Planck constant

Origin of the 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).^[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, ${\displaystyle h}$ , 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 frequency ν at absolute temperature T is given by

${\displaystyle B_{\nu }(\nu ,T)={\frac {2h\nu ^{3}}{c^{2}}}{\frac {1}{e^{\frac {h\nu }{k_{\mathrm {B} }T}}-1}}}$ ,

where ${\displaystyle k_{\text{B}}}$ is the Boltzmann constant, ${\displaystyle h}$ is the Planck constant, and ${\displaystyle c}$ is the speed of light in the medium, whether material or vacuum.^[9][10][11]

The spectral radiance of a body, ${\displaystyle B_{\nu }}$ , 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 ${\displaystyle \lambda }$ instead of per unit frequency. In this case, it is given by

${\displaystyle B_{\lambda }(\lambda ,T)={\frac {2hc^{2}}{\lambda ^{5}}}{\frac {1}{e^{\frac {hc}{\lambda k_{\mathrm {B} }T}}-1}}}$ ,

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 ${\displaystyle B_{\nu }}$ are W·sr⁻¹·m⁻²·Hz⁻¹, while those of ${\displaystyle B_{\lambda }}$ 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 U_N [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":

${\displaystyle E=hf.}$

Planck was able to calculate the value of ${\displaystyle h}$ 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 ${\displaystyle k_{\text{B}}}$ 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]

Photoelectric effect

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:

${\displaystyle E=hf.}$

Einstein's postulate was later proven experimentally: the constant of proportionality between the frequency of incident light ${\displaystyle f}$ and the kinetic energy of photoelectrons ${\displaystyle E}$ was shown to be equal to the Planck constant ${\displaystyle h}$ .^[20]

Atomic structure

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 ${\displaystyle E_{n}}$

${\displaystyle E_{n}=-{\frac {hcR_{\infty }}{n^{2}}},}$

where ${\displaystyle c}$ is the speed of light in vacuum, ${\displaystyle R_{\infty }}$ is an experimentally determined constant (the Rydberg constant) and ${\displaystyle n\in \{1,2,3,...\}}$ . Once the electron reached the lowest energy level (${\displaystyle n=1}$ ), 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 ${\displaystyle R_{\infty }}$ in terms of other fundamental constants.

Bohr also introduced the quantity ${\displaystyle \hbar ={\frac {h}{2\pi }}}$ , 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 ${\displaystyle J}$ is the total angular momentum of a system with rotational invariance, and ${\displaystyle J_{z}}$ the angular momentum measured along any given direction, these quantities can only take on the values

${\displaystyle {\begin{aligned}J^{2}=j(j+1)\hbar ^{2},\qquad &j=0,{\tfrac {1}{2}},1,{\tfrac {3}{2}},\ldots ,\\J_{z}=m\hbar ,\qquad \qquad \quad &m=-j,-j+1,\ldots ,j.\end{aligned}}}$

Uncertainty principle

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, ${\displaystyle \Delta x}$ , and the uncertainty in their momentum, ${\displaystyle \Delta p_{x}}$ , obey

${\displaystyle \Delta x\,\Delta p_{x}\geq {\frac {\hbar }{2}},}$

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 ${\displaystyle {\hat {x}}}$ and the momentum operator ${\displaystyle {\hat {p}}}$ :

${\displaystyle [{\hat {p}}_{i},{\hat {x}}_{j}]=-i\hbar \delta _{ij},}$

where ${\displaystyle \delta _{ij}}$ is the Kronecker delta.

Photon energy

The Planck relation connects the particular photon energy E with its associated wave frequency f:

${\displaystyle E=hf.}$

This energy is extremely small in terms of ordinarily perceived everyday objects.

Since the frequency f, wavelength λ, and speed of light c are related by ${\displaystyle f={\frac {c}{\lambda }}}$ , the relation can also be expressed as

${\displaystyle E={\frac {hc}{\lambda }}.}$

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 wavelength λ of the particle is given by

${\displaystyle \lambda ={\frac {h}{p}},}$

where p denotes the linear momentum of a particle, such as a photon, or any other elementary particle.

The energy of a photon with angular frequency ω = 2πf is given by

${\displaystyle E=\hbar \omega ,}$

while its linear momentum relates to

${\displaystyle p=\hbar k,}$

where k is an angular wavenumber.

These two relations are the temporal and spatial parts of the special relativistic expression using 4-vectors.

