Planck constant

SI defining constant
Name	Symbol	Value	Unit		Expressed in terms of SI base units
Planck constant	h	6.626 070 15 × 10−34	J s		kg m2 s−1
Definition The Planck constant, symbol h, is a fundamental constant of nature. It is the proportionality constant that relates the energy carried by a photon to its associated wave frequency. The numerical value of the Planck constant, symbol h, is defined to be exactly 6.626 070 15 × 10−34 when expressed in the unit joule second, J s, or kg m2 s−1.

Quantum mechanics

In quantum mechanics, the Planck–Einstein relation states that the energy, E, of a photon is directly proportional to the frequency, ν, of its associated wave:

When expressed in the form of an equation, the Planck constant, h, is the proportionality constant:

Using SI coherent units,

where:

• E is energy in joules, symbol J,
• ν is frequency in hertz, symbol Hz,
• h is the Planck constant, in kg m² s⁻¹.

Equivalence of mass and energy

Einstein’s principle of the equivalence of mass and energy describes the relation between energy, E, and mass, m, where c is the speed of light in vacuum:

Using SI coherent units,

where:

• E is energy in joules, symbol J,
• m is mass in kilograms, symbol kg,
• c is the speed of light in vacuum, in metres per second, symbol m s⁻¹.

The Planck constant and mass

Combining the two above expressions for energy gives the relation between the Planck constant and mass:

Using SI coherent units,

where:

• m is mass in kilograms, symbol kg,
• ν is frequency in hertz, or reciprocal seconds, symbol s⁻¹,
• c is the speed of light in vacuum, in metres per second, symbol m s⁻¹,
• h is the Planck constant, in kg m² s⁻¹.

The relation between the Planck constant and mass forms the basis for the definition of the kilogram.

The Planck constant and spin angular momentum of light

The component of the angular momentum of light associated with a photon’s quantum spin and the rotation between its polarisation degrees of freedom is known as the spin angular momentum of light.

When a beam of light is circularly polarised, each of its photons has a spin angular momentum equal to ±ħ, where ħ is the reduced Planck constant.

The SI derived unit used to express the Planck constant, the joule second, symbol J s, is equivalent to the joule per hertz, symbol J Hz^-1.

The reduced Planck constant, ħ, relates to the Planck constant in the same way as the hertz relates to the radian per second – one hertz being equal to one complete cycle, or 2π radians, per second.