The Planck constant, or Planck's constant, denoted by ,[1] is a fundamental
physical constant[1] of foundational importance in
quantum mechanics: a
photon's energy is equal to its
frequency multiplied by the Planck constant, and the
wavelength of a
matter wave equals the Planck constant divided by the associated particle momentum.
The constant was postulated by
Max Planck in 1900 as a
proportionality constant needed to explain experimental
black-body radiation.[2] Planck later referred to the constant as the "quantum of
action".[3] 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.[4] The SI units are defined in such a way that, when the Planck constant is expressed in SI units, it has the exact value = 6.62607015×10−34 J⋅Hz−1.[5][6] It is often used with units of
electronvolt (eV), which corresponds to the SI unit per
elementary charge.
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).[7] This mathematical expression is now known as Planck's law.
In the last years of the 19th century, Max Planck was investigating the problem of black-body radiation first 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.[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.
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,[2] 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, , which is thought to be for Hilfsgrösse (auxiliary variable),[8] and subsequently became known as the Planck constant. The expression formulated by Planck showed that the spectral radiance of a body for
frequencyν at
absolute temperatureT is given by
The
spectral radiance of a body, , describes the amount of energy it emits at different radiation frequencies. It is the power emitted per unit area of the body, per unit solid angle of emission, per unit frequency. The spectral radiance can also be expressed per unit
wavelength instead of per unit frequency. In this case, it is given by
,
showing how radiated energy emitted at shorter wavelengths increases more rapidly with temperature than energy emitted at longer wavelengths.[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. The
SI units of are W·
sr−1·
m−2·
Hz−1, while those of are W·sr−1·m−3.
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.[2] To save his theory, Planck resorted to using the then-controversial theory of
statistical mechanics,[2] which he described as "an act of desperation".[13] One of his new boundary conditions was
to interpret UN [the vibrational energy of N oscillators] not as a continuous, infinitely divisible quantity, but as a discrete quantity composed of an integral number of finite equal parts. 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. 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, 6.55×10−34 J⋅s, is within 1.2% of the currently defined value.[2] He also made the first determination of the
Boltzmann constant from the same data and theory.[15]
Development and application
The black-body problem was revisited in 1905, when
Lord Rayleigh and
James Jeans (on the one hand) and
Albert Einstein (on the other hand) independently proved that classical electromagnetism could never account for the observed spectrum. These proofs are commonly known as the "
ultraviolet catastrophe", a name coined by
Paul Ehrenfest in 1911. They contributed greatly (along with Einstein's work on the
photoelectric effect) in convincing physicists that Planck's postulate of quantized energy levels was more than a mere mathematical formalism. The first
Solvay Conference in 1911 was devoted to "the theory of radiation and quanta".[16]
The photoelectric effect is the emission of electrons (called "photoelectrons") from a surface when light is shone on it. It was first observed by
Alexandre Edmond Becquerel in 1839, although credit is usually reserved for
Heinrich Hertz,[17] who published the first thorough investigation in 1887. Another particularly thorough investigation was published by
Philipp Lenard (Lénárd Fülöp) in 1902.[18] Einstein's 1905 paper[19] discussing the effect in terms of light quanta would earn him the Nobel Prize in 1921,[17] after his predictions had been confirmed by the experimental work of
Robert Andrews Millikan.[20] The Nobel committee awarded the prize for his work on the photo-electric effect, rather than relativity, both because of a bias against purely theoretical physics not grounded in discovery or experiment, and dissent amongst its members as to the actual proof that relativity was real.[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. The energy transferred by a wave in a given time is called its
intensity. The light from a theatre spotlight is more intense than the light from a domestic lightbulb; that is to say that the spotlight gives out more energy per unit time and per unit space (and hence consumes more electricity) than the ordinary bulb, even though the color of the light might be very similar. Other waves, such as sound or the waves crashing against a seafront, also have their intensity. However, the energy account of the photoelectric effect did not seem to agree with the wave description of light.
The "photoelectrons" emitted as a result of the photoelectric effect have a certain
kinetic energy, which can be measured. 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).[23] Assuming the frequency is high enough to cause the photoelectric effect, a rise in intensity of the light source causes more photoelectrons to be emitted with the same kinetic energy, rather than the same number of photoelectrons to be emitted with higher kinetic energy.[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. 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 .[20]
It was
John William Nicholson in 1912 who introduced h-bar into the theory of the atom which was the first quantum and nuclear atom and the first to quantize angular momentum as h/2π.[24][25][26][27][28]Niels Bohr quoted him in his 1913 paper of the Bohr model of the atom.[29] The influence of the work of Nicholson's nuclear quantum atomic model on Bohr's model has been written about by many historians.[30][31][28]
Niels Bohr introduced the third quantized model of the atom in 1913, in an attempt to overcome a major shortcoming of
Rutherford's classical model. The first quantized model of the atom was introduced in 1910 by
Arthur Erich Haas and was discussed at the 1911 Solvay conference.[24][29] In classical electrodynamics, a charge moving in a circle should radiate electromagnetic radiation. If that charge were to be an electron orbiting a
nucleus, the radiation would cause it to lose energy and spiral down into the nucleus. 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 . Once the electron reached the lowest energy level (), it could not get any closer to the nucleus (lower energy). 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. At first, Bohr thought that this was the angular momentum of each electron in an atom: this proved incorrect and, despite developments by
Sommerfeld and others, an accurate description of the electron angular momentum proved beyond the Bohr model. 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. 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
The Planck constant also occurs in statements of
Werner Heisenberg's uncertainty principle. 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. There are several other such pairs of physically measurable
conjugate variables which obey a similar rule. One example is time vs. energy. 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.
