Planck's quantum of action

Physical constant
Surname	Planck's quantum of action
Formula symbol	${\displaystyle h}$
Size type	effect
value
SI	6th.626 070 15e-34 J s
Uncertainty  (rel.)	(exactly)
Gauss	6th.626 070 15e-27 erg p
Planck units	${\displaystyle 2\pi }$
Sources and Notes
Source SI value: CODATA  2018 ( direct link )

Memorial plaque - Humboldt University of Berlin

The Planck's constant, or the Planck constant , the ratio of energy ( ) and the frequency ( ) of a photon corresponding to the formula . The same relationship applies generally between the energy of a particle or physical system and the frequency of its quantum mechanical phase . ${\displaystyle h}$ ${\displaystyle E}$ ${\displaystyle f}$ ${\displaystyle E=h\cdot f}$

The discovery of the quantum of action by Max Planck in 1899 and 1900 founded quantum physics . The quantum of action links properties that were previously ascribed either only to particles or only to waves in classical physics . This makes it the basis of the wave-particle dualism of modern physics .

At that time, Planck regarded the quantum of action as the third of the fundamental natural constants in physics, alongside the gravitational constant and the speed of light . Together these constants form the basis of the natural system of Planck units . He gave the constant he discovered the name "elementary quantum of action" because it plays a decisive role in "elementary oscillation processes" and according to the definition (see above) results as the quotient of an energy and a frequency, which is why it has the same dimension as the physical one Size has an effect .

definition

For every physical system that can oscillate harmoniously, Planck's quantum of action is always the same ratio of the smallest possible energy expenditure to the oscillation frequency. Larger energy conversions are only possible if they are integer multiples of this smallest amount of energy. In addition, in quantum mechanics it applies to every physical system that the ratio of its total energy content to the frequency of its quantum mechanical phase is. ${\displaystyle h}$ ${\displaystyle h}$

Planck's quantum of action has the dimension of energy times time , which is called effect . It receives its universal meaning through its appearance in the basic equations of quantum physics ( Schrödinger equation , Heisenberg equation of motion , Dirac equation ).

Some general consequences

• Every harmonic oscillation (with frequency , angular frequency ) can only absorb or release energy in discrete amounts that are integral multiples of the oscillation quantum . ${\displaystyle f}$ ${\displaystyle \omega =2\pi f}$ ${\displaystyle \Delta E=hf\equiv \hbar \omega }$
• Every physical system can only change its angular momentum (more precisely: the projection of the angular momentum vector onto an arbitrary straight line) by integer multiples of . ${\displaystyle J}$ ${\displaystyle {\vec {J}}}$ ${\displaystyle \hbar \equiv {\tfrac {h}{2\pi }}}$
• Each physical system with pulse is a matter wave having the wavelength assigned. ${\displaystyle p}$ ${\displaystyle \lambda ={\tfrac {h}{p}}}$
• In every physical system, the energy and angular frequency of its quantum mechanical phase fulfill the equation . ${\displaystyle E}$ ${\displaystyle \omega }$ ${\displaystyle E=\hbar \omega }$
• Any two variables of a physical system that are canonically conjugated to one another (e.g. position and momentum of a particle, or generalized position and generalized momentum , e.g. angle of rotation and angular momentum) fulfill an uncertainty relation , according to which they are not in any state of the Systems can both have well-defined values ​​at the same time. Rather applies to the variations of both variables of values: . ${\displaystyle x}$ ${\displaystyle p}$ ${\displaystyle \sigma _{x},\,\sigma _{p}}$ ${\displaystyle \sigma _{x}\cdot \sigma _{p}\geq {\tfrac {\hbar }{2}}}$

Value and mark

Planck's quantum of action is one of the natural constants that are used to define the basic units in the International System of Units and have therefore been defined with an exact value. Its value is 6.626 070 15 · 10 ⁻³⁴  Js. The value of Planck's quantum of action is still exact if it is given in electron volts per hertz, but the numerical value as the quotient of the two exact numbers 6.62607015 · 10 ⁻³⁴ and 1.602176634 · 10 ⁻¹⁹ does not have a finite representation of decimal places and does not have to therefore be abbreviated with ...: 4.135 667 696 92 ...e^-15 eV s.

