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[edit] 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[edit] 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[edit] 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

  • [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[edit] 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
separate symbol: [102]: 428  [103]: 549  [104]: 508  [105]: 230  [106]: 458  [107][108]: 276  [109][110][111].[112]
6. “Planck constant”. The NIST Reference on Constants, Units, and Uncertainty.
NIST. 20 May 2019. Archived from the original on 2022-05-27. Retrieved 2023-09-03.
7. ^ Jump up to:a b c d e f Planck, Max (1901), “Ueber das Gesetz der Energieverteilung im Normalspectrum” (PDF), Ann. Phys., 309 (3): 553–63, Bibcode:1901AnP…309..553P,
doi:10.1002/andp.19013090310, archived (PDF) from the original on 2012-06-10, retrieved 2008-12-15. English translation: “On the Law of Distribution of Energy in the Normal Spectrum”. Archived from the original on 2008-04-18.”. “On the Law of Distribution
of Energy in the Normal Spectrum” (PDF). Archived from the original (PDF) on 2011-10-06. Retrieved 2011-10-13.
8. ^ “Max Planck Nobel Lecture”. Archived from the original on 2023-07-14. Retrieved 2023-07-14.
9. ^ Le Système international d’unités
[The International System of Units] (PDF) (in French and English) (9th ed.), International Bureau of Weights and Measures, 2019, p. 131, ISBN 978-92-822-2272-0
10. ^ “2018 CODATA Value: Planck constant”. The NIST Reference on Constants, Units, and
Uncertainty. NIST. 20 May 2019. Retrieved 2021-04-28.
11. ^ “Resolutions of the 26th CGPM” (PDF). BIPM. 2018-11-16. Archived from the original (PDF) on 2018-11-19. Retrieved 2018-11-20.
12. ^ Jump up to:a b Bitter, Francis; Medicus, Heinrich A.
(1973). Fields and particles. New York: Elsevier. pp. 137–144.
13. ^ Boya, Luis J. (2004). “The Thermal Radiation Formula of Planck (1900)”. arXiv:physics/0402064v1.
14. ^ Planck, M. (1914). The Theory of Heat Radiation. Masius, M. (transl.) (2nd
ed.). P. Blakiston’s Son. pp. 6, 168. OL 7154661M.
15. ^ Chandrasekhar, S. (1960) [1950]. Radiative Transfer (Revised reprint ed.). Dover. p. 8. ISBN 978-0-486-60590-6.
16. ^ Rybicki, G. B.; Lightman, A. P. (1979). Radiative Processes in Astrophysics.
Wiley. p. 22. ISBN 978-0-471-82759-7. Archived from the original on 2020-07-27. Retrieved 2020-05-20.
17. ^ Shao, Gaofeng; et al. (2019). “Improved oxidation resistance of high emissivity coatings on fibrous ceramic for reusable space systems”.
Corrosion Science. 146: 233–246. arXiv:1902.03943. doi:10.1016/j.corsci.2018.11.006. S2CID 118927116.
18. ^ Kragh, Helge (1 December 2000), Max Planck: the reluctant revolutionary,, archived from the original on 2009-01-08
19. ^
Kragh, Helge (1999), Quantum Generations: A History of Physics in the Twentieth Century, Princeton University Press, p. 62, ISBN 978-0-691-09552-3, archived from the original on 2021-12-06, retrieved 2021-10-31
20. ^ Planck, Max (2 June 1920), The
Genesis and Present State of Development of the Quantum Theory (Nobel Lecture), archived from the original on 15 July 2011, retrieved 13 December 2008
21. ^ Previous Solvay Conferences on Physics, International Solvay Institutes, archived from the
original on 16 December 2008, retrieved 12 December 2008
22. ^ Jump up to:a b See, e.g., Arrhenius, Svante (10 December 1922), Presentation speech of the 1921 Nobel Prize for Physics, archived from the original on 4 September 2011, retrieved 13
December 2008
23. ^ Jump up to:a b c Lenard, P. (1902), “Ueber die lichtelektrische Wirkung”, Annalen der Physik, 313 (5): 149–98, Bibcode:1902AnP…313..149L, doi:10.1002/andp.19023130510, archived from the original on 2019-08-18, retrieved 2019-07-03
24. ^
Einstein, Albert (1905), “Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt” (PDF), Annalen der Physik, 17 (6): 132–48, Bibcode:1905AnP…322..132E, doi:10.1002/andp.19053220607, archived (PDF) from the
original on 2011-07-09, retrieved 2009-12-03
25. ^ Jump up to:a b c Millikan, R. A. (1916), “A Direct Photoelectric Determination of Planck’s h”, Physical Review, 7 (3): 355–88, Bibcode:1916PhRv….7..355M, doi:10.1103/PhysRev.7.355
26. ^ Isaacson,
Walter (2007-04-10), Einstein: His Life and Universe, Simon and Schuster, ISBN 978-1-4165-3932-2, archived from the original on 2020-01-09, retrieved 2021-10-31, pp. 309–314.
