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Philosophers

Mortimer Adler
Rogers Albritton
Alexander of Aphrodisias
Samuel Alexander
William Alston
Anaximander
G.E.M.Anscombe
Anselm
Louise Antony
Thomas Aquinas
Aristotle
David Armstrong
Harald Atmanspacher
Robert Audi
Augustine
J.L.Austin
A.J.Ayer
Alexander Bain
Mark Balaguer
Jeffrey Barrett
William Belsham
Henri Bergson
George Berkeley
Isaiah Berlin
Richard J. Bernstein
Bernard Berofsky
Robert Bishop
Max Black
Susanne Bobzien
Emil du Bois-Reymond
Hilary Bok
Laurence BonJour
George Boole
Émile Boutroux
F.H.Bradley
C.D.Broad
Michael Burke
C.A.Campbell
Joseph Keim Campbell
Rudolf Carnap
Carneades
Ernst Cassirer
David Chalmers
Roderick Chisholm
Chrysippus
Cicero
Randolph Clarke
Samuel Clarke
Anthony Collins
Antonella Corradini
Diodorus Cronus
Jonathan Dancy
Donald Davidson
Mario De Caro
Democritus
Daniel Dennett
Jacques Derrida
René Descartes
Richard Double
Fred Dretske
John Dupré
John Earman
Laura Waddell Ekstrom
Epictetus
Epicurus
Herbert Feigl
John Martin Fischer
Owen Flanagan
Luciano Floridi
Philippa Foot
Alfred Fouilleé
Harry Frankfurt
Richard L. Franklin
Michael Frede
Gottlob Frege
Peter Geach
Edmund Gettier
Carl Ginet
Alvin Goldman
Gorgias
Nicholas St. John Green
H.Paul Grice
Ian Hacking
Ishtiyaque Haji
Stuart Hampshire
W.F.R.Hardie
Sam Harris
William Hasker
R.M.Hare
Georg W.F. Hegel
Martin Heidegger
Heraclitus
R.E.Hobart
Thomas Hobbes
David Hodgson
Shadsworth Hodgson
Baron d'Holbach
Ted Honderich
Pamela Huby
David Hume
Ferenc Huoranszki
William James
Lord Kames
Robert Kane
Immanuel Kant
Tomis Kapitan
Jaegwon Kim
William King
Hilary Kornblith
Christine Korsgaard
Saul Kripke
Andrea Lavazza
Keith Lehrer
Gottfried Leibniz
Leucippus
Michael Levin
George Henry Lewes
C.I.Lewis
David Lewis
Peter Lipton
C. Lloyd Morgan
John Locke
Michael Lockwood
E. Jonathan Lowe
John R. Lucas
Lucretius
Alasdair MacIntyre
Ruth Barcan Marcus
James Martineau
Storrs McCall
Hugh McCann
Colin McGinn
Michael McKenna
Brian McLaughlin
John McTaggart
Paul E. Meehl
Uwe Meixner
Alfred Mele
Trenton Merricks
John Stuart Mill
Dickinson Miller
G.E.Moore
Thomas Nagel
Friedrich Nietzsche
John Norton
P.H.Nowell-Smith
Robert Nozick
William of Ockham
Timothy O'Connor
Parmenides
David F. Pears
Charles Sanders Peirce
Derk Pereboom
Steven Pinker
Plato
Karl Popper
Porphyry
Huw Price
H.A.Prichard
Protagoras
Hilary Putnam
Willard van Orman Quine
Frank Ramsey
Ayn Rand
Michael Rea
Thomas Reid
Charles Renouvier
Nicholas Rescher
C.W.Rietdijk
Richard Rorty
Josiah Royce
Bertrand Russell
Paul Russell
Gilbert Ryle
Jean-Paul Sartre
Kenneth Sayre
T.M.Scanlon
Moritz Schlick
Arthur Schopenhauer
John Searle
Wilfrid Sellars
Alan Sidelle
Ted Sider
Henry Sidgwick
Walter Sinnott-Armstrong
J.J.C.Smart
Saul Smilansky
Michael Smith
Baruch Spinoza
L. Susan Stebbing
Isabelle Stengers
George F. Stout
Galen Strawson
Peter Strawson
Eleonore Stump
Francisco Suárez
Richard Taylor
Kevin Timpe
Mark Twain
Peter Unger
Peter van Inwagen
Manuel Vargas
John Venn
Kadri Vihvelin
Voltaire
G.H. von Wright
David Foster Wallace
R. Jay Wallace
W.G.Ward
Ted Warfield
Roy Weatherford
William Whewell
Alfred North Whitehead
David Widerker
David Wiggins
Bernard Williams
Timothy Williamson
Ludwig Wittgenstein
Susan Wolf

