Planckn.

1905 Nature 13 July 244/1 In Nature, May 18, I gave a calculation of the coefficient of complete radiation at a given absolute temperature for waves of great length.., and it appeared that the result was eight times as great as that deduced from Planck's formula.

1923 R. Glazebrook Dict. Appl. Physics IV. 563/2 Throughout the spectrum from 0·5 μ to 50 μ Planck's formula fits the observed spectral energy distribution more closely than any other equation yet proposed.

1974 G. Reece tr. F. Hund Hist. Quantum Theory ii. 27 By comparing the measured energy densities..with Planck's formula it was possible to calculate h and h/k.

1989 Sci. Amer. May 96/2 The Planck formula gives the amount of radiation that an object emits per unit area into a unit solid angle—its surface brightness.

1905 Nature 27 July 294/2 Planck's law is in good agreement with experiment if h is given a value different from zero.

1933 A. W. Barton Text Bk. Heat ii. 51 Suppose the pendulum were to obey Planck's law; then its energy could only have certain discrete values, and so only certain amplitudes differing from one another by a finite amount would be permitted.

1955 W. Heisenberg in W. Pauli Niels Bohr & Devel. Physics 14 Bohr explained to him [sc. Schrödinger] that not even Planck's Law could be understood without the quantum jumps.

1974 R. E. Turner & D. S. Betts Introd. Statistical Mech. xii. 148 The Planck law describes the distributions of energy among the infinite range of possible frequencies.

2001 Jrnl. Res. Nat. Inst. Standards & Technol. (Nexis) 106 151 The determination of the spectral radiance of a blackbody source can be used to infer its temperature using the Planck law.

1909 Sci. Abstr. A. 12 315 (heading) Planck's radiation law.

1935 L. Pauling & E. B. Wilson Introd. Quantum Mech. xi. 301 The density of radiant energy is known to be given by Planck's radiation law as [etc.].

1962 O. Struve Universe 109 If we assume that 10¹⁶° was reached at a frequency of 60 megacycles per second, corresponding to a wave length of 5 meters, then Planck's radiation law indicates that the total amount of energy received on the earth from this active region was 10⁻¹² watt per square meter per cycle per second.

1989 R. Penrose Emperor's New Mind (1991) vi. 230 Planck was able to get theoretical agreement with the observed dependence of intensity with frequency—now called the Planck radiation law.

1910 London, Edinb., & Dublin Philos. Mag. 6th Ser. 20 244 Corpuscles which are emitted by bodies when exposed to ultra-violet light. Ladenburg found that the maximum energy of these corpuscles..was proportional to n the frequency of the light, being of the order h′n/2π where h′ is Planck's constant.

1940 S. Glasstone Text-bk. Physical Chem. i. 73 Every particle is associated with a wave.., and the wave length (λ) is related to the mechanical momentum (mυ) by λ = h/mυ, where h is the Planck constant.

1960 B. Chalmers & A. G. Quarrell Physical Exam. Metals (ed. 2) xvi. 749 This is known as the spin of the particle and is measured in terms of h/2π..where h is Planck's constant.

2018 M. de Gosson Emergence of Quantum from Classical vii. 215 The quantum world is very sensitive to the value of Planck's constant.

1911 Sci. Abstr. A. 14 404 It is concluded that the Einstein formula for specific heat is confirmed and also the Planck radiation formula.

1974 G. Reece tr. F. Hund Hist. Quantum Theory xi. 145 The Bose statistics of light quanta was thus the same as that earlier applied by Planck for energy quanta..and thus led to the Planck radiation formula.

1996 Amer. Jrnl. Physics 64 649 Planck's radiation formula is used to estimate the dimensionless efficiency of incandescent lamps as a function of filament temperature.

1911 Bull. Bureau of Standards (U.S.) 7 393 Planck's equation for the intensity of radiation J, of wave length λ, from a black body at the absolute temperature θ,..appears to represent the results of all known observations.

1966 M. Ferro-Luzzi tr. E. Fermi Molecules, Crystals & Quantum Statistics vii. 223 From the Planck equation, we can easily deduce Stefan's law, which also can be obtained by thermodynamic arguments alone.

1984 J. C. Polkinghorne Quantum World (BNC) 7 According to Planck's equation the quantity of energy depended directly upon the frequency.

1914 Philos. Trans. (Royal Soc.) A. 214 227 Beginning with the ‘Rutherford’ model of the atom, and introducing Planck's quantum, Dr Bohr has derived a formula for the hydrogen spectrum which excludes these lines.

1954 Sci. Monthly Jan. 223 In more recent times physicists have not hesitated to call on their creative imagination for constructs like, say, Planck's quantum of action that have no direct connection with experience.

1989 F. Close in P. Davies New Physics xiv. 399/1 Quantum mechanics restricts the allowed magnitudes to be integer multiples of a basic amount known as Planck's quantum.

1917 Science 28 Sept. 299/1 It will be noted that this objection applies with equal force to the Planck oscillator which maintains energy even at the absolute zero.

1920 Sci. Abstr. A. 23 423 If the nuclei are related in a quasi-elastic manner to the equilibrium positions, then the energy values for a diatomic non-rotating molecule are the same as for a Planck oscillator, namely: E = nhω, where n denotes the quantum number for the atomic oscillation.

1966 tr. S.-I. Tomonaga's Quantum Mech. II. vi. 111 Each term in Eq. (47·9′) has the form of the Hamiltonian of a Planck oscillator.

1991 D. Layzer Cosmogenesis vi. 97 Heisenberg himself had carried out the first such calculation, predicting the energy levels of a Planck oscillator, and a few months later Wolfgang Pauli succeeded in calculating the energy levels of the hydrogen atom.

1955 J. A. Wheeler in Physical Rev. 97 514/1 The characteristic length, L* = (ℏG/c³)^½ = 1.63 × 10⁻³³ cm.]

1964 J. A. Wheeler in C. De Witt & B. De Witt Relativity, Groups & Topol. 509 For these fluctuations there is only a single physical constant that is relevant—the Planck length L* = (ℏG/c³)^½.

1970 Jrnl. Physics A 3 60 A vaguer but more physical motivation is to be found in the significance of the so-called ‘Planck length’,..which constitutes a sort of limit beyond which the concepts of measurement..have no meaning.

1991 C. A. Ronan Nat. Hist. Universe 174/1 The size of open strings is roughly equal to the Planck length, a distance of only 10⁻³² millimetres, equivalent to a hundred-billion-billionth of the diameter of the atomic nucleus.

1993 N.Y. Times 30 Mar. c10/4 Perhaps when a black hole evaporates, it leaves behind what is called a remnant... The remnant, it turned out, would take the form of a new kind of elementary particle that would be just the Planck length in size yet would weigh one ten-thousandth of a gram.

1964 J. A. Wheeler in C. De Witt & B. De Witt Relativity, Groups & Topol. 508 Only when the mass of an object exceeds the Planck mass (ℏc/G)^½ = 2.18 × 10⁻⁵ g does its static field become large enough to show up against the background.

1971 Progress Theoret. Physics 46 984 The limiting radius and mass of the collapsing extreme relativistic Fermi gas are roughly given by N^2/5L* and N^2/5m*, respectively, where N is the number of particles in the gas and m*..is the Planck mass.

1981 D. Wilkinson in B. Dixon From Creation to Chaos (1993) 324 It is..clear that if we attempt to discuss concentrated masses of greater than a certain magnitude called the Planck mass of about 10¹⁹GeV we shall simply not know how to do it.

1989 J. Silk Big Bang (rev. ed.) vi. 118 The Planck mass..represents the ultimate building block of nature.

1967 E. R. Harrison in Nature 8 July 152/1 We obtain the Planck density ρ ∼ c⁵/G²ℏ ∼ 10⁹⁵ g cm⁻³ when the age of the universe is t* ∼ λ/c or..∼ 10⁻⁴⁴.

1971 Progress Theoret. Physics 46 984 For any given number N of particles one can obtain a configuration with mass as close to zero as one pleases by prescribing a sufficiently high particle density (of the order of the Planck density ∼ 10⁹⁴ g/cm³ for small N).

1992 S. P. Maran Astron. & Astrophysics Encycl. 147/1 The scope of such theories is limited at present by quantum effects that are expected to become significant..at some density not exceeding the Planck density.

1994 Sci. Amer. Nov. 32/2 The current laws of physics..took hold only after the density of the universe dropped below the so-called Planck density, which equals about 10⁹⁴ grams per cubic centimeter.

1968 Science 14 June 1200/3 The geometry of space at the Planck scale of distances.

1980 Philos. Trans. (Royal Soc.) A. 296 286 I have outlined how naïve quantum gravitational effects predict metric and density fluctuations of unit amplitude on the Planck scale ca. 10⁻⁵ g.

1989 J. Silk Big Bang (rev. ed.) vi. 118 At the Planck scale, we need an entirely novel concept in order to describe the state of matter.

1997 Sci. Amer. Apr. 42/3 Very little is known with absolute certainty about collisions at energies beyond the Planck scale, but there is a good educated guess.

1976 Nature 6 May 20/1 If the particle creation occurs near the Planck time, the entropy produced is consistent..with that required by the Einstein equations.

1978 Proc. Royal Soc. A. 361 523 [This] will become of the same order of magnitude as ρ_rad when t is about 10¹⁰ Planck times.

1980 Philos. Trans. (Royal Soc.) A. 296 276 Constant curvature fluctuations arising at the Planck time would be just right.

1988 J. Trefil Dark Side of Universe xi. 153 In one version of the superstring theory, which is particularly appealing from an aesthetic point of view, the equations seem to suggest that at the Planck time the universe split into two separate parts.

1994 Science 13 May 1000/1 Over the Planck time, a mere 10⁻⁴³ second later, the wormhole closes.

2000 New Scientist 3 June 24/1 There is a point known as the Planck time—within 10⁻⁴³ seconds of the big bang—when gravity is comparable in strength to the other forces of nature.

1979 Physics Lett. B. 80 360 Grand unified models of elementary particle interactions suggest that there was an early epoch during the big bang while the temperature was slightly less than the Planck temperature, during which no known interactions were in thermal equilibrium.

1989 A. Salam in P. Davies New Physics xviii. 489/2 The system of preonic fields should undergo a phase transition at Planck temperature.

2000 L. Smolin Three Roads to Quantum Gravity iv. 63 There is a fundamental Planck temperature, which is likely to be the hottest anything can get.

1980 Progress Theoret. Physics 64 1498 The conjecture that the homogeneity of the Universe might be explained by assuming G ∼ T⁻² for T > Planck Energy is analyzed.

1988 Nature 25 Feb. 674/3 Einstein's theory of gravity arises in its low-energy limit (with energies much less than the Planck energy 10¹⁹Gev).

1997 Sci. Amer. Apr. 42/1 If the particles have a ‘Planck energy’ of about 10²⁸ electron volts, so much energy—and therefore mass—becomes concentrated in a tiny volume that gravitational forces outweigh all others.