27. ^ “The Nobel Prize in Physics 1921”. Archived from
the original on 2018-07-03. Retrieved 2014-04-23.
28. ^ *Smith, Richard (1962). “Two Photon Photoelectric Effect”. Physical Review. 128 (5): 2225. Bibcode:1962PhRv..128.2225S. doi:10.1103/PhysRev.128.2225.
a. Smith, Richard (1963). “Two-Photon
Photoelectric Effect”. Physical Review. 130 (6): 2599. Bibcode:1963PhRv..130.2599S. doi:10.1103/PhysRev.130.2599.4.
29. ^ Jump up to:a b Heilbron, John L. (2013). “The path to the quantum atom”. Nature. 498 (7452): 27–30. doi:10.1038/498027a. PMID
23739408. S2CID 4355108.
30. ^ Nicholson, J. W. (1911). “The spectrum of Nebulium”. Monthly Notices of the Royal Astronomical Society. 72: 49. Bibcode:1911MNRAS..72…49N. doi:10.1093/mnras/72.1.49.
31. ^ *Nicholson, J. W. (1911). “The Constitution
of the Solar Corona I”. Monthly Notices of the Royal Astronomical Society. 72: 139. Bibcode:1911MNRAS..72..139N. doi:10.1093/mnras/72.2.139.
a. Nicholson, J. W. (1912). “The Constitution of the Solar Corona II”. Monthly Notices of the Royal Astronomical
Society. 72 (8): 677–693. doi:10.1093/mnras/72.8.677.
b. Nicholson, J. W. (1912). “The Constitution of the Solar Corona III”. Monthly Notices of the Royal Astronomical Society. 72 (9): 729–740. doi:10.1093/mnras/72.9.729.
32. ^ Nicholson, J. W.
(1912). “On the new nebular line at λ4353”. Monthly Notices of the Royal Astronomical Society. 72 (8): 693. Bibcode:1912MNRAS..72..693N. doi:10.1093/mnras/72.8.693.
33. ^ Jump up to:a b McCormmach, Russell (1966). “The Atomic Theory of John William
Nicholson”. Archive for History of Exact Sciences. 3 (2): 160–184. doi:10.1007/BF00357268. JSTOR 41133258. S2CID 120797894.
34. ^ Jump up to:a b Bohr, N. (1913). “On the constitution of atoms and molecules”. The London, Edinburgh, and Dublin Philosophical
Magazine and Journal of Science. 6th series. 26 (151): 1–25. Bibcode:1913PMag…26..476B. doi:10.1080/14786441308634955. Archived from the original on 2023-03-07. Retrieved 2023-07-23.
35. ^ Hirosige, Tetu; Nisio, Sigeko (1964). “Formation of Bohr’s
theory of atomic constitution”. Japanese Studies in History of Science. 3: 6–28.
36. ^ 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,
37. ^ Giuseppe Morandi; F. Napoli; E. Ercolessi (2001), Statistical mechanics: an intermediate course, World Scientific, p. 84, ISBN 978-981-02-4477-4, archived from the original on 2021-12-06, retrieved 2021-10-31
38. ^ ter Haar, D. (1967).
