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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”
Constant Symbol Value Status
Speed of light cc 299 792 458 m s−1299\,792\,458\ \mathrm{m\,s^{-1}} exact
Planck constant hh 6.62607015×10−34 J s6.62607015\times10^{-34}\ \mathrm{J\,s} exact
Reduced Planck constant ℏ=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 magnitude ee 1.602176634×10−19 C1.602176634\times10^{-19}\ \mathrm C exact
Boltzmann constant kBk_B 1.380649×10−23 J K−11.380649\times10^{-23}\ \mathrm{J\,K^{-1}} exact
Avogadro constant NAN_A 6.02214076×1023 mol−16.02214076\times10^{23}\ \mathrm{mol^{-1}} exact
Cesium hyperfine frequency ΔνCs\Delta\nu_{\mathrm{Cs}} 9 192 631 770 Hz9\,192\,631\,770\ \mathrm{Hz} exact

Common Measured Constants

Section titled “Common Measured Constants”
Constant Symbol Rounded Value Typical Use
Electron mass mem_e 9.1093837139×10−31 kg9.1093837139\times10^{-31}\ \mathrm{kg} atomic, magnetic, and nonrelativistic electron scales
Proton mass mpm_p 1.67262192595×10−27 kg1.67262192595\times10^{-27}\ \mathrm{kg} reduced masses and nuclear scales
Fine-structure constant α\alpha 7.2973525643×10−37.2973525643\times10^{-3} electromagnetic coupling strength
Inverse fine-structure constant α−1\alpha^{-1} 137.035999177137.035999177 precision atomic physics
Bohr radius a0a_0 5.29177210544×10−11 m5.29177210544\times10^{-11}\ \mathrm m hydrogenic length scale
Bohr magneton μB\mu_B 9.2740100657×10−24 J T−19.2740100657\times10^{-24}\ \mathrm{J\,T^{-1}} electron spin and orbital magnetic moments
Rydberg constant R∞R_\infty 1.0973731568157×107 m−11.0973731568157\times10^7\ \mathrm{m^{-1}} hydrogenic spectra
Hartree energy EhE_h 4.3597447222060×10−18 J4.3597447222060\times10^{-18}\ \mathrm J atomic units
Vacuum permittivity ϵ0\epsilon_0 8.8541878188(14)×10−12 F m−18.8541878188(14)\times10^{-12}\ \mathrm{F\,m^{-1}} SI electromagnetic formulas
Vacuum permeability μ0\mu_0 1.25663706127×10−6 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”