ħ

The symbol $\hbar$ denotes the reduced Planck constant:

$$\hbar=\frac{h}{2\pi}.$$

It converts among action, quantum phase, angular frequency and energy, wave number and momentum, and dimensionless commutators. It is pronounced “h-bar”; the standard mathematical glyph is shown above.

Default Meaning

This documentation keeps $\hbar$ explicit unless a page declares natural or atomic units. A missing $\hbar$ should never be inferred silently from context when dimensions or convention comparisons matter.

The mathematical type is a positive universal physical constant. It is a scalar, not an operator.

Units

$\hbar$ has dimensions of action:

$$[\hbar]=M L^2T^{-1}.$$

Equivalent SI units are

$$\mathrm{J\,s} =\mathrm{kg\,m^2\,s^{-1}}.$$

These are also the dimensions of angular momentum.

Current SI Value

The SI fixes the Planck constant exactly:

$$h =6.626\,070\,15\times10^{-34}\ \mathrm{J\,s} \quad\text{exactly}.$$

Therefore

$$\hbar =\frac{ 6.626\,070\,15\times10^{-34} }{2\pi}\ \mathrm{J\,s}$$

is exact through the defining relation. Its decimal expansion does not terminate:

$$\hbar =1.054\,571\,817\ldots \times10^{-34}\ \mathrm{J\,s}.$$

In electron-volt units,

$$\hbar =6.582\,119\,569\ldots \times10^{-16}\ \mathrm{eV\,s}.$$

The ellipses matter. The finite displayed decimals are convenient representations of a value defined by the exact quotient $h/(2\pi)$.

Common Forms

Relation	Role
$E=\hbar\omega$	Converts angular frequency to energy
$p=\hbar k$	Converts wave number to momentum
$[\hat x,\hat p]=i\hbar I$	Sets the canonical quantum scale
$\hat p=-i\hbar\nabla$	Momentum in position representation
$i\hbar\,\partial_t\lvert\psi\rangle=H\lvert\psi\rangle$	Schrödinger evolution
$U(t)=e^{-iHt/\hbar}$	Time evolution for time-independent $H$
$e^{iS/\hbar}$	Action as a dimensionless quantum phase
$L^2\lvert\ell m\rangle=\hbar^2\ell(\ell+1)\lvert\ell m\rangle$	Angular-momentum spectrum
$E_n=\hbar\omega(n+1/2)$	Harmonic-oscillator spectrum

Every exponential or trigonometric argument must be dimensionless. The factors $Ht/\hbar$, $S/\hbar$, $px/\hbar$, and $Et/\hbar$ pass this check.

Ordinary and Angular Frequency

Planck’s constant $h$ pairs naturally with ordinary frequency $\nu$, while $\hbar$ pairs with angular frequency $\omega$:

$$E=h\nu=\hbar\omega, \qquad \omega=2\pi\nu.$$

Likewise,

$$p=\frac{h}{\lambda}=\hbar k, \qquad k=\frac{2\pi}{\lambda}.$$

Mixing $h$ with $\omega$ or $\hbar$ with $\nu$ without the corresponding $2\pi$ is a common source of errors.

Commutators and Uncertainty

For a canonical pair,

$$[\hat q,\hat p]=i\hbar I.$$

The Robertson lower bound then gives

$$\Delta q\,\Delta p \ge\frac{\hbar}{2}$$

in states for which the relevant domains and variances are well defined.

$\hbar$ does not mean that every uncertainty product equals $\hbar/2$. Equality occurs only for special states and operator pairs.

Natural and Atomic Units

In many relativistic and field-theory calculations,

$$\hbar=c=1.$$

Then action and angular momentum are dimensionless in the chosen unit bookkeeping, while time and inverse energy share dimensions. The physical constant has not vanished; the unit system uses it as a conversion factor equal to one.

Hartree atomic units also set

$$\hbar=m_e=e=4\pi\epsilon_0=1.$$

Natural units and atomic units identify different collections of quantities with one. A formula written with $\hbar=1$ does not by itself reveal which other constants have also been suppressed.

Natural Units and Atomic Units provide the translation tables.

Restoring ħ

Dimensional analysis is the primary restoration tool:

1. identify the physical dimensions of every quantity;
2. require phases and exponential arguments to be dimensionless;
3. use $E=\hbar\omega$ and $p=\hbar k$ to translate frequency and wave-number conventions;
4. restore other suppressed constants, such as $c$ or $k_{\mathrm B}$, at the same time;
5. verify a known limit or canonical commutator.

There is no universal rule that inserts one factor of $\hbar$ for every derivative or loop without reference to field normalizations and conventions. Restoring only the visible phase factor can leave hidden unit choices elsewhere in a formula.

Canonical Home

Units and Constants owns the site-wide explicit-$\hbar$ policy and dimensional ledger. Hbar Conventions is the quick translation card.

Canonical Commutation Relations develops the operator role, and Fourier-Transform Conventions fixes the $p/\hbar$ and wave-number kernels.

Other Names and Forms

• reduced Planck constant;
• Dirac constant, an older name;
• h-bar, the spoken name;
• $\hbar$, the standard mathematical form;
• $\hslash$, an alternate glyph encountered in some typography.

The ordinary Planck constant $h$ is a different symbol related by $h=2\pi\hbar$.

Convention Warnings

• Do not confuse $h$ with $\hbar$; they differ by $2\pi$.
• Ordinary frequency $\nu$ and angular frequency $\omega$ differ by $\omega=2\pi\nu$.
• A decimal value ending after nine shown digits is rounded or truncated even though the SI relation is exact.
• Setting $\hbar=1$ is a unit convention, not a classical limit.
• The classical limit is organized by dimensionless action ratios such as $S/\hbar$, not by physically varying a dimensional constant in isolation.
• $\hbar$ has units even when it multiplies a dimensionless quantum number.
• Probability amplitudes and density operators do not acquire units merely because $\hbar$ appears elsewhere in the theory.
• Restoring $\hbar$ without restoring the page’s other suppressed constants can remain dimensionally inconsistent.

References

• Bureau International des Poids et Mesures, SI defining constants, accessed 2026-08-19.
• P. J. Mohr et al., CODATA recommended values of the fundamental physical constants: 2022, Reviews of Modern Physics 97, 025002 (2025).
• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.