The Old Quantum Theory. Pergamon Press. p. 133. ISBN 978-0-08-012101-7.
39. ^ Einstein, Albert (2003), “Physics and Reality” (PDF), Daedalus, 132 (4): 24, doi:10.1162/001152603771338742, S2CID 57559543, 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.
40. ^
41. ^ Le Système international d’unités
[The International System of Units] (PDF) (in French and English) (9th ed.), International Bureau of Weights and Measures, 2019, ISBN 978-92-822-2272-0
42. ^ “The Feynman Lectures on Physics Vol. II Ch. 19: The Principle of Least Action”.
Retrieved 2023-11-03.
43. ^ “2018 CODATA Value: Avogadro constant”. The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
44. ^ Jump up to:a b “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.
45. ^ Lyth, David H.; Liddle, Andrew R. (11 June 2009). The Primordial Density Perturbation: Cosmology, Inflation and the Origin of Structure.
Cambridge University Press. ISBN 978-1-139-64374-0.
46. ^ Huang, Kerson (26 April 2010). Quantum Field Theory: From Operators to Path Integrals. John Wiley & Sons. ISBN 978-3-527-40846-7.
47. ^ Schmitz, Kenneth S. (11 November 2016). Physical
Chemistry: Concepts and Theory. Elsevier. ISBN 978-0-12-800600-9.
48. ^ Chabay, Ruth W.; Sherwood, Bruce A. (20 November 2017). Matter and Interactions. John Wiley & Sons. ISBN 978-1-119-45575-2.
49. ^ Schwarz, Patricia M.; Schwarz, John H. (25
March 2004). Special Relativity: From Einstein to Strings. Cambridge University Press. ISBN 978-1-139-44950-2.
50. ^ Lévy-Leblond, Jean-Marc (2002). “The meanings of Planck’s constant” (PDF). In Beltrametti, E.; Rimini, A.; Robotti, Nadia (eds.).
One Hundred Years of H: Pavia, 14-16 September 2000. Italian Physical Society. ISBN 978-88-7438-003-9. Archived from the original (PDF) on 2023-10-14.
51. ^ Shu, Frank (1982). The Physical Universe: An Introduction to Astronomy. University Science
Books. ISBN 978-0-935702-05-7.
52. ^ “Bohr Radius — from Eric Weisstein’s World of Physics”. Archived from the original on 2023-10-14. Retrieved 14 October 2023.
53. ^ Jump up to:a b Rennie, Richard; Law, Jonathan,
eds. (2017). “Planck constant”. A Dictionary of Physics. Oxford Quick Reference (7th ed.). Oxford, UK: OUP Oxford. ISBN 978-0198821472.
54. ^ The International Encyclopedia of Physical Chemistry and Chemical Physics. Pergamon Press. 1960.
55. ^
Vértes, Attila; Nagy, Sándor; Klencsár, Zoltán; Lovas, Rezso György; Rösch, Frank (10 December 2010). Handbook of Nuclear Chemistry. Springer Science & Business Media. ISBN 978-1-4419-0719-6.
56. ^ Bethe, Hans A.; Salpeter, Edwin E. (1957). “Quantum
Mechanics of One- and Two-Electron Atoms”. In Flügge, Siegfried (ed.). Handbuch der Physik: Atome I-II. Springer.
57. ^ Lang, Kenneth (11 November 2013). Astrophysical Formulae: A Compendium for the Physicist and Astrophysicist. Springer Science
& Business Media. ISBN 978-3-662-11188-8.
58. ^ 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. doi:10.1142/9789812776273_0011. ISBN 978-981-238-235-1.
59. ^ Fox, Mark (14 June 2018). A Student’s Guide to Atomic Physics. Cambridge University
Press. ISBN 978-1-316-99309-5.
60. ^ Kleiss, Ronald (10 June 2021). Quantum Field Theory: A Diagrammatic Approach. Cambridge University Press. ISBN 978-1-108-78750-5.
61. ^ Zohuri, Bahman (5 January 2021). Thermal Effects of High Power Laser Energy
on Materials. Springer Nature. ISBN 978-3-030-63064-5.
