# 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:

$E \propto \nu$

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

Using SI coherent units,

$E = h \nu$

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,

$E = m c^2$

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,

$m c^2 = h \nu\\ \\ \\ m \mspace{14mu} = \dfrac{h \nu}{c^2}$

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.

$1 \ \text{rad s}^{-1} = \dfrac{1}{2 \pi} \ \text{Hz}\\ \\ \\ \hbar \mspace{56mu} = \dfrac{h}{2 \pi}\\ \\ \\ \hbar \mspace{56mu} = \dfrac{6.626\ 070\ 15 \times 10^{-34}}{2 \pi} \ \text{J Hz}^{-1}\\ \\ \\ \hbar \mspace{56mu} = 1.054\ 571\ 817\ \text{...} \times 10^{-34} \ \text{J Hz}^{-1}$