# planck constant

• Applying this new approach to Wien’s displacement law showed that the “energy element” must be proportional to the frequency of the oscillator, the first version of what is
now sometimes termed the “Planck–Einstein relation”: Planck was able to calculate the value of from experimental data on black-body radiation: his result, J⋅s, is within 1.2% of the currently defined value.

• The size of these “packets” of energy, which would later be named photons, was to be the same as Planck’s “energy element”, giving the modern version of the Planck–Einstein
relation: Einstein’s postulate was later proven experimentally: the constant of proportionality between the frequency of incident light and the kinetic energy of photoelectrons was shown to be equal to the Planck constant .

• [citation needed] Reduced Planck constant ℏ In many applications, the Planck constant naturally appears in combination with as , which can be traced to the fact that in these
applications it is natural to use the angular frequency (in radians per second) rather than plain frequency (in cycles per second or hertz).

• [32] Eventually, following upon Planck’s discovery, it was speculated that physical action could not take on an arbitrary value, but instead was restricted to integer multiples
of a very small quantity, the “[elementary] quantum of action”, now called the Planck constant.

• Planck’s constant was formulated as part of Max Planck’s successful effort to produce a mathematical expression that accurately predicted the observed spectral distribution
of thermal radiation from a closed furnace (black-body radiation).

• Let us call each such part the energy element ε; — Planck, On the Law of Distribution of Energy in the Normal Spectrum[2] With this new condition, Planck had imposed the quantization
of the energy of the oscillators, “a purely formal assumption … actually I did not think much about it …” in his own words,[14] but one that would revolutionize physics.

• 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,[2] which gave a simple empirical formula for long wavelengths.

• Photon energy The Planck relation connects the particular photon energy E with its associated wave frequency f: This energy is extremely small in terms of ordinarily
perceived everyday objects.

• [37] Equivalently, the order of the Planck constant reflects the fact that everyday objects and systems are made of a large number of microscopic particles.

• [12] Planck’s law may also be expressed in other terms, such as the number of photons emitted at a certain wavelength, or the energy density in a volume of radiation.

• Significance of the value The Planck constant is one of the smallest constants used in physics.

• This view has been replaced by fully modern quantum theory, in which definite trajectories of motion do not even exist; rather, the particle is represented by a wavefunction
spread out in space and in time.

• When the product of energy and time for a physical event approaches the Planck constant, quantum effects dominate.

• Many of the most important equations, relations, definitions, and results of quantum mechanics are customarily written using the reduced Planck constant rather than the Planck
constant , including the Schrödinger equation, momentum operator, canonical commutation relation, Heisenberg’s uncertainty principle, and Planck units.

• Approaching this problem, Planck hypothesized that the equations of motion for light describe a set of harmonic oscillators, one for each possible frequency.

• The expression formulated by Planck showed that the spectral radiance of a body for frequency ν at absolute temperature T is given by , where is the Boltzmann constant, is
the Planck constant, and is the speed of light in the medium, whether material or vacuum.

• In many cases, such as for monochromatic light or for atoms, quantization of energy also implies that only certain energy levels are allowed, and values in between are forbidden.

• In modern terms, if is the total angular momentum of a system with rotational invariance, and the angular momentum measured along any given direction, these quantities can
only take on the values Uncertainty principle Main article: Uncertainty principle The Planck constant also occurs in statements of Werner Heisenberg’s uncertainty principle.

• Bohr solved this paradox with explicit reference to Planck’s work: an electron in a Bohr atom could only have certain defined energies where is the speed of light in vacuum,
is an experimentally determined constant (the Rydberg constant) and .

• [2] He also made the first determination of the Boltzmann constant from the same data and theory.

• [21][22] Before Einstein’s paper, electromagnetic radiation such as visible light was considered to behave as a wave: hence the use of the terms “frequency” and “wavelength”
to characterize different types of radiation.

• An amount of light more typical in everyday experience (though much larger than the smallest amount perceivable by the human eye) is the energy of one mole of photons; its
energy can be computed by multiplying the photon energy by the Avogadro constant, [38], with the result of 216 kJ, about the food energy in three apples.

• This fixed value is used to define the Si unit of mass, the kilogram: “the kilogram […] is defined by taking the fixed numerical value of h to be when expressed in the unit
J⋅s, which is equal to, where the metre and the second are defined in terms of speed of light c and duration of hyperfine transition of the ground state of an unperturbed caesium-133 atom ΔνCs.

• [2] Planck later referred to the constant as the “quantum of action”.

• The energy transferred by a wave in a given time is called its intensity.

• This reflects the fact that on a scale adapted to humans, where energies are typical of the order of kilojoules and times are typical of the order of seconds or minutes, the
Planck constant is very small.

• Planck tried to find a mathematical expression that could reproduce Wien’s law (for short wavelengths) and the empirical formula (for long wavelengths).

• This approach also allowed Bohr to account for the Rydberg formula, an empirical description of the atomic spectrum of hydrogen, and to account for the value of the Rydberg
constant in terms of other fundamental constants.

• Bohr also introduced the quantity , now known as the reduced Planck constant or Dirac constant, as the quantum of angular momentum.

• However, the energy account of the photoelectric effect did not seem to agree with the wave description of light.

• [33] This was a significant conceptual part of the so-called “old quantum theory” developed by physicists including Bohr, Sommerfeld, and Ishiwara, in which particle trajectories
exist but are hidden, but quantum laws constrain them based on their action.

• Related to this is the concept of energy quantization which existed in old quantum theory and also exists in altered form in modern quantum physics.

• The inverse relationship between the uncertainty of the two conjugate variables forces a tradeoff in quantum experiments, as measuring one quantity more precisely results
in the other quantity becoming imprecise.

