Science > Physics > Gap in Einstein's Early Argument for Existence of Photons
| Topic: |
Science > Physics |
| User: |
"Sam Wormley" |
| Date: |
04 Oct 2005 06:14:30 PM |
| Object: |
Gap in Einstein's Early Argument for Existence of Photons |
Physics Today: Letters
Gap in Einstein's Early Argument for Existence of Photons
http://www.physicstoday.org/vol-58/iss-10/p17.html
During this the centennial year of Albert Einstein's epochal papers
from his miraculous year, I was prompted to read again his paper on the
quantum nature of radiation.[1] Surprisingly, I found a gap in his
earliest argument for the existence of photons, a gap that apparently
has remained unnoticed up to the present.
Einstein's argument was based on his proof that for sufficiently large
frequencies the entropy of thermal radiation varies logarithmically
with the volume, in the same manner it does in an ideal gas. He
concluded that "monochromatic radiation of low density (within the
range of validity of Wien's radiation formula) behaves
thermodynamically as if it consisted of mutually independent quanta
[photons] of magnitude hv[1] where v is the frequency of the
radiation and h is Planck's constant. In a footnote Einstein also gave
a derivation, based on his entropy--volume relation, for the
familiar pressure--volume formula of an ideal gas. But substituting
in this formula Einstein's relation for the number of photons--that
is, the energy of the monochromatic thermal radiation divided by
hv --results in an incorrect expression for the pressure. This
pressure is one-third of the energy density of the radiation, as was
originally shown by Maxwell for the case of electromagnetic waves.
Moreover, a kinetic-theory calculation of the pressure shows that this
relation is also valid for photons.
Maxwell's relation also played a fundamental role in Boltzmann's
derivation of the dependence of thermal radiation on the fourth power
of the temperature, and in Wien's derivation of the general dependence
of the thermal energy density on both frequency and temperature. I have
not found any evidence, however, that Einstein's contemporaries noticed
the problem with calculating the pressure of monochromatic thermal
radiation from Einstein's entropy--volume relation, nor has this
problem been pointed out in commentaries about this paper.[2,3]
What is the solution to the pressure paradox? Einstein considered the
volume dependence of the radiation entropy for fixed frequency v,
but in an actual thermodynamic process, where the volume V is changed
by moving a piston, the frequency does not remain fixed, because the
wavelength is proportional to the linear dimensions of the cavity.
Hence, the frequency v varies as V^-1/3, giving the entropy an
additional volume dependence that was not discussed by Einstein. It can
be readily verified that this addition leads to a thermodynamic
derivation of Maxwell's relation for the pressure of isotropic
radiation, filling in a long-standing gap in Einstein's earliest
thermodynamic argument for the existence of photons.
References
1. A. Einstein, Ann. der Phys. (Leipzig) 17, 132 (1905); reproduced in
The Collected Papers of Albert Einstein, vol. 2, J. Stachel, ed.,
Princeton U. Press, Princeton, NJ (1989), p. 150.
2. M. Klein, The Natural Philosopher 2, 59 (1963).
3. J. Stachel, ed., Einstein's Miraculous Year: Five Papers That
Changed the Face of Physics, Princeton U. Press, Princeton, NJ (1998),
p. 177.
Michael Nauenberg
(michael@physics.ucsc.edu)
University of California
Santa Cruz
.
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| User: "" |
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| Title: Re: Gap in Einstein's Early Argument for Existence of Photons |
05 Oct 2005 04:02:35 AM |
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Sam Wormley wrote:
Physics Today: Letters
Gap in Einstein's Early Argument for Existence of Photons
http://www.physicstoday.org/vol-58/iss-10/p17.html
During this the centennial year of Albert Einstein's epochal papers
from his miraculous year, I was prompted to read again his paper on the
quantum nature of radiation.[1] Surprisingly, I found a gap in his
earliest argument for the existence of photons, a gap that apparently
has remained unnoticed up to the present.
The gap certainly was noticed by De Broglie, Schroedinger,
Heisenburg,
Dirac, and Born. Which is why they reinvented Probability Theory
to explain it.
Since the Planck wavelength doesn't exist, as
Einstein assumed it did, to explain the
photo-electric effect.
.
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