The fundamental constant ħ, equal to the Planck constant divided by 2π, central to quantum mechanics.The reduced Planck constant, written as ħ (h-bar), is the Planck constant h divided by 2π. It is the fundamental unit of action in quantum mechanics, setting the scale at which quantum effects become dominant. While the Planck constant appears in the energy–frequency relation E = hν, the reduced form naturally emerges in angular momentum, wave mechanics, and the Schrödinger equation, where phases advance by 2π per wavelength.
Introduced by Paul Dirac in his 1926 papers on quantum mechanics, who denoted it as h/2π to simplify equations by removing the recurring 2π factor.
In quantum mechanics, ħ quantizes angular momentum: any system's angular momentum is an integer or half-integer multiple of ħ. It also appears in the uncertainty principle Δx·Δp ≥ ħ/2, and in the de Broglie relation p = ħk, where k is the wave number. The factor 2π arises because a full cycle of a wave corresponds to 2π radians, making ħ the natural unit for phase-space volume.
Used in all quantum mechanical calculations, from atomic energy levels and semiconductor physics to quantum computing and the standard model of particle physics.
The value of ħ is approximately 1.0545718 × 10⁻³⁴ joule-seconds, so small that a macroscopic object like a baseball has an angular momentum of about 10³⁴ ħ.
| E_n | variable | Energy of the nth level — energy joule |
| n | variable | Quantum number (non-negative integer) — quantum number |
| \hbar | constant | Reduced Planck constant — reduced planck constant = 1.054571817e-34 joule second |
| \omega | constant | Angular frequency of the oscillator — angular frequency radian per second |
| \Delta x | variable | Standard deviation of position — position uncertainty metre |
| \Delta p | variable | Standard deviation of momentum — momentum uncertainty kilogram metre per second |
| \hbar | constant | Reduced Planck constant — reduced planck constant = 1.054571817e-34 joule second |
| T | variable | tunneling probability — transmission probability |
| z1 | variable | left turning point of the barrier — position metre |
| z2 | variable | right turning point of the barrier — position metre |
| m | variable | mass of the tunneling particle — mass kilogram |
| V | variable | potential energy as a function of position — potential energy joule |
| E | variable | energy of the particle — energy joule |
| ħ | constant | fundamental quantum of action divided by 2π — reduced planck constant = 1.054571817e-34 joule second |
| ∫ | operator | sums the integrand over the interval — integral |
| √ | operator | takes the non-negative root — square root |
| exp | operator | raises e to the given power — exponential |
| ρ | variable | density matrix of the open quantum system — density matrix |
| t | variable | time — time second |
| H | variable | Hamiltonian of the system — hamiltonian joule |
| L_n | variable | Lindblad operators (jump operators) for each dissipation channel — lindblad operator |
| i | constant | imaginary unit, square root of -1 — imaginary unit = 1 |
| ħ | constant | Planck constant divided by 2π — reduced planck constant = 1.054571817e-34 joule second |
| d/dt | operator | derivative with respect to time — time derivative |
| [H, ρ] | operator | commutator of H and ρ — commutator |
| {L_n† L_n, ρ} | operator | anticommutator of L_n† L_n and ρ — anticommutator |
| ∑_n | operator | sum over all dissipation channels — sum over channels |
| P | variable | Detection probability of a particle — probability |
| M | variable | Mass of the particle — mass kilogram |
| a | variable | Proper acceleration of the detector — acceleration metre per second squared |
| c | constant | Speed of light in vacuum — speed of light = 299792458 metre per second |
| ħ | constant | Reduced Planck constant — reduced planck constant = 1.054571817e-34 joule second |
| ψ | variable | Quantum wave function — wave function metre to the minus 3 over 2 |
| R | variable | Real amplitude of the wave function — probability amplitude metre to the minus 3 over 2 |
| S | variable | Phase of the wave function (action) — action joule second |
| ℏ | constant | Fundamental constant setting the scale of quantum phenomena — reduced planck constant = 1.054571817e-34 joule second |
| e^{iS/ℏ} | operator | Complex phase factor encoding the oscillatory behaviour — complex exponential |
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