Until the change in the definition of the base units on May 20, 2019, h had to be determined experimentally and was therefore subject to a measurement uncertainty. The value was 6.626 070 040 (81) · 10 ⁻³⁴  Js, with the number in ^brackets indicating the estimated uncertainty (1 standard uncertainty ) for the mean value and referring to the last two decimal digits given.

Reduced Planck's quantum of action

Because frequencies are often given as angular frequency instead of frequency , in many equations the reduced Planck constant (pronounced: “h across”) is used instead of the quantum of action . Thus applies: . It is also by Paul Dirac as Dirac constant refers to and is its value: ${\displaystyle \textstyle \omega =2\pi f}$ ${\displaystyle \textstyle f}$ ${\displaystyle \textstyle h}$ ${\displaystyle \textstyle \hbar ={\frac {h}{2\pi }}}$ ${\displaystyle \textstyle hf=\hbar \omega }$

${\displaystyle {\begin{aligned}\hbar &=1{,}054\,571\,817\ldots \cdot 10^{-34}\,\mathrm {J\,s} \\&=6{,}582\,119\,569\ldots \cdot 10^{-16}\,\mathrm {eV\,s} \end{aligned}}}$

Often the product is required with the speed of light , which expresses a universal connection between the energy and length scale due to its dimension of energy times length . In the units customary in nuclear physics , the following applies: ${\displaystyle \hbar c}$ ${\displaystyle c}$

${\displaystyle \hbar c=197{,}326\,9804\ldots \mathrm {MeV\,fm} }$

Since it is also precisely defined, the product is also exact. ${\displaystyle c}$ ${\displaystyle \hbar c}$

character

In Unicode , the symbols for Planck's quantum of action and for the reduced Planck's quantum of action are in position U + 210E ( ℎ ) and U + 210F ( ℏ ).

History for discovery and reception

Thermal radiation I (Planck 1899)

Max Planck had discovered a new natural constant in 1899 when he developed a thermodynamic description of the thermal radiation of black bodies , also known as cavity radiation . According to Kirchhoff's law , the spectrum of thermal radiation and its temperature dependency, such as is visible on charcoal when the transition from red heat to white heat, should be exactly the same for all ideal black bodies, completely independent of their other properties. The calculation of the spectrum was therefore considered an outstanding unsolved problem in theoretical physics.

In the high-frequency (i.e. short-wave) range, the measured values ​​show a characteristic decrease towards higher frequencies. According to Wien's radiation law, this can be well represented by an exponential factor ( frequency, temperature, a fixed parameter), but this formula contradicts any theoretical derivation from classical physics. However, Planck was able to provide a novel theoretical derivation. To do this, he analyzed the thermal equilibrium between the walls of a cavity and the electromagnetic waves inside it. He modeled the walls as a collection of emitting and absorbing oscillators and selected a novel, suitable formula with two free parameters and for their entropy . Due to the generally applicable derivation, these parameters now have universal significance. turned out to be the parameter mentioned above in Wien's radiation law, as the product of with Boltzmann's constant . For , which was later renamed to, Planck stated the value , only 4% above the current value for . Planck also recognized that these new constants, together with the gravitational constant and the speed of light, form a system of universal natural constants, from which universal units can be formed for length, mass, time and temperature, the Planck units . ${\displaystyle e^{-af/T}}$ ${\displaystyle f}$ ${\displaystyle T}$ ${\displaystyle a}$ ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle a}$ ${\displaystyle k_{\mathrm {B} }}$ ${\displaystyle b}$ ${\displaystyle h}$ ${\displaystyle b=6{,}885\cdot 10^{-27}\,\mathrm {erg\,s} }$ ${\displaystyle h}$

Thermal radiation II (Planck 1900)