62. ^ Balian, Roger (26 June 2007). From Microphysics to Macrophysics: Methods and Applications of Statistical Physics. Volume II. Springer Science & Business Media. ISBN 978-3-540-45480-9.
63. ^
Chen, C. Julian (15 August 2011). Physics of Solar Energy. John Wiley & Sons. ISBN 978-1-118-04459-9.
64. ^ “Dirac h”. Britannica. Archived from the original on 2023-02-17. Retrieved 2023-09-27.
65. ^ Shoenberg, D. (3 September 2009). Magnetic
Oscillations in Metals. Cambridge University Press. ISBN 978-1-316-58317-3.
66. ^ Powell, John L.; Crasemann, Bernd (5 May 2015). Quantum Mechanics. Courier Dover Publications. ISBN 978-0-486-80478-1.
67. ^ Dresden, Max (6 December 2012). H.A.
Kramers Between Tradition and Revolution. Springer Science & Business Media. ISBN 978-1-4612-4622-0.
68. ^ Johnson, R. E. (6 December 2012). Introduction to Atomic and Molecular Collisions. Springer Science & Business Media. ISBN 978-1-4684-8448-9.
69. ^
Garcia, Alejandro; Henley, Ernest M. (13 July 2007). Subatomic Physics (3rd ed.). World Scientific Publishing Company. ISBN 978-981-310-167-8.
70. ^ Holbrow, Charles H.; Lloyd, James N.; Amato, Joseph C.; Galvez, Enrique; Parks, M. Elizabeth (14
September 2010). Modern Introductory Physics. New York: Springer Science & Business Media. ISBN 978-0-387-79080-0.
71. ^ Polyanin, Andrei D.; Chernoutsan, Alexei (18 October 2010). A Concise Handbook of Mathematics, Physics, and Engineering Sciences.
CRC Press. ISBN 978-1-4398-0640-1.
72. ^ Dowling, Jonathan P. (24 August 2020). Schrödinger’s Web: Race to Build the Quantum Internet. CRC Press. ISBN 978-1-000-08017-9.
73. ^ Landau, L. D.; Lifshitz, E. M. (22 October 2013). Quantum Mechanics:
Non-Relativistic Theory. Elsevier. ISBN 978-1-4831-4912-7.
74. ^ Griffiths, David J.; Schroeter, Darrell F. (20 November 2019). Introduction to Quantum Mechanics. Cambridge University Press. ISBN 978-1-108-10314-5.
75. ^ “Planck’s constant”.
The Great Soviet Encyclopedia (1970-1979, 3rd ed.). The Gale Group.
76. ^ Itzykson, Claude; Zuber, Jean-Bernard (20 September 2012). Quantum Field Theory. Courier Corporation. ISBN 978-0-486-13469-7.
77. ^ Kaku, Michio (1993). Quantum Field Theory:
A Modern Introduction. Oxford University Press. ISBN 978-0-19-507652-3.
78. ^ Bogoli︠u︡bov, Nikolaĭ Nikolaevich; Shirkov, Dmitriĭ Vasilʹevich (1982). Quantum Fields. Benjamin/Cummings Publishing Company, Advanced Book Program/World Science Division.
ISBN 978-0-8053-0983-6.
79. ^ Aitchison, Ian J. R.; Hey, Anthony J. G. (17 December 2012). Gauge Theories in Particle Physics: A Practical Introduction: From Relativistic Quantum Mechanics to QED, Fourth Edition. CRC Press. ISBN 978-1-4665-1299-3.
80. ^
de Wit, B.; Smith, J. (2 December 2012). Field Theory in Particle Physics, Volume 1. Elsevier. ISBN 978-0-444-59622-2.
81. ^ Brown, Lowell S. (1992). Quantum Field Theory. Cambridge University Press. ISBN 978-0-521-46946-3.
82. ^ Buchbinder, Iosif
L.; Shapiro, Ilya (March 2021). Introduction to Quantum Field Theory with Applications to Quantum Gravity. Oxford University Press. ISBN 978-0-19-883831-9.