• [44]: 104  Because the fundamental equations look simpler when written using as opposed to , it is usually rather than that gives the most reliable results when used in order-of-magnitude
estimates.

• [18] Einstein’s explanation for these observations was that light itself is quantized; that the energy of light is not transferred continuously as in a classical wave, but
only in small “packets” or quanta.

• This expression included a constant, , which is thought to be for Hilfsgrösse (auxiliary variable),[8] and subsequently became known as the Planck constant.

• The constant was postulated by Max Planck in 1900 as a proportionality constant needed to explain experimental black-body radiation.

• The correct quantization rules for electrons – in which the energy reduces to the Bohr model equation in the case of the hydrogen atom – were given by Heisenberg’s matrix
mechanics in 1925 and the Schrödinger wave equation in 1926: the reduced Planck constant remains the fundamental quantum of angular momentum.

• Planck constant: Common symbols: SI unit : joule per hertz (joule seconds) ; Other units: electronvolt per hertz (electronvolt seconds); Reduced Planck constant: Common symbols:
SI unit: joule-seconds; Other units: electronvolt-seconds History Origin of the constant Main article: Planck’s law Plaque at the Humboldt University of Berlin: “Max Planck, who discovered the elementary quantum of action h, taught here
from 1889 to 1928.

• This kinetic energy (for each photoelectron) is independent of the intensity of the light,[18] but depends linearly on the frequency;[20] and if the frequency is too low (corresponding
to a photon energy that is less than the work function of the material), no photoelectrons are emitted at all, unless a plurality of photons, whose energetic sum is greater than the energy of the photoelectrons, acts virtually simultaneously
(multiphoton effect).

• Classical physics cannot explain either quantization of energy or the lack of classical particle motion.

• Given numerous particles prepared in the same state, the uncertainty in their position, , and the uncertainty in their momentum, , obey where the uncertainty is given as the
standard deviation of the measured value from its expected value.

• For this reason, it is often useful to absorb that factor of 2π into the Planck constant by introducing the reduced Planck constant[39][40]: 482  (or reduced Planck’s constant[41]: 5
[42]: 788 ), equal to the Planck constant divided by [39] and denoted by (pronounced h-bar[43]: 336 ).

• [7]: 141  Also around this time, but unknown to Planck, Lord Rayleigh had derived theoretically a formula, now known as the Rayleigh–Jeans law, that could reasonably predict
long wavelengths but failed dramatically at short wavelengths.

Works Cited

[‘As examples, the preceding reference shows what happens when one uses dimensional analysis to obtain estimates for the ionization energy and the size of a hydrogen atom. If we use the Gaussian units, then the relevant parameters that determine the ionization
energy are the mass of the electron , the electron charge , and either the Planck constant or the reduced Planck constant (since and have the same dimensions, they will enter the dimensional analysis in the same way). One obtains that must be proportional
to if we used , and to is we used . In an order-of-magnitude estimate, we take that the constant of proportionality is 1. Now, the actual correct answer is ;[46]: 45  therefore, if we choose to use as one of our parameters, our estimate
will off by a factor of 2, whereas if we choose to use , it will be off by a factor of . Similarly for the estimate of the size of a hydrogen atom: depending on whether we use or as one of the parameters, we get either or . The latter happens to be
exactly correct,[47] whereas the estimate using is off by a factor of .
2. ^ Notable examples of such usage include Landau and Lifshitz[68]: 20  and Giffiths,[69]: 3  but there are many others, e.g.[70][71]: 449  [72]: 284
[73]: 3  [74]: 365  [75]: 14  [76]: 18  [77]: 4  [78]: 138  [79]: 251  [80]: 1  [81]: 622  [82]: xx  [83]: 20  [84]: 4  [85]: 36
[86]: 41  [87]: 199  [88]: 846  [89][90][91]: 25  [92]: 653
3. ^ Some sources[96][97]: 169  [98]: 180  claim that John William Nicholson discovered the quantization of angular
momentum in units of in his 1912 paper,[99] so prior to Bohr. True, Bohr does credit Nicholson for emphasizing “the possible importance of the angular momentum in the discussion of atomic systems in relation to Planck’s theory.”[100]: 15
However, in his paper, Nicholson deals exclusively with the quantization of energy, not angular momentum—with the exception of one paragraph in which he says, if, therefore, the constant of Planck has, as Sommerfeld has suggested, an atomic significance,
it may mean that the angular momentum of an atom can only rise or fall by discrete amounts when electrons leave or return. It is readily seen that this view presents less difficulty to the mind than the more usual interpretation, which is believed
to involve an atomic constitution of energy itself,[99]: 679  and with the exception of the following text in the summary: in the present paper, the suggested theory of the coronal spectrum has been put upon a definite basis which is in
accord with the recent theories of emission of energy by bodies. It is indicated that the key to the physical side of these theories lies in the fact that an expulsion or retention of an electron by any atom probably involves a discontinuous change
in the angular momentum of the atom, which is dependent on the number of electrons already present.[99]: 692  The literal combination does not appear in that paper. A biographical memoir of Nicholson[101] states that Nicholson only “later”
realized that the discrete changes in angular momentum are integral multiples of , but unfortunately the memoir does not say if this realization occurred before or after Bohr published his paper, or whether Nicholson ever published it.
4. ^ Bohr
denoted by the angular momentum of the electron around the nucleus, and wrote the quantization condition as , where is a positive integer. (See the Bohr model.)
5. ^ Here are some papers that are mentioned in[98] and in which appeared without a
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The question is first: How can one assign a discrete succession of energy values Hσ to a system specified in the sense of classical mechanics (the energy function is a given function of the coordinates qr and the corresponding momenta pr)? The Planck
constant h relates the frequency Hσ/h to the energy values Hσ. It is therefore sufficient to give to the system a succession of discrete frequency values.
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