| Planck constant | |
|---|---|
| Name | Planck constant |
| Quantity | action / angular momentum |
| SI unit | joule second (J·s) |
| Value | 6.62607015×10^−34 |
| Dimension | M L^2 T^−1 |
| Named after | Max Planck |
Planck constant
The Planck constant is a fundamental physical constant that relates the energy of a quantum to the frequency of its associated electromagnetic wave. It is central to Quantum mechanics and underpins the quantization of energy, momentum, and angular momentum in microscopic systems. As a bridge between classical and quantum descriptions, the constant figures in laws and instruments across physics, chemistry, and modern technology.
The Planck constant, conventionally denoted h, defines the proportionality between the energy E of a photon and its frequency ν via the relation E = hν. Its reduced form, the reduced Planck constant or ħ (h-bar), equals h/2π and appears in the canonical commutation relations of quantum theory, e.g., [x, p] = iħ. The constant sets the scale at which quantum effects become significant compared to classical behavior and defines fundamental units of action and phase space volume in statistical mechanics and quantum field theory. Its small magnitude explains why macroscopic objects exhibit classical trajectories, while microscopic particles require probabilistic descriptions.
The Planck constant emerged from the work of Max Planck in 1900 while studying black-body radiation at the University of Berlin. To resolve the ultraviolet catastrophe predicted by classical Rayleigh–Jeans law, Planck introduced the hypothesis that oscillators exchange energy in discrete quanta proportional to frequency. This ad hoc assumption, later supported by Albert Einstein's explanation of the photoelectric effect, inaugurated the quantum era. Subsequent developments by Niels Bohr, Werner Heisenberg, and Erwin Schrödinger incorporated h into atomic models, matrix mechanics, and wave mechanics, respectively. Major experiments at institutions such as the Kaiser Wilhelm Institute (now part of the Max Planck Society) and later national metrology laboratories refined the constant's value and interpretation.
In SI units, the Planck constant has dimensions of action (energy × time) with exact numerical value since the 2019 redefinition of the International System of Units (SI): h = 6.62607015×10^−34 J·s. The relation to angular frequency ω is E = ħω. The constant enters formulas for quantized energy levels (E_n = n h ν) in simple harmonic oscillators, the de Broglie relation p = h/λ connecting momentum p and wavelength λ, and the quantization of magnetic flux in superconductors (Φ_0 = h/2e). In solid state physics and spectroscopy, h converts between frequency units and energy scales (e.g., electronvolt uses e and h to relate to Hz). Mathematically it provides the fundamental scale in Fourier transforms that relate conjugate variables.
Planck's constant is woven into the axioms and formalism of quantum theory. It appears explicitly in Heisenberg's uncertainty principle Δx Δp ≥ ħ/2, setting a lower bound on simultaneous knowledge of conjugate variables. In quantum electrodynamics and quantum chromodynamics, ħ determines loop expansion parameters and perturbative scales. In the formulation of quantum operators, commutators are proportional to iħ, fixing the normalization of canonical variables and ensuring consistency with classical Poisson brackets via correspondence principles. The constant also figures in modern topics such as quantum information (quantum of action in qubit manipulations) and the definition of coherent states in quantum optics.
Precise determination of h has historically relied on experiments linking mechanical, electrical, and quantum phenomena. Early approaches used black-body radiation and photoelectric measurements; later techniques included X-ray crystal density methods and watt-balance (now Kibble) experiments that equate mechanical power to electromagnetic power using the Josephson effect and quantum Hall effect. The Kibble balance, developed at institutions like the National Physical Laboratory (UK) and the National Institute of Standards and Technology (NIST), provided the data leading to the fixed numerical value adopted in the 2019 SI revision. Alternate methods use atom-recoil measurements with cold atoms and interferometry performed at research centers such as LKB (Laboratoire Kastler Brossel) and national metrology institutes.
Beyond foundational theory, the Planck constant underlies many technologies. The Josephson junction provides a voltage standard via frequency-to-voltage conversion dependent on h, while the quantum Hall effect yields resistance standards tied to h/e^2. Semiconductor device physics, laser operation, and photovoltaics depend on photon energy quantization. In precision measurement, the fixed value of h stabilizes the kilogram and thereby impacts pharmaceutical, aerospace, and industrial calibration practices. Research in quantum computing, quantum sensing, and nanotechnology continues to exploit phenomena governed by h to design devices with unprecedented sensitivity and coherence times.
Planck constant is one of the cornerstones among the set of fundamental physical constants including the speed of light c, the elementary charge e, and the Boltzmann constant k_B. Its exact SI value since 2019 reflects a policy by organizations such as the General Conference on Weights and Measures (CGPM) and the International Bureau of Weights and Measures (BIPM) to anchor units in unchanging constants. This decision harmonizes measurement science at laboratories like PTB (Germany), LNE (France), and NMIJ (Japan), ensuring international coherence and continuity in science, commerce, and national standards.
Category:Physical constants Category:Quantum mechanics Category:Max Planck