Reduced Planck constant, ℏ

The Planck constant divided by 2π, read aloud as h-bar. It is the natural unit of angular momentum, and it takes the place of h wherever a frequency is measured in radians per second rather than cycles.

ℏ = 1.054571817… × 10⁻³⁴ J·s

Exact, derived. The decimal runs on past ten significant figures, so it is cut off there and marked …, the way NIST prints it, rather than rounded.

In other units

Reduced Planck constant in SI units and in other units
Quantity	Value
ℏ	1.054571817… × 10⁻³⁴ J·s
ℏ	6.582119569… × 10⁻¹⁶ eV·s
ℏc	197.3269804… eV·nm

ℏ in eV·s

1.054571817… × 10⁻³⁴ J·s ÷ (1.602176634 × 10⁻¹⁹ J/eV)
= 6.582119569… × 10⁻¹⁶ eV·s

ℏc in eV·nm

1.054571817… × 10⁻³⁴ J·s × 299,792,458 m/s ÷ (1.602176634 × 10⁻¹⁹ J/eV) × 10⁹ nm/m
= 197.3269804… eV·nm

The same digits read in MeV·fm, the form nuclear physics uses, since both units are 10⁻⁹ eV·m.

Where the value comes from

ℏ is defined as h/2π, so it became exact on the day h did. Exact is not the same as short: 2π is irrational, so ℏ has no terminating decimal, and every printed value, this one included, stops somewhere.

It exists because 2π keeps appearing. Written with ℏ, a quantum of angular frequency ω carries energy ℏω, the component of orbital angular momentum along any axis comes in whole multiples of ℏ, and an electron’s spin component is plus or minus ℏ/2.

ℏ from h

ℏ = h ÷ 2π
= 6.62607015 × 10⁻³⁴ ÷ 6.283185307…
= 1.054571817… × 10⁻³⁴ J·s

Which form to use

Use ℏ with angular frequency and h with ordinary frequency: E = ℏω = hf, because ω = 2πf.

The uncertainty principle, Δx Δp ≥ ℏ/2, and the Schrödinger equation are written with ℏ. With energies in electronvolts, 6.582 × 10⁻¹⁶ eV·s is the convenient form, and ℏc ≈ 197.3 eV·nm is the standard shortcut in atomic and nuclear physics.

Common mistakes

Using h where ℏ belongs, or the reverse. They differ by 2π, about 6.28, and checking units will not catch it, because both are in J·s.
Writing the uncertainty principle as Δx Δp ≥ h/2. The bound is ℏ/2, which is h/4π, so the h/2 version is 2π times too loose.
Comparing a rounded value with NIST’s and finding the last digit differs. Rounded to ten figures ℏ is 1.054571818 × 10⁻³⁴ J·s; NIST cuts it off instead, at 1.054571817… × 10⁻³⁴ J·s, which is the form this page uses. Both describe the same exact number.

Tools that use it

Bohr Model and Hydrogen Spectrum Simulator quantises each orbit’s angular momentum as nℏ, which fixes the electron’s speed.

Related constants

Planck constant, h = 6.62607015 × 10⁻³⁴ J·s
Elementary charge, e = 1.602176634 × 10⁻¹⁹ C
Speed of light in vacuum, c = 299,792,458 m/s