83. ^ Jaffe, Arthur (25 March 2004). “9. Where does quantum field theory fit into the big
picture?”. In Cao, Tian Yu (ed.). Conceptual Foundations of Quantum Field Theory. Cambridge University Press. ISBN 978-0-521-60272-3.
84. ^ Cabibbo, Nicola; Maiani, Luciano; Benhar, Omar (28 July 2017). An Introduction to Gauge Theories. CRC Press.
ISBN 978-1-4987-3452-3.
85. ^ Casalbuoni, Roberto (6 April 2017). Introduction To Quantum Field Theory (Second ed.). World Scientific Publishing Company. ISBN 978-981-314-668-6.
86. ^ Das, Ashok (24 July 2020). Lectures On Quantum Field Theory
(2nd ed.). World Scientific. ISBN 978-981-12-2088-3.
87. ^ Desai, Bipin R. (2010). Quantum Mechanics with Basic Field Theory. Cambridge University Press. ISBN 978-0-521-87760-2.
88. ^ Donoghue, John; Sorbo, Lorenzo (8 March 2022). A Prelude to
Quantum Field Theory. Princeton University Press. ISBN 978-0-691-22348-3.
89. ^ Folland, Gerald B. (3 February 2021). Quantum Field Theory: A Tourist Guide for Mathematicians. American Mathematical Soc. ISBN 978-1-4704-6483-7.
90. ^ Fradkin, Eduardo
(23 March 2021). Quantum Field Theory: An Integrated Approach. Princeton University Press. ISBN 978-0-691-14908-0.
91. ^ Gelis, François (11 July 2019). Quantum Field Theory. Cambridge University Press. ISBN 978-1-108-48090-1.
92. ^ Greiner, Walter;
Reinhardt, Joachim (9 March 2013). Quantum Electrodynamics. Springer Science & Business Media. ISBN 978-3-662-05246-4.
93. ^ Liboff, Richard L. (2003). Introductory Quantum Mechanics (4th ed.). San Francisco: Pearson Education. ISBN 978-81-317-0441-7.
94. ^
Barut, A. O. (1 August 1978). “The Creation of a Photon: A Heuristic Calculation of Planck’s Constant ħ or the Fine Structure Constant α”. Zeitschrift für Naturforschung A. 33 (8): 993–994. Bibcode:1978ZNatA..33..993B. doi:10.1515/zna-1978-0819.
S2CID 45829793.
95. ^ Kocia, Lucas; Love, Peter (12 July 2018). “Measurement contextuality and Planck’s constant”. New Journal of Physics. 20 (7): 073020. arXiv:1711.08066. Bibcode:2018NJPh…20g3020K. doi:10.1088/1367-2630/aacef2. S2CID 73623448.
96. ^
Humpherys, David (28 November 2022). “The Implicit Structure of Planck’s Constant”. European Journal of Applied Physics. 4 (6): 22–25. doi:10.24018/ejphysics.2022.4.6.227. S2CID 254359279.
97. ^ Bais, F. Alexander; Farmer, J. Doyne (2008). “The
Physics of Information”. In Adriaans, Pieter; van Benthem, Johan (eds.). Philosophy of Information. Handbook of the Philosophy of Science. Vol. 8. Amsterdam: North-Holland. arXiv:0708.2837. ISBN 978-0-444-51726-5.
98. ^ Hirota, E.; Sakakima, H.;
Inomata, K. (9 March 2013). Giant Magneto-Resistance Devices. Springer Science & Business Media. ISBN 978-3-662-04777-4.
99. ^ Gardner, John H. (1988). “An Invariance Theory”. Encyclia. 65: 139.
100. ^ Levine, Raphael D. (4 June 2009). Molecular
Reaction Dynamics. Cambridge University Press. ISBN 978-1-139-44287-9.
101. ^ Heilbron, John L. (June 2013). “The path to the quantum atom”. Nature. 498 (7452): 27–30. doi:10.1038/498027a. PMID 23739408. S2CID 4355108.
102. ^ McCormmach, Russell
(1 January 1966). “The atomic theory of John William Nicholson”. Archive for History of Exact Sciences. 3 (2): 160–184. doi:10.1007/BF00357268. JSTOR 41133258. S2CID 120797894.