New measurements contradicted Vienna's radiation law and thus also the interpretation found by Planck. They showed that in the low-frequency (i.e. long-wave, infrared) part of the thermal radiation, the intensity initially increases towards higher frequencies before it decreases again in accordance with Wien's radiation law. This increase corresponded well to the Rayleigh-Jeans law , as it had been derived without further assumptions from classical electrodynamics and the uniform distribution theorem of statistical mechanics. However, this law also predicted an unlimited increase in intensity with further increasing frequency, which was known as an ultraviolet disaster and contradicted the older measurements in the high-frequency part of the spectrum (see above). Planck found (literally) “a happily guessed interpolating formula” that now agreed excellently with all measurements (including new ones made afterwards). Theoretically, he could only derive this result, known as Planck's law of radiation , by experimentally interpreting the exponential factor of Wien's law as the Boltzmann factor known from kinetic gas theory and using it for the different discrete energy levels depending on the frequency . He took the letter of auxiliary size. The comparison with Wien's formula showed that it is precisely the product mentioned . ${\displaystyle e^{-{\tfrac {\Delta E}{k_{\mathrm {B} }T}}}}$ ${\displaystyle \Delta E}$ ${\displaystyle f}$ ${\displaystyle \Delta E=hf}$ ${\displaystyle h}$ ${\displaystyle h}$ ${\displaystyle b=ak_{\mathrm {B} }}$

Planck thus assigned the oscillators the new property that they could only change their energy in finite steps of size . He thus introduced for the first time a quantization of an apparently continuously variable quantity, an idea that was completely alien to physics at the time, when the atomic hypothesis was still being fiercely attacked. But all attempts to find a theoretical derivation without the assumption of discrete energy conversions failed. Initially, Planck did not consider the discontinuous character of the energy exchange to be a property of the supposedly well-understood light waves, but attributed it exclusively to the emission and absorption processes in the material of the cavity walls. With great delay he was awarded the Nobel Prize in 1918 for the discovery of quantization . ${\displaystyle \Delta E=hf}$

h and the light quanta

In 1905 Albert Einstein analyzed the photoelectric effect , which is also incompatible with classical physics. Einstein was one of the few physicists who recognized and made use of the fundamental importance of Planck's work early on. He was able to explain the effect with the help of the light quantum hypothesis , according to which light also has quantum properties. Accordingly, contrary to Planck's view at the time, electromagnetic radiation itself consists of particle-like objects, the light quanta, the energy of which is given by the equation depending on the frequency of  the light wave . This equation was later called Einstein's equation for the light quantum. This was the first time he recognized the wave-particle dualism , a new problem for physics. Not least because of this, this analysis also took years to establish itself. In 1921 it brought Einstein the Nobel Prize. ${\displaystyle f}$ ${\displaystyle E=hf}$

See also : Determination of h with the photoelectric effect

h and the specific heat of solid bodies

For Albert Einstein in 1907, the quantization of the vibrational energy was also the key to explaining another misunderstood phenomenon, the decrease in the specific heat of solid bodies towards low temperatures. At higher temperatures, on the other hand, the measured values ​​mostly agreed well with the value predicted by Dulong-Petit based on classical physics. Einstein assumed that the thermal energy in the solid body is in the form of oscillations of the atoms around their position of rest, and that this purely mechanical type of oscillations can only be excited in energy levels. Since the amounts of energy fluctuating between the individual atoms in the thermal equilibrium are of the order of magnitude , it was possible to differentiate between “high” temperatures ( ) and “low” temperatures ( ). Then the quantization has no visible effects at high temperatures, while it hinders the absorption of thermal energy at low temperatures. The formula that Einstein was able to derive from this idea matched the data measured at that time perfectly (after a suitable definition for each solid body). Nevertheless, for a long time it was still doubted that Planck's constant could be important not only for electromagnetic waves, but also in the field of mechanics. ${\displaystyle \Delta E=hf}$ ${\displaystyle k_{\mathrm {B} }T}$ ${\displaystyle k_{\mathrm {B} }T>hf}$ ${\displaystyle k_{\mathrm {B} }T<hf}$ ${\displaystyle f}$

