| reduced Planck constant | |
|---|---|
| Name | Reduced Planck constant |
| Also known as | Dirac constant |
| Quantity | action / angular momentum |
| Units | J·s (joule-second) |
| Si unit | kilogram metre^2 second^−1 |
| Named after | Paul Dirac |
reduced Planck constant
The reduced Planck constant (symbol: ħ), also called the Dirac constant, is the Planck constant divided by 2π and serves as a fundamental scale factor relating energy and angular frequency in quantum mechanics. It simplifies many equations by converting ordinary Planck-constant appearances into angular units, and underlies the quantization of action and angular momentum in microscopic systems.
The reduced Planck constant is defined as ħ = h / (2π), where Planck constant h is a fundamental physical constant introduced in the context of black-body radiation. The conventional symbol for the reduced constant is the lowercase Greek letter "h-bar" (ħ). In many theoretical treatments, especially those employing angular frequencies ω rather than cyclic frequencies ν, ħ appears naturally in expressions for energy E = ħω, the canonical commutation relations [x,p] = iħ, and the Schrödinger equation. The first mention of the symbol ħ and the eponym "Dirac constant" is associated with Paul Dirac's formulation of quantum theory.
The origin of ħ follows from the historical introduction of h by Max Planck in 1900 to quantify black-body radiation. Subsequent development of quantum theory by Albert Einstein (photoelectric effect), Niels Bohr (atomic model), and later the formulation of matrix mechanics by Werner Heisenberg and wave mechanics by Erwin Schrödinger led to widespread use of h. Paul Dirac popularized the use of h/(2π) in his work on relativistic quantum mechanics and quantum electrodynamics; the reduced constant became standard in texts by Dirac and in the formalism of Dirac's canonical quantization. Major institutions such as International Bureau of Weights and Measures and codifications in the SI reflect the constant's place in modern physics.
ħ appears in central equations of quantum theory. In nonrelativistic Schrödinger equation, iħ ∂ψ/∂t = Ĥψ, ħ sets the scale between temporal evolution and energy via the Hamiltonian Ĥ. In canonical quantization, the fundamental commutation relation [x̂, p̂] = iħ encodes Heisenberg uncertainty, directly linking ħ to the Heisenberg uncertainty principle. In angular momentum theory, eigenvalues are integer or half-integer multiples of ħ. In quantum field theory, ħ controls loop expansions and perturbative order; setting ħ→0 recovers classical mechanics through the classical limit. In quantum electrodynamics and quantum chromodynamics, ħ combines with other constants (e.g., elementary charge, speed of light) in coupling constants and action integrals.
Physically, ħ has dimensions of action (energy × time) or angular momentum. Its SI unit is joule-second (J·s), identical to that of h but scaled by 2π. Numerically, ħ ≈ 1.054571817×10^−34 J·s under the 2019 SI redefinition where h has an exact value. ħ quantifies the quantum of phase-space area in units of action: minimal phase-space cell volumes are of order ħ per degree of freedom in semiclassical quantization rules such as the Bohr–Sommerfeld quantization and WKB approximation. In thermodynamic and statistical treatments, ħ appears in expressions for partition functions and density of states, and in condensed-matter phenomena it sets scales like the quantum Hall effect conductance quantum when combined with e^2/h.
By definition ħ = h/(2π). In practice, many theoretical frameworks adopt natural units setting ħ = 1 to simplify algebra; this is common in high-energy physics at institutions such as CERN and in textbooks by Steven Weinberg and Richard Feynman. In Planck units, combinations of ħ, G, and speed of light c yield the Planck length, Planck time, and Planck energy, central to discussions of quantum gravity and Planck scale. The redefinition of the SI base units fixed h exactly, which consequently fixes ħ's value via the relation to h. Use of ħ versus h often distinguishes angular-frequency formulations from cycles-per-second formulations.
Experimental determination of h (and thus ħ) historically relied on measurements including the photoelectric effect, black-body radiation, and the Josephson and von Klitzing effects. Modern determinations use the Kibble balance (formerly watt balance) and X-ray crystal density (XRCD) methods performed at institutions such as the National Institute of Standards and Technology (NIST) and the International Bureau of Weights and Measures (BIPM). The 2019 SI revision fixed h to an exact numerical value, thereby defining ħ exactly in SI terms. Prior to that, uncertainty budgets involved contributions from electrical metrology (Josephson constant), quantum Hall measurements, and mass determinations in the XRCD method.
ħ plays an essential role across quantum technologies and theoretical frameworks. In atomic and molecular physics it determines spectral line spacings and selection rules; in quantum optics it appears in descriptions of photon energy and coherent states; in solid-state physics it enters Bloch oscillations, tunnelling rates, and band structure calculations. Quantum information theory employs ħ implicitly in gate and qubit dynamics; superconducting qubits and trapped ion systems are engineered with energy-level separations characterized by ħω. In high-energy physics and cosmology, ħ contributes to action integrals and path-integral phases in formulations by Richard Feynman and underlies semiclassical approximations in black-hole thermodynamics and inflationary perturbations. The constant's presence also frames philosophical questions about quantization, discreteness, and the emergence of classicality.
Category:Quantum mechanics Category:Physical constants Category:Planck constant