Scientists

Michael Arbib
Bernard Baars
Gregory Bateson
John S. Bell
Charles Bennett
Ludwig von Bertalanffy
Susan Blackmore
Margaret Boden
David Bohm
Niels Bohr
Ludwig Boltzmann
Emile Borel
Max Born
Satyendra Nath Bose
Walther Bothe
Hans Briegel
Leon Brillouin
Stephen Brush
Henry Thomas Buckle
S. H. Burbury
Donald Campbell
Anthony Cashmore
Eric Chaisson
Jean-Pierre Changeux
Arthur Holly Compton
John Conway
John Cramer
E. P. Culverwell
Charles Darwin
Terrence Deacon
Louis de Broglie
Max Delbrück
Abraham de Moivre
Paul Dirac
Hans Driesch
John Eccles
Arthur Stanley Eddington
Paul Ehrenfest
Albert Einstein
Hugh Everett, III
Franz Exner
Richard Feynman
R. A. Fisher
Joseph Fourier
Lila Gatlin
Michael Gazzaniga
GianCarlo Ghirardi
J. Willard Gibbs
Nicolas Gisin
Paul Glimcher
Thomas Gold
A.O.Gomes
Brian Goodwin
Joshua Greene
Jacques Hadamard
Patrick Haggard
Stuart Hameroff
Augustin Hamon
Sam Harris
Hyman Hartman
John-Dylan Haynes
Martin Heisenberg
Werner Heisenberg
John Herschel
Jesper Hoffmeyer
E. T. Jaynes
William Stanley Jevons
Roman Jakobson
Pascual Jordan
Ruth E. Kastner
Stuart Kauffman
Martin J. Klein
Simon Kochen
Stephen Kosslyn
Ladislav Kovàč
Rolf Landauer
Alfred Landé
Pierre-Simon Laplace
David Layzer
Benjamin Libet
Seth Lloyd
Hendrik Lorentz
Josef Loschmidt
Ernst Mach
Donald MacKay
Henry Margenau
James Clerk Maxwell
Ernst Mayr
Ulrich Mohrhoff
Jacques Monod
Emmy Noether
Abraham Pais
Howard Pattee
Wolfgang Pauli
Massimo Pauri
Roger Penrose
Steven Pinker
Colin Pittendrigh
Max Planck
Susan Pockett
Henri Poincaré
Daniel Pollen
Ilya Prigogine
Hans Primas
Adolphe Quételet
Juan Roederer
Jerome Rothstein
David Ruelle
Erwin Schrödinger
Aaron Schurger
Claude Shannon
David Shiang
Herbert Simon
Dean Keith Simonton
B. F. Skinner
Roger Sperry
John Stachel
Henry Stapp
Tom Stonier
Antoine Suarez
Leo Szilard
William Thomson (Kelvin)
Peter Tse
Vlatko Vedral
Heinz von Foerster
John von Neumann
John B. Watson
Daniel Wegner
Steven Weinberg
Paul A. Weiss
John Wheeler
Wilhelm Wien
Norbert Wiener
Eugene Wigner
E. O. Wilson
H. Dieter Zeh
Ernst Zermelo
Wojciech Zurek

Presentations

Biosemiotics
Free Will
Mental Causation
James Symposium
 
Albert Einstein - Emission and Absorption of Radiation in Quantum Theory
Sixteen years ago, when Planck created quantum theory by deriving his radiation formula, he took the following approach. He calculated the mean energy Ē of a resonator as a function of temperature according to his newly found quantum-theoretic basic principles, and determined from this the radiation density ρ as a function of frequency ν and temperature. He accomplished this by deriving — based upon electromagnetic considerations — a relation between radiation density and resonator energy Ē:

Ē = c3 ρ / 8 π ν2         (1)

His derivation was of unparalleled boldness, but found brilliant confirmation. Not only the radiation formula proper and the calculated value of the elementary quantum in it was confirmed, but also the quantum-theoretically calculated value of Ē was confirmed by later investigations on specific heat. In this manner, equation (1), originally found by electromagnetic reasoning, was also confirmed. However, it remained unsatisfactory that the electromagnetic-mechanical analysis, which led to (1), is incompatible with quantum theory, and it is not surprising that Planck himself and all theoreticians who work on this topic incessantly tried to modify the theory such as to base it on noncontradictory foundations.

Since Bohr's theory of spectra has achieved its great successes, it seems no longer doubtful that the basic idea of quantum theory must be maintained. It so appears that the uniformity of the theory must be established such that the electromagneto-mechanical considerations, which led Planck to equation (1), are to be replaced by quantum-theoretical contemplations on the interaction between matter and radiation. In this endeavor I feel galvanized by the following consideration which is attractive both for its simplicity and generality.

§1. PLANCK's Resonator in a Field of Radiation

The behavior of a monochromatic resonator in a field of radiation, according to the classical theory, can be easily understood if one recalls the manner of treatment that was first used in the theory of Brownian movement. Let E be the energy of the resonator at a given moment in time; we ask for the energy after time τ has elapsed. Hereby, τ is assumed to be large compared to the period of oscillation of the resonator, but still so small that the percentage change of E during τ can be treated as infinitely small. Two kinds of change can be distinguished. First the change

Δ1E = - A Eτ

effected by emission; and second, the change Δ2E caused by the work done by the electric field on the resonator. This second change increases with the radiation density and has a "chance"-dependent value and a "chance"-dependent sign. An electromagnetic, statistical consideration yields the mean-value relation

< Δ2E > = - B ρ τ

The constants A and B can be calculated in known manner. We call Δ1E the energy change due to emitted radiation, Δ2E the energy change due to incident radiation. Since the mean value of E, taken over many resonators, is supposed to be independent of time, there has to be

< E + Δ1E + Δ2E > = Ē

or

Ē = - (B / A) ρ

One obtains relation (1) if one calculates B and A for the monochromatic resonator in the known way with the help of electromagnetism and mechanics. We now want to undertake corresponding considerations, but on a quantum-theoretical basis and without specialized suppositions about the interaction between radiation and those structures which we want to call "molecules."

§2. Quantum Theory and Radiation

We consider a gas of identical molecules that are in static equilibrium with thermal radiation. Let each molecule be able to assume only a discrete sequence Z1, Z2, etc., of states with energy values ε1, ε2, respectively. Then it follows in known manner and in analogy to statistical mechanics, or directly from Boltzmann's principle, or finally from thermodynamic considerations, that the probability Wn of state Zn (or the relative number of molecules which were in state Zn) is given by

Wn = pn e- εn / k T         (2)

where k is the well-known Boltzmann constant. pn is the statistical "weight" of state Zn, i.e., a constant that is characteristic of the quantum state of the molecule but independent of the gas temperature T.

We shall now assume that a molecule can go from state Zn to state Zm by absorbing radiation of the distinct frequency ν = νnm ; and likewise from state Zm to state Zn by emitting such radiation. The radiation energy involved is εm - εn. In general, this is possible for any combination of two indices m and n. With respect to any of these elementary processes there must be a statistical equilibrium in thermal equilibrium. Therefore, we can confine ourselves to a single elementary process belonging to a distinct pair of indices (n,m).

At the thermal equilibrium, as many molecules per time unit will change from state Zn to state Zm under absorption of radiation, as molecules will go from state Zm to state Zn with emission of radiation. We shall state simple hypotheses about these transitions, where our guiding principle is the limiting case of classical theory, as it has been briefly outlined above.