h and the phase space cell

Many laws of thermodynamics, e.g. B. to the specific heat of gases and solids, but also to the irreversible increase in entropy and the shape of the state of equilibrium achieved thereby, had been given a mechanical interpretation by statistical mechanics (especially by Ludwig Boltzmann and Josiah Willard Gibbs ). Statistical mechanics is based on the assumption of the disordered movement of an extremely large number of atoms or molecules and uses statistical methods to determine the most probable values ​​of macroscopically measurable quantities (such as density, pressure, etc.) in order to characterize the state of equilibrium. To do this, the total amount of all possible states of all particles must first be mathematically recorded in a state or phase space . If a certain macroscopic state is determined, then all particle states in which the system shows this macroscopic state form a partial volume in phase space. The size of each such partial volume is used to determine the probability with which the macroscopic state in question will occur. Mathematically, a volume integral has to be formed, and for this purpose the definition of a volume element, also called a phase space cell, is needed temporarily and as an auxiliary variable . In the end result, however, the phase space cell should no longer appear. If possible, one lets their size in the obtained formula shrink to zero (like differential sizes in general in infinitesimal calculus ), if not, one sees them as an undesirable parameter (which e.g. determines an unknown additive constant) and just tries to consider such conclusions that are independent of the phase space cell (e.g. differences in which the constant cancels out). If the entropy of a gas is calculated in this way, the constant is called the chemical constant . Otto Sackur noticed to his surprise in 1913 that the phase space cell had to be given a certain size so that the chemical constant matched the measured values. The phase space cell (per particle and per space dimension of its movement) must be exactly the size . He gave his publication the title The universal meaning of the so-called Planck quantum of action and Max Planck called it “of fundamental importance” if the daring hypothesis should prove to be true that this result applies regardless of the type of gas. This was the case. ${\displaystyle h}$

What is fundamental about this result is, in particular, that a deep reason for the phenomenon of quantization begins to appear here, which, however, only became clear in full years later with the quantum statistics of radiation. A phase space cell can also be defined for oscillations, and then Einstein's formula shows that the phase space cell for the light quantum also has the following size : The physical size that is decisive for the size of the phase space cell is the effect here , with an oscillation it is the effect the product of energy and period , so it follows . ${\displaystyle E=hf}$ ${\displaystyle h}$ ${\displaystyle E}$ ${\displaystyle T={\tfrac {1}{f}}}$ ${\displaystyle ET=h}$

h and the size of the atoms

Classical physics must fail to explain the stable size of atoms. Because if it could explain a certain size, a z. B. half the size of an atom is then just as possible according to the same laws. In other words: The basic formulas of classical physics do not contain enough natural constants to be able to derive from them a formula for a quantity with the dimension of a length. The quantum of action can close this gap, as Planck himself noted in 1899 when he first presented the Planck units (see above). But because the prevailing opinion was that the quantum of action should not be introduced into mechanics, the first attempt to use it to explain the atomic radius was not made until 1910 by Arthur Erich Haas . T. ridiculed. Haas assumed that an electron was circling in the field of a positive charge and put the rotational frequency and the binding energy of this system in relation to one another . This results in a radius in the range of the atomic radii known from chemistry and kinetic gas theory. ${\displaystyle +e}$ ${\displaystyle f}$ ${\displaystyle E}$ ${\displaystyle E=hf}$

In 1913, Niels Bohr had more success; he assumed the same image in his atomic model , but also introduced circular orbits of different energies and, above all, the emission of light quanta when the quantum leap from one orbit to the other. The correspondence with the measured wavelengths, which he only obtained through a quantum condition that could hardly be justified ( with the new principal quantum number ), quickly made the model famous. The main role of the quantum of action in the internal structure of the atoms was proven. The quantum condition was quickly recognized as quantum angular momentum, because the circular path to the main quantum number can be defined by the condition that the angular momentum of the electron has the value . ${\displaystyle E={\tfrac {n}{2}}hf}$ ${\displaystyle n=1,2,3,\dotsc }$ ${\displaystyle n}$ ${\displaystyle L=nh/2\pi }$

This great progress made Bohr's atomic model the decisive starting point for further developments, although further similarly great progress did not materialize for years. In particular, attempts to understand atoms with several electrons have failed.

