67262192595\times10^{-27}\ \mathrm{kg}reduced masses and nuclear scales
Fine-structure constantα\alpha7.2973525643×10−37.2973525643\times10^{-3}electromagnetic coupling strength
Inverse fine-structure constantα−1\alpha^{-1}137.035999177137.035999177precision atomic physics
Bohr radiusa0a_05.29177210544×10−11 m5.29177210544\times10^{-11}\ \mathrm mhydrogenic length scale
Bohr magnetonμB\mu_B9.2740100657×10−24 J T−19.2740100657\times10^{-24}\ \mathrm{J\,T^{-1}}electron spin and orbital magnetic moments
Rydberg constantR∞R_\infty1.0973731568157×107 m−11.0973731568157\times10^7\ \mathrm{m^{-1}}hydrogenic spectra
Hartree energyEhE_h4.3597447222060×10−18 J4.3597447222060\times10^{-18}\ \mathrm Jatomic units
Vacuum permittivityϵ0\epsilon_08.8541878188(14)×10−12 F m−18.8541878188(14)\times10^{-12}\ \mathrm{F\,m^{-1}}SI electromagnetic formulas
Vacuum permeabilityμ0\mu_01.25663706127×10−6 N A−21.25663706127\times10^{-6}\ \mathrm{N\,A^{-2}}SI magnetic formulas

How to Use This Table

Section titled “How to Use This Table”

Use exact constants directly when defining units. Use rounded measured constants for estimates, page examples, and order-of-magnitude checks. For precision work, retrieve the current value and uncertainty from the official source.

Common Mistakes

Section titled “Common Mistakes”
  • Treating ℏ\hbar as an independently exact decimal rather than h/(2π)h/(2\pi).
  • Forgetting that ee in this table is positive, while the electron charge is −e-e.
  • Using μ0=4π×10−7 N A−2\mu_0=4\pi\times10^{-7}\ \mathrm{N\,A^{-2}} as exact after the 2019 SI redefinition.
  • Mixing SI Coulomb formulas with Gaussian-unit formulas.
Section titled “Canonical Links”

Sources and Verification

Section titled “Sources and Verification”