The Planck constant, or Planck's constant, denoted by h {\displaystyle h} , is a fundamental physical constant of foundational importance in quantum mechanics: a photon's energy is equal to its frequency multiplied by the Planck constant, and a particle's momentum is equal to the wavenumber of the associated matter wave (the reciprocal of its wavelength) multiplied by the Planck constant. The constant was postulated by Max Planck in 1900 as a proportionality constant needed to explain experimental black-body radiation. Planck later referred to the constant as the "quantum of action". In 1905, Albert Einstein associated the "quantum" or minimal element of the energy to the electromagnetic wave itself. Max Planck received the 1918 Nobel Prize in Physics "in recognition of the services he rendered to the advancement of Physics by his discovery of energy quanta". In metrology, the Planck constant is used, together with other constants, to define the kilogram, the SI unit of mass. The SI units are defined such that the Planck constant has the exact value h {\displaystyle h} = 6.62607015×10−34 J⋅Hz−1 when it is expressed in SI units. The closely-related reduced Planck constant, denoted ℏ {\textstyle \hbar } (h-bar), equal to the Planck constant divided by 2π: ℏ = h 2 π {\textstyle \hbar ={\frac {h}{2\pi }}} , is commonly used in quantum physics equations. It relates the energy of a photon to its angular frequency, and the linear momentum of a particle to the angular wavenumber of its associated matter wave. As h {\displaystyle h} has an exact defined value, the value of ℏ {\textstyle \hbar } can be calculated to arbitrary precision: ℏ {\displaystyle \hbar } = 1.054571817...×10−34 J⋅s. As a proportionality constant in relationships involving angular quantities, the unit of ℏ {\textstyle \hbar } may be given as J·s/rad, with the same numerical value, as the radian is the natural dimensionless unit of angle.
History
Origin of the constant
The Planck constant was formulated as part of Max Planck's successful effort to produce a mathematical expression that accurately predicted the observed spectral distribution of black-body radiation. This expression is known as Planck's law. In the last years of the 19th century, Max Planck was investigating the problem of black-body radiation posed by Kirchhoff some 40 years earlier. Every physical body spontaneously and continuously emits electromagnetic radiation. There was no expression or explanation for the overall shape of the observed emission spectrum. At the time, Wien's law fit the data for short wavelengths and high temperatures, but failed for long wavelengths. Also around this time, but unknown to Planck, Lord Rayleigh had derived theoretically a formula, later known as the Rayleigh–Jeans law, that could reasonably predict long wavelengths but failed dramatically at short wavelengths. Approaching this problem, Planck hypothesized that the equations of motion for light describe a set of harmonic oscillators, one for each possible frequency. He examined how the entropy of the oscillators varied with the temperature of the body, trying to match Wien's law, and was able to derive an approximate mathematical function for the black-body spectrum, which gave a simple empirical formula for long wavelengths. Planck tried to find a mathematical expression that could reproduce Wien's law (for short wavelengths) and the empirical formula (for long wavelengths). This expression included a constant, h {\displaystyle h} , which is thought to be for Hilfsgröße (auxiliary quantity), and subsequently became known as the Planck constant. The expression formulated by Planck showed that the spectral radiance per unit frequency of a body for frequency ν at absolute temperature T is given by
B ν ( ν , T ) d ν = 2 h ν 3 c 2 1 e h ν k B T − 1 d ν , {\displaystyle B_{\nu }(\nu ,T)d\nu ={\frac {2h\nu ^{3}}{c^{2}}}{\frac {1}{e^{\frac {h\nu }{k_{\mathrm {B} }T}}-1}}d\nu ,}
where k B {\displaystyle k_{\text{B}}} is the Boltzmann constant, h {\displaystyle h} is the Planck constant, and c {\displaystyle c} is the speed of light in the medium, whether material or vacuum. Planck soon realized that his solution was not unique. There were several different solutions, each of which gave a different value for the entropy of the oscillators. To save his theory, Planck resorted to using the then-controversial theory of statistical mechanics, which he described as "an act of desperation". One of his new boundary conditions was
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