h and the matter waves

The success of Bohr's atomic model since 1913 was due in large part to Bohr's quantum condition, which intervenes heavily in mechanics from the outside by allowing the electron only a few of the mechanically possible paths. Due to the continuing difficulties with the further development of atomic theory, possibilities were sought to redesign the mechanics itself in such a way that it takes the quantum condition into account from the outset. The previous quantum theory was to be replaced by real quantum mechanics . Louis de Broglie made the greatest step before the real beginning of quantum mechanics in 1924 when he discovered material particles, e.g. B. attributed to electrons, wave properties . He transferred the relationship between momentum and wavelength found for photons to the matter wave of the electron, which he imagined . With that he extended the wave-particle dualism to particles. As an immediate success it turns out that the Bohr circular orbit to the main quantum number has exactly the circumference , so the matter wave of the electron can form a standing wave on it. Without being able to say much about this matter wave, Erwin Schrödinger found a formula for the propagation of this wave in a force field in early 1926, with which he founded wave mechanics . For the stationary states of the hydrogen atom, he was able to precisely calculate the known results with this Schrödinger equation without additional quantum conditions. In addition, known errors in the Bohr model have been fixed, e.g. B. that the atom is flat or that the angular momentum cannot be. The only natural constant that appears in the Schrödinger equation is the quantum of action . The same applies to the equation that Werner Heisenberg obtained a few months earlier from a “quantum theoretical reinterpretation of kinematic and mechanical relationships”, with which he founded matrix mechanics . Both approaches are mathematically equivalent and are seen as the basic equations of actual quantum mechanics. What remains, however, are the difficulties of getting a picture of the quantum mechanical concepts and processes that is compatible with the wave-particle dualism. ${\displaystyle p=h/\lambda }$ ${\displaystyle {\vec {p}}}$ ${\displaystyle \lambda }$ ${\displaystyle n}$ ${\displaystyle n\lambda }$ ${\displaystyle L=0\hbar }$ ${\displaystyle h}$

Angular momentum

For Planck, the term “quantum of effect” was initially motivated solely by the physical dimension of energy times time of the constant , which is referred to as effect . However, the classical mechanical orbital angular momentum has the same dimension, and in general has also been shown to be the natural constant that is decisive for angular momentum. ${\displaystyle h}$ ${\displaystyle {\vec {l}}={\vec {r}}\times {\vec {p}}}$ ${\displaystyle \hbar }$

In the atomic model established by Niels Bohr in 1913 , after it was expanded to include Bohr-Sommerfeld's atomic model in 1917 , the orbital angular momentum vector of the electron appears as a doubly quantized quantity. According to the amount, as in Bohr's model, it can only assume integer multiples of : with the angular momentum quantum number . In addition, the condition applies that the projection of the angular momentum vector of the length onto a coordinate axis can only assume values where the magnetic quantum number is an integer (see directional quantization ) and is limited to the range from to . For the orbits to the principal quantum number can have all values . ${\displaystyle {\vec {l}}={\vec {r}}\times {\vec {p}}}$ ${\displaystyle \hbar }$ ${\displaystyle |{\vec {l}}|=l\hbar }$ ${\displaystyle l}$ ${\displaystyle l\hbar }$ ${\displaystyle m\hbar }$ ${\displaystyle m}$ ${\displaystyle -l}$ ${\displaystyle +l}$ ${\displaystyle n}$ ${\displaystyle l}$ ${\displaystyle l=1,2,\dotsc ,n}$

In the quantum mechanics founded by Werner Heisenberg and Erwin Schrödinger in 1925 , the same quantification of the orbital angular momentum results in that it is represented by the operator . However, the absolute value of the angular momentum vector now has the length . In addition, according to quantum mechanical calculations, the orbital angular momentum quantum numbers belong to the electron states with principal quantum numbers in the hydrogen atom , so these are 1 smaller than in the Bohr-Sommerfeld model. This agrees with all of the observations. ${\displaystyle {\hat {\vec {l}}}={\hat {\vec {r}}}\times {\hat {\vec {p}}}}$ ${\displaystyle |{\vec {l}}|={\sqrt {l(l+1)}}\hbar }$ ${\displaystyle n}$ ${\displaystyle l=0,1,\dotsc ,n-1}$