${\displaystyle P^{\mu }=\left({\frac {E}{c}},{\vec {p}}\right)=\hbar K^{\mu }=\hbar \left({\frac {\omega }{c}},{\vec {k}}\right).}$

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]: 373Related 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.

The Planck constant has the same dimensions as action and as angular momentum. In SI units, the Planck constant is expressed with the unit joule per hertz (J⋅Hz⁻¹) or joule-second (J⋅s).

$${\displaystyle h=\mathrm {6.626\ 070\ 15\times 10^{-34}{J\cdot s}} }$$

$${\displaystyle \hbar ={h \over 2\pi }=\mathrm {1.054\ 571\ 817\times 10^{-34}\ {J\cdot s}} =\mathrm {6.582\ 119\ 569...\times 10^{-16}\ {eV\cdot s}} .}$$

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 light c and duration of hyperfine transition of the ground state of an unperturbed caesium-133 atom Δν_Cs."^[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, N_A = 6.02214076×10²³ mol^−1[38], with the result of 216 kJ, about the food energy in three apples.^{[citation needed]}

In many applications, the Planck constant ${\textstyle h}$ naturally appears in combination with ${\textstyle 2\pi }$ as ${\textstyle h/(2\pi )}$ , 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 ${\textstyle 2\pi }$ ^[39] and denoted by ${\textstyle \hbar }$ (pronounced h-bar^[43]: 336).

Many of the most important equations, relations, definitions, and results of quantum mechanics are customarily written using the reduced Planck constant ${\textstyle \hbar }$ rather than the Planck constant ${\textstyle h}$ , including the Schrödinger equation, momentum operator, canonical commutation relation, Heisenberg's uncertainty principle, and Planck units.^[44]: 104

Because the fundamental equations look simpler when written using ${\textstyle \hbar }$ as opposed to ${\textstyle h}$ , it is usually ${\textstyle \hbar }$ rather than ${\textstyle h}$ that gives the most reliable results when used in order-of-magnitude estimates.^{[45]: 8–9  [a]}

Names

The reduced Planck constant is known by many other names: the rationalized Planck constant^{[48]: 726  [49]: 10  [50]: -} (or rationalized Planck's constant^{[51]: 334  [52]: ix  [53]: 112} ), the Dirac constant^{[54]: 275  [48]: 726  [55]: xv} (or Dirac's constant^{[56]: 148  [57]: 604  [58]: 313}), the Dirac ${\textstyle h}$ ^{[59][60]: xviii} (or Dirac's ${\textstyle h}$ ^[61]: 17 ), the Dirac ${\textstyle \hbar }$ ^[62]: 187 (or Dirac's ${\textstyle \hbar }$ ^{[63]: 273  [64]: 14} ), and h-bar.^{[65]: 558 [66]: 561} It is also common to refer to this ${\textstyle \hbar }$ as “Planck's constant”^{[67]: 55  [b]} while retaining the relationship ${\textstyle \hbar \,{=}h/(2\pi )}$ .

Symbols

By far the most common symbol for the reduced Planck constant is ${\textstyle \hbar }$ . However, there are some sources that denote it by ${\textstyle h}$ instead, in which case they usually refer to it as the “Dirac ${\textstyle h}$ ”^{[93]: 43  [94]: 151} (or “Dirac's ${\textstyle h}$ ”^[95]: 21).

History

The combination ${\textstyle h/(2\pi )}$ first made its appearance^[c] in Niels Bohr's 1913 paper,^[100]: 15 where it was denoted by ${\textstyle M_{0}}$ .^[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: ${\textstyle K}$ in the case of Schrödinger,^[113] and ${\textstyle h}$ in the case of Dirac.^[114] Dirac continued to use ${\textstyle h}$ in this way until 1930,^[115]: 291 when he introduced the symbol ${\textstyle \hbar }$ in his book The Principles of Quantum Mechanics.^{[115]: 291  [116]}

• CODATA 2018
• International System of Units
• Introduction to quantum mechanics
• List of scientists whose names are used in physical constants
• Planck units
• Wave–particle duality

Citations

Sources

• "The role of the Planck constant in physics" – presentation at 26th CGPM meeting at Versailles, France, November 2018 when voting took place.
• “The Planck constant and its units” – presentation at the 35th Symposium on Chemical Physics at the University of Waterloo, Waterloo, Ontario, Canada, November 3 2019.