In addition to some assumptions underlying the interpretation of certain values in the quantum mechanical formulation, one of the fundamental cornerstones to the entire theory lies in the
commutator relationship between the
position operator and the
momentum operator:
This energy is extremely small in terms of ordinarily perceived everyday objects.
Since the frequency f,
wavelengthλ, and
speed of lightc are related by , the relation can also be expressed as
de Broglie wavelength
In 1923,
Louis de Broglie generalized the Planck–Einstein relation by postulating that the Planck constant represents the proportionality between the momentum and the quantum wavelength of not just the photon, but the quantum wavelength of any particle. This was confirmed by experiments soon afterward. This holds throughout the quantum theory, including
electrodynamics. The
de Broglie wavelengthλ of the particle is given by
These two relations are the temporal and spatial parts of the special relativistic expression using
4-vectors.
Statistical mechanics
Classical
statistical mechanics requires the existence of h (but does not define its value).[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.[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. 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.[34]: 373
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. Classical physics cannot explain quantization of energy.
Since 2019, the numerical value of the Planck constant has been fixed, with a
finite decimal representation. 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 6.62607015×10−34 when expressed in the unit J⋅s, which is equal to kg⋅m2⋅s−1, where the
metre and the
second are defined in terms of
speed of lightc and duration of
hyperfine transition of the
ground state of an unperturbed
caesium-133 atom ΔνCs."[36] Technologies of mass
metrology such as the
Kibble balance measure refine the value of kilogram applying fixed value of the Planck constant.
Significance of the value
The Planck constant is one of the smallest constants used in physics. 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. When the
product of energy and time for a physical event approaches the Planck constant, quantum effects dominate.[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. For example, in
green light (with a
wavelength of 555
nanometres or a frequency of 540 THz) each
photon has an energy E = hf = 3.58×10−19 J. That is a very small amount of energy in terms of everyday experience, but everyday experience is not concerned with individual photons any more than with individual atoms or molecules. 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, NA = 6.02214076×1023 mol−1[38], with the result of 216 kJ, about the food energy in three apples.[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). 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).
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.[45]: 8–9 [a]
By far the most common symbol for the reduced Planck constant is . However, there are some sources that denote it by instead, in which case they usually refer to it as the “Dirac ”[93]:
43[94]:
151 (or “Dirac's ”[95]:
21).
History
The combination first made its appearance[c] in
Niels Bohr's 1913 paper,[100]:
15 where it was denoted by .[d] For the next 15 years, the combination continued to appear in the literature, but normally without a separate symbol.[e] Then, in 1926, in their seminal papers,
Schrödinger and
Dirac again introduced special symbols for it: in the case of Schrödinger,[113] and in the case of Dirac.[114] Dirac continued to use in this way until 1930,[115]:
291 when he introduced the symbol in his book The Principles of Quantum Mechanics.[115]:
291[116]
^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 if 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 .
^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.
^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.)
^
ab"Planck constant". The NIST Reference on Constants, Units, and Uncertainty.
NIST. 20 May 2019.
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^Hirosige, Tetu; Nisio, Sigeko (1964). "Formation of Bohr's theory of atomic constitution". Japanese Studies in History of Science. 3: 6–28.
^J. L. Heilbron, A History of Atomic Models from the Discovery of the Electron to the Beginnings of Quantum Mechanics, diss. (University of California, Berkeley, 1964).
^Einstein, Albert (2003),
"Physics and Reality"(PDF), Daedalus, 132 (4): 24,
doi:
10.1162/001152603771338742,
S2CID57559543, archived from
the original(PDF) on 2012-04-15, 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.
^
ab"reduced Planck constant". The NIST Reference on Constants, Units, and Uncertainty.
NIST. 20 May 2019.
Archived from the original on 2023-04-08. Retrieved 2023-09-03.
^Galgani, L.; Carati, A.; Pozzi, B. (December 2002). "The Problem of the Rate of Thermalization, and the Relations between Classical and Quantum Mechanics". In Fabrizio, Mauro; Morro, Angelo (eds.). Mathematical Models and Methods for Smart Materials, Cortona, Italy, 25 – 29 June 2001. pp. 111–122.
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^McCormmach, Russell (1 January 1966). "The atomic theory of John William Nicholson". Archive for History of Exact Sciences. 3 (2): 160–184.
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^Schwarzschild, K. (1916). "Zur Quantenhypothese". Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin: 548–568.
^Ehrenfest, P. (June 1917). "XLVIII. Adiabatic invariants and the theory of quanta". The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 33 (198): 500–513.
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^Landé, A. (June 1919). "Das Serienspektrum des Heliums". Physikalische Zeitschrift. 20: 228–234.
^Bohr, N. (October 1920). "Über die Serienspektra der Elemente". Zeitschrift für Physik. 2 (5): 423–469.
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^Heisenberg, Werner (December 1922). "Zur Quantentheorie der Linienstruktur und der anomalen Zeemaneflekte". Zeitschrift für Physik. 8 (1): 273–297.
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