In addition to the orbital angular momentum, the particles (as well as particle systems) can also have spin , which is an intrinsic angular momentum around their own center of gravity, often referred to as. Spin is also expressed in units of . There are particles whose spin is an integral multiple of ( bosons ), but there are also particles with half- integer multiples of ( fermions ). The distinction between the two types of particles, bosons and fermions, is fundamental in physics. The expansion from only integer to half-integer quantum numbers of the angular momentum results from the properties of the quantum mechanical spin operator . Its three components fulfill the same commutation relations as the components of the orbital angular momentum operator . The same applies to the orbital angular momentum, but this does not apply to the spin. ${\displaystyle {\vec {s}}}$ ${\displaystyle \hbar }$ ${\displaystyle \hbar }$ ${\displaystyle \hbar }$ ${\displaystyle {\hat {\vec {s}}}}$ ${\displaystyle {\hat {\vec {l}}}}$ ${\displaystyle {\hat {\vec {l}}}\cdot {\hat {\vec {p}}}=0}$

Uncertainty relation

In Heisenberg's commutation relation , the (reduced) Planck quantum of action occurs as the value of the commutator between position and momentum operator :

${\displaystyle \left[{\hat {X}},{\hat {P}}\right]\;=\;\mathrm {i} \hbar }$

As a result, Heisenberg's uncertainty principle applies to the product of position and momentum uncertainty

${\displaystyle \Delta x\,\Delta p\;\geq \;{\frac {\hbar }{2}}\;.}$

Von Klitzing's constant

→ Main article : Von Klitzing constant

The Von Klitzing constant (with the elementary charge ) occurs in the quantum Hall effect . Their value was until May 19, 2019 . This constant could be measured extremely accurately. Therefore, analogous to the modern definition of the speed of light, it could serve to trace the determination of Planck's constant back to very precise resistance measurements. ${\displaystyle R_{\mathrm {K} }={\tfrac {h}{e^{2}}}}$ ${\displaystyle e}$ ${\displaystyle R_{\mathrm {K} }\approx 25\ 812{,}807\ 4555(59)\ \Omega }$ ${\displaystyle h}$

With the new definition of the SI units on May 20, 2019, this constant also got an exact value of:

${\displaystyle R_{\mathrm {K} }={\frac {6{,}626\,070\,15\cdot 10^{-34}\mathrm {J\cdot s} }{(1{,}602\,176\,634\cdot 10^{-19}\mathrm {C} )^{2}}}=25\ 812{,}807\ 45\ldots \ \Omega .}$

literature

• Domenico Giulini: "Long live the brazenness!" - Albert Einstein and the foundation of quantum theory. online (PDF; 453 kB). In: Herbert Hunziker: The young Einstein and Aarau. Birkhäuser 2005, ISBN 3-7643-7444-6 .
• Enrico G. Beltrametti: One Hundred Years of h. Italian Physical Soc., Bologna 2002, ISBN 88-7438-003-8 .

Web links

Wiktionary: Effect quantum  - explanations of meanings, word origins, synonyms, translations

• Armin Hermann: How Max Planck came up with the idea of ​​quantum theory.
• Jörg Resag: What is the quantum of action?
• h-determination with LEDs. Simple experiment to determine Planck's quantum of action at student level.
• Diode lighting test (PDF; 208 kB) Determination of Planck's quantum of action by lighting different colored light emitting diodes.

