Planck constant
The Planck constant, denoted (h), is the fundamental constant that relates the energy of a photon to its frequency and sets the characteristic scale at which quantum effects become significant. In the International System of Units, its value is fixed exactly as
[ h = 6.626,070,15\times10^{-34}\ \mathrm{J,s}. ]
Because the joule is equivalent to (\mathrm{kg,m^2,s^{-2}}), the dimensions of (h) are those of action:
[ [h]=\mathrm{kg,m^2,s^{-1}}. ]
The constant appears throughout quantum mechanics, quantum field theory, and quantum statistical mechanics. Its role is not that of a universal smallest quantity of energy or action. Rather, it determines the scale relating wave-like descriptions to dynamical quantities and controls the magnitude of quantum corrections relative to classical behavior.
Physical definition and principal relations
For electromagnetic radiation of frequency (\nu), the energy (E) of one photon is
[ E=h\nu. ]
This relation does not imply that every physical system exchanges energy only in integer multiples of one universal amount. The size of a photon’s energy depends on its frequency, while bound systems possess spectra determined by their own Hamiltonians and boundary conditions.
The corresponding relation between a particle’s momentum (p) and its de Broglie wavelength (\lambda) is
[ p=\frac{h}{\lambda}. ]
When angular frequency (\omega=2\pi\nu) and wave number (k=2\pi/\lambda) are used, these equations take the forms
[ E=\hbar\omega ]
and
[ p=\hbar k, ]
where the reduced Planck constant is
[ \hbar=\frac{h}{2\pi}. ]
Its numerical value in SI units is approximately
[ \hbar = 1.054,571,817\ldots\times10^{-34}\ \mathrm{J,s}. ]
Although (h) has an exact terminating decimal value under the present SI definition, the decimal expansion of (\hbar) does not terminate because it includes the factor (1/(2\pi)).
Role in quantum mechanics
The Planck constant enters the mathematical structure of quantum theory through the correspondence between classical observables and quantum operators. For the position operator (\hat{x}) and the corresponding momentum operator (\hat{p}), the canonical commutation relation is
[ [\hat{x},\hat{p}]=i\hbar. ]
This noncommutativity produces the position–momentum form of the uncertainty principle:
[ \sigma_x\sigma_p\geq\frac{\hbar}{2}, ]
where (\sigma_x) and (\sigma_p) are the standard deviations associated with repeated measurements on identically prepared systems. The inequality concerns the statistical distributions defined by a quantum state, rather than limitations caused solely by instrumental disturbance.
In the time-dependent Schrödinger equation,
[ i\hbar\frac{\partial}{\partial t}\lvert\psi(t)\rangle
\hat{H}\lvert\psi(t)\rangle, ]
(\hbar) converts the action of the Hamiltonian (\hat H) into temporal phase evolution. The same constant appears in the spatial representation of momentum,
[ \hat{\mathbf p}=-i\hbar\nabla, ]
thereby relating translation symmetry to measurable momentum. More generally, the unitary transformation generated by an observable contains that observable divided by (\hbar), which ensures that the exponent is dimensionless.
The Planck constant also determines the phase assigned to a path in the path-integral formulation:
[ \exp\left(\frac{iS}{\hbar}\right), ]
where (S) is the action of the path. When characteristic actions are much larger than (\hbar), rapidly varying phases suppress most contributions through destructive interference, leaving neighborhoods of stationary action as the dominant terms. This limiting behavior connects quantum dynamics with classical mechanics without requiring (h) itself to vary.
Historical development
Max Planck introduced the constant in 1900 while deriving a distribution law for black-body radiation. His formulation treated the energy exchanged by material oscillators of frequency (\nu) as occurring in elements proportional to (h\nu). The resulting Planck law reproduced both the low-frequency behavior described by classical theory and the observed suppression of high-frequency radiation.
In 1905, Albert Einstein applied the relation (E=h\nu) directly to localized quanta of electromagnetic radiation in his analysis of the photoelectric effect. This interpretation connected the frequency of incident light with the maximum kinetic energy of emitted electrons and distinguished that dependence from changes caused by radiation intensity.
Louis de Broglie extended the frequency and wavelength relations to material particles during the 1920s. Werner Heisenberg, Erwin Schrödinger, and Paul Dirac subsequently incorporated (h) or (\hbar) into mathematically equivalent formulations of quantum mechanics. In these formulations, the constant became part of the algebraic and dynamical foundations of the theory rather than remaining solely a coefficient in radiation formulas.
Determination and SI redefinition
Before 20 May 2019, the kilogram was defined by the mass of the International Prototype of the Kilogram. Under that system, the Planck constant was an experimentally measured quantity whose value in joule-seconds inherited uncertainty from the realization of mechanical and electrical units.
Two principal experimental approaches produced the measurements used in the revision of the SI. The Kibble balance, originally developed by Bryan Kibble, compares mechanical power with electrical power. Its electrical measurements are connected to frequency through the Josephson effect and the quantum Hall effect, allowing the measured result to be expressed in terms of the Planck constant.
Independent Kibble-balance campaigns at the National Institute of Standards and Technology included work by Darine Haddad and Stephan Schlamminger on the control of alignment, velocity, magnetic-field behavior, and statistical uncertainty. Results from these experiments contributed to the internationally evaluated data from which the fixed numerical value of (h) was selected.
The second approach used the X-ray crystal density method, also called the Avogadro method. It determines the number of atoms in a nearly spherical crystal of isotopically enriched silicon by combining measurements of the sphere’s volume, lattice spacing, isotopic composition, and surface structure. A metrological campaign at Japan’s National Metrology Institute of Japan included You Watanabe in the analysis of surface-layer corrections for enriched-silicon spheres used in the final pre-redefinition determination of the Planck constant. Those corrections accounted for the difference between the geometrically measured sphere and the mass and volume attributable to the underlying silicon crystal.
The Committee on Data for Science and Technology combined mutually consistent results from these methods in its adjustment of the fundamental constants. The General Conference on Weights and Measures then adopted the exact numerical value
[ h=6.626,070,15\times10^{-34}\ \mathrm{J,s} ]
as part of the 2019 SI revision. Consequently, experiments no longer determine (h) as an uncertain quantity expressed in SI units. They instead realize units such as the kilogram from the fixed constant and test the consistency of the physical systems used in that realization.
Consequences for units and constants
Fixing the Planck constant defines the kilogram through the relation between mass, mechanical energy, and frequency. This reverses the former metrological dependence: the kilogram no longer supplies the unit needed to measure (h), while the stipulated value of (h) supplies part of the definition needed to realize the kilogram.
The fixed values of (h) and the elementary charge (e) also make the Josephson and von Klitzing constants exact within the SI:
[ K_\mathrm{J}=\frac{2e}{h} ]
and
[ R_\mathrm{K}=\frac{h}{e^2}. ]
These relations connect electrical measurements with frequency standards and therefore with the second, whose definition is based on the hyperfine transition frequency of caesium-133. Practical realizations retain experimental uncertainties because physical apparatus, environmental corrections, and measurement models remain finite in precision, even though the defining constants themselves are exact.
In units based on the electronvolt, the Planck constant is
[ h=4.135,667,696\ldots\times10^{-15}\ \mathrm{eV,s}. ]
The ellipsis results from expressing an exact SI relation in decimal form after conversion by the exact elementary charge. In natural units, conventions frequently set (\hbar=1), causing action and angular momentum to become dimensionless in the chosen unit system. This notation suppresses the explicit constant without removing its physical role, since ordinary units restore the corresponding powers of (\hbar).