h-bar (ħ)

How to read it

The IUPAC Green Book lists ħ straight after h, the Planck constant, and defines it as h/2π. The SI has fixed h at exactly 6.626 070 15 × 10⁻³⁴ J s (BIPM SI Brochure), so ħ is exact too: NIST’s 2022 CODATA table gives it as 1.054 571 817… × 10⁻³⁴ J s, with the uncertainty marked “exact” and the dots standing for digits that carry on. Physicists write ħ where an equation would otherwise carry h and a 2π. OpenStax’s University Physics writes de Broglie’s relations as p = ħk and E = ħω, and Heisenberg’s uncertainty principle as Δx Δp ≥ ħ/2: the product of the uncertainties in a particle’s position and momentum, in OpenStax’s words, “can never be less than one-half of the reduced Planck constant”. Unicode gives ħ two other names, “reduced Planck constant” and “Dirac constant”.

Also searched as: Dirac constant, ħ, ℏ, h bar, h bar symbol, h-bar symbol, h with a line through it, Planck constant over 2 pi, reduced Planck constant.

ħ as a unit

In quantum chemistry ħ is also the unit that angular momentum is counted in. The Green Book gives the spin of an electron in units of ħ, and in atomic units it measures all angular momenta as multiples of ħ, just as it measures charges as multiples of the elementary charge e.

Look-alikes

There are two characters for it. Unicode names ℏ (U+210F) “PLANCK CONSTANT OVER TWO PI” and treats it as a font variant of ħ (U+0127), the Latin small letter h with stroke, which is a letter of Maltese and of the International Phonetic Alphabet. Unicode cross-refers both to the Cyrillic letter tshe, ћ (U+045B). Without the bar, h is the Planck constant itself, and Unicode’s ℎ (U+210E), named “PLANCK CONSTANT”, is in its chart’s words simply a mathematical italic h, named that way for legacy reasons.