Individual evidence

1. ↑ It follows from that which applies in natural units . ${\displaystyle \hbar =1}$ ${\displaystyle h=2\pi }$
2. ↑ ^{a b} Max Planck (note: a = Boltzmann constant (temp.), B = effective quantum, f = gravitational constant, c = speed of light): About irreversible radiation processes. Session reports of the Royal PREUSSIAN Academy of Sciences at Berlin. 1899 - First half volume (Publ. Of the Royal Academy of Sciences, Berlin 1899). Page 479-480.
3. ↑ Michael Bonitz: Max Planck, the quantum of effectiveness and modern physics. (PDF).
4. ^ Günter Sturm: 100 years of quantum theory. At: quanten.de. Special edition December 14, 2000.
5. ↑ Max Planck: Lectures on the theory of thermal radiation. Verlag Joh.Amb.Barth, Leipzig 1906, p. 154.
6. ↑ Max Planck: On the history of the discovery of the physical quantum of action. Naturwissenschaften Vol. 31, No. 14 (1943), pp. 153-159.
7. ↑ CODATA Value: Planck constant in eV / Hz. In: The NIST Reference on Constants, Units and Uncertainty. National Institute of Standards and Technology, accessed April 15, 2020 . 
8. ↑ ^{a b} Fundamental Physical Constants - Extensive Listing. (PDF; 128 kB) Archive 2014. In: The NIST Reference on Constants, Units and Uncertainty. National Institute of Standards and Technology, 2014, p. 1 , accessed on May 23, 2019 . 
9. ↑ The one-character notation for the reduced Planck quantum of action was introduced in 1926 by P. A. M. Dirac. A short section on history can be found e.g. B. in M. Jammer: The Conceptual Development of Quantum Mechanics. McGraw-Hill, New York 1966, p. 294. Original Dirac paper: P. A. M. Dirac: Quantum mechanics and a preliminary investigation of the hydrogen atom. Proc. Roy. Soc. A, 110 (1926), pp. 561-579.
10. ↑ CODATA Value: reduced Planck constant. In: The NIST Reference on Constants, Units and Uncertainty. National Institute of Standards and Technology, accessed April 15, 2020 . 
11. ↑ J. Bleck-Neuhaus: Elementary particles. 2nd edition, Springer Verlag 2013, ISBN 978-3-642-32578-6 , pages 43-45.
12. ↑ CODATA Value: reduced Planck constant times c in MeV fm. In: The NIST Reference on Constants, Units and Uncertainty. National Institute of Standards and Technology, accessed April 15, 2020 .  
13. ↑ Abraham Pais: Introducing Atoms and their Nuclei. In: Laurie M. Brown et al. a. (Ed.): Twentieth Century Physics. Vol. I, IOP Publishing Ltd. AIP Press. Inc. 1995, ISBN 0-7503-0353-0 .
14. ↑ Max Planck: About irreversible radiation processes. Session reports of the Royal PREUSSIAN Academy of Sciences at Berlin. 1899 - First half volume (Publ. Of the Royal Academy of Sciences, Berlin 1899). Page 440-480.
15. ↑ M. Planck: On the theory of the law of energy distribution in the normal spectrum. Negotiations of the German Physical Society 2 (1900) No. 17, pp. 237–245, Berlin (presented on December 14, 1900).
16. ↑ Albert Einstein: About a heuristic point of view concerning the generation and transformation of light. Annalen der Physik, 17 (1905), p. 133 and p. 143. (Online document: PDF ).
17. ↑ Otto Sackur: The universal meaning of the so-called Planck quantum of action . In: Annals of Physics . tape 345 , 1913, pp. 67 , doi : 10.1002 / andp.19133450103 .
18. ↑ Max Planck: The current importance of the quantum hypothesis for the kinetic gas theory. Phys. Magazine Vol. 14 (1913) p. 258.
19. ^ Max Jammer: The Conceptual Development of Quantum Mechanics . McGraw-Hill, New York 1966.
20. ↑ E. Schrödinger: Quantization as an eigenvalue problem I. Annalen der Physik 79 (1926), pp. 361–376.
21. ^ W. Heisenberg: About quantum theoretical reinterpretation of kinematic and mechanical relationships. In: Journal of Physics. Volume 33, 1925, pp. 879-893.
22. ↑ Cornelius Noack : Comments on the quantum theory of orbital angular momentum . In: Physical sheets . tape 41 , no. 8 , 1985, pp. 283–285 ( see homepage [PDF; 154 kB ; accessed on November 26, 2012]).
23. ↑ CODATA Value: von Klitzing constant. In: The NIST Reference on Constants, Units and Uncertainty. National Institute of Standards and Technology, accessed May 23, 2019 .