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absorbs all incident radiation) at a given temperature is a
universal function of the temperature of the body and wave-
length of the radiation. He inferred that equilibrium thermal
radiation in a cavity with walls maintained at a certain tem-
perature behaves like radiation emitted by a black body at
the same temperature.
Heinrich Friedrich Weber, Einstein s physics professor at
the Eidgenössische Technische Hochschule (ETH), was one
of those who attempted to determine the universal black-
body radiation function. He made measurements of the en-
ergy spectrum and proposed an empirical formula for the
167
PART FOUR
distribution function. He showed that, as a consequence of
his formula, »m = constant/T (where »m is the wavelength
with the maximum intensity of the distribution), thus an-
ticipating Wilhelm Wien s formulation of the displacement
law for black-body radiation. Weber described his work in
a course at the ETH given during the winter semester of
1898 1899, which Einstein took.
Einstein soon started to think seriously about the problem
of radiation. By the spring of 1901, he was closely following
Planck s work on black-body radiation. Originally, Planck had
hoped to explain the irreversibility of physical processes by
studying electromagnetic radiation; ultimately, he came to
recognize that this could not be done without introducing
statistical elements into the argument. In a series of papers
between 1897 and 1900, Planck utilized Maxwell s electro-
dynamics to develop a theory of thermal radiation in inter-
action with one or more identical, charged harmonic oscil-
lators within a cavity. He was only able to account for the
radiation s irreversible approach to thermal equilibrium by
employing methods analogous to those that Ludwig Boltz-
mann had used in kinetic theory. Planck introduced the no-
tion of natural (that is, maximally disordered) radiation,
which he defined by analogy with Boltzmann s definition of
molecular chaos. Using Maxwell s theory, Planck derived a
relation between the average energy of a charged oscil-
lator of frequency ½ in equilibrium with thermal radiation
and the energy density per unit frequency interval Á½ of the
radiation at the same frequency:
c3
½ = Á½ (1)
8À½2
where c is the speed of light.
168
EARLY WORK ON QUANTUM HYPOTHESIS
Planck calculated the average energy of an oscillator by
making assumptions about the entropy of the oscillators that
enabled him to derive Wien s law for the energy density
of the black-body spectrum, which originally seemed well
supported by the experimental evidence. But by the turn of
the century new observations showed systematic deviations
from Wien s law for large values of »T .
Planck presented a new energy density distribution for-
mula that agreed closely with observations over the entire
spectrum: 1
8Àh½3 1
Á½ = (2)
c3 eh½/kT - 1
In this expression, now known as Planck s law or Planck s
formula, k = R/N is Boltzmann s constant, where R is the
gas constant, N is Avogadro s (or Loschmidt s) number, and
h is a new constant (later called Planck s constant). To de-
rive this formula, Planck calculated the entropy of the os-
cillators, using what Einstein later called the Boltzmann
principle : S = k ln W where S is the entropy of a macro-
scopic state of the system, the probability of which is W .
Following Boltzmann, Planck took the probability of a state
to be proportional to the number of complexions, or possi-
ble microconfigurations of the system corresponding to that
state. He calculated this number by dividing the total en-
ergy of the state into a finite number of elements of equal
magnitude, and counting the number of possible ways of
distributing these energy elements among the individual os-
cillators. If the size