103. ^ Jump up to:a b Mehra, Jagdish; Rechenberg, Helmut (3 August 1982).
The Historical Development of Quantum Theory. Vol. 1. Springer New York. ISBN 978-0-387-90642-3.
104. ^ Jump up to:a b c Nicholson, J. W. (14 June 1912). “The Constitution of the Solar Corona. II”. Monthly Notices of the Royal Astronomical Society.
Oxford University Press. 72 (8): 677–693. doi:10.1093/mnras/72.8.677. ISSN 0035-8711.
105. ^ Jump up to:a b Bohr, N. (July 1913). “I. On the constitution of atoms and molecules”. The London, Edinburgh, and Dublin Philosophical Magazine and Journal
of Science. 26 (151): 1–25. doi:10.1080/14786441308634955.
106. ^ Wilson, W. (1956). “John William Nicholson 1881-1955”. Biographical Memoirs of Fellows of the Royal Society. 2: 209–214. doi:10.1098/rsbm.1956.0014. JSTOR 769485.
107. ^ Sommerfeld,
A. (1915). “Zur Theorie der Balmerschen Serie” (PDF). Sitzungsberichte der mathematisch-physikalischen Klasse der K. B. Akademie der Wissenschaften zu München. 33 (198): 425–458. doi:10.1140/epjh/e2013-40053-8.
108. ^ Schwarzschild, K. (1916). “Zur
Quantenhypothese”. Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin: 548–568.
109. ^ 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. doi:10.1080/14786440608635664.
110. ^ Landé, A. (June 1919). “Das Serienspektrum des Heliums”. Physikalische Zeitschrift. 20: 228–234.
111. ^ Bohr, N. (October 1920). “Über die Serienspektra
der Elemente”. Zeitschrift für Physik. 2 (5): 423–469. doi:10.1007/BF01329978.
112. ^ Stern, Otto (December 1921). “Ein Weg zur experimentellen Prüfung der Richtungsquantelung im Magnetfeld”. Zeitschrift für Physik. 7 (1): 249–253. doi:10.1007/BF01332793.
113. ^
Heisenberg, Werner (December 1922). “Zur Quantentheorie der Linienstruktur und der anomalen Zeemaneflekte”. Zeitschrift für Physik. 8 (1): 273–297. doi:10.1007/BF01329602.
114. ^ Kramers, H. A.; Pauli, W. (December 1923). “Zur Theorie der Bandenspektren”.
Zeitschrift für Physik. 13 (1): 351–367. doi:10.1007/BF01328226.
115. ^ Born, M.; Jordan, P. (December 1925). “Zur Quantenmechanik”. Zeitschrift für Physik. 34 (1): 858–888. doi:10.1007/BF01328531.
116. ^ Dirac, P. A. M. (December 1925). “The
fundamental equations of quantum mechanics”. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character. 109 (752): 642–653. doi:10.1098/rspa.1925.0150.
117. ^ Born, M.; Heisenberg, W.; Jordan,
P. (August 1926). “Zur Quantenmechanik. II”. Zeitschrift für Physik. 35 (8–9): 557–615. doi:10.1007/BF01379806.
118. ^ Schrödinger, E. (1926). “Quantisierung als Eigenwertproblem”. Annalen der Physik. 384 (4): 361–376. doi:10.1002/andp.19263840404.
119. ^
Dirac, P. A. M. (October 1926). “On the theory of quantum mechanics”. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character. 112 (762): 661–677. doi:10.1098/rspa.1926.0133.
120. ^ Jump up
to:a b Mehra, Jagdish; Rechenberg, Helmut (2000). The Historical Development of Quantum Theory. Vol. 6. New York: Springer.
121. ^ Dirac, P. A. M. (1930). The Principles of Quantum Mechanics (1st ed.). Oxford, U.K.: Clarendon.
b. Barrow, John
D. (2002), The Constants of Nature; From Alpha to Omega – The Numbers that Encode the Deepest Secrets of the Universe, Pantheon Books, ISBN 978-0-375-42221-8
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