We shall distinguish here also two types of transitions:

a) Emission of Radiation. This will be a transition from state Zm to state Zn with emission of the radiation energy εm - εn. This transition will take place without external influence. One can hardly imagine it to be other than similar to radioactive reactions. The number of transitions per time unit will have to be put at

Amn Nm,

where Amn is a constant that is characteristic of the combination of the states Zm and Zn, and Nm is the number of molecules in state Zm.

b) Incidence of Radiation. Incidence is determined by the radiation within which the molecule resides; let it be proportional to the radiation density ρ of the effective frequency. In case of the resonator it may cause a loss in energy as well as an increase in energy; that is, in our case, it may cause a transition ZnZm as well as a transition ZmZn. The number of transitions
ZnZm per unit time is then

Bnm Nn ρ,

and the number of transitions ZmZn is to be expressed as

Bmn Nm ρ,

where Bnm, Bmn are constants related to the combination of states Zn, Zm.

As a condition for the statistical equilibrium between the reactions ZnZm and ZmZn one finds, therefore, the equation

Amn Nm + Bmn Nm ρ = Bnm Nn ρ         (3)

Equation (2), on the other hand, yields

Nn / Nm = (pn / pm) e ( εm - εn ) / k T         (4)

From (3) and (4) follows

Amn pm = ρ ( Bnm pn e ( εm - εn ) / k T - Bmn pm )         (5)

ρ is the radiation density of that frequency which is emitted with the transition ZmZn and is absorbed with ZnZm. Our equation shows the relation between T and ρ at this frequency.
If we postulate that ρ must approach infinity with ever increasing T, then we necessarily have

Bnm pn = Bmn pm         (6)

Introducing the abbreviation

Amn / Bmn pm = αmn,         (7)

one finds

ρ = αmn / ( e ( εm - εn ) / k T - 1 )         (5a)

Einstein has derived Planck's blackbody law
This is Planck's relation between ρ and T with the constants left indeterminate. The constants Amn and Bmn could be calculated directly if we possessed a modified version of electrodynamics and mechanics that is in compliance with the quantum hypothesis.

Einstein derives Bohr's second postulate
The fact that ρ must be a universal function of T and ν implies that αmn and εm - εn cannot depend upon the specific constitution of the molecule, but only upon the effective frequency ν. From Wien's law follows furthermore that αmn must be proportional to the third power, and εm - εn to the first power of ν . Consequently, one has

εm - εn =         (8)

where is a constant.

While the three hypotheses concerning emission and incidence of radiation lead to Planck's radiation formula, I am of course very willing to admit that this does not elevate them to confirmed results. But the simplicity of the hypotheses, the generality with which the analysis can be carried out so effortlessly, and the natural connection to Planck's linear oscillator (as a limiting case of classical electrodynamics and mechanics) seem to make it highly probable that these are basic traits of a future theoretical representation. The postulated statistical law of emission is nothing but Rutherford's law of radioactive decay, and the law expressed by (8), in conjunction with (5a), is identical with the second basic hypothesis in Bohr's theory of spectra — this too speaks in favor of the theory presented here.

§3. Remark on the Photochemical Law of Equivalence

The photochemical law of equivalence falls in line with our train of thoughts in the following manner. Let there be a gas of such low temperature that the thermal radiation of frequency ν, which leads from state Zm to state Zn, does not practically occur.

According to (2) and (5a), the state Zm will be quite rare compared to state Zn, and we shall assume that almost all gas molecules are in state Zn. Aside from the previously considered process ZmZn, let the molecule in state Zm also have the capability of another elementary "chemical" process, e.g., monomolecular dissociation. Let us furthermore assume that the reaction rate of this dissociation is large compared to the rate of occurrence of the reaction ZmZn.

What will happen now if we irradiate the gas with the effective frequency? Under absorption of the radiation energy εm - εn = , molecules will continually go from state Zn to state Zm. Only a very small fraction of these molecules will return to state Zn by emission or absorption. Most, by far, will suffer chemical dissociation, corresponding to the postulated higher reaction rate of this process. This means that per dissociating molecule, we will practically find that the radiation energy has been absorbed, just as the law of equivalence demands.

The essence of this interpretation is that molecular dissociation is achieved by the absorption of light via the quantum state Zm, but not directly without this intermediate state. In consequence, one need not distinguish between a chemically effective and a chemically ineffective absorption of radiation. The absorption of light and the chemical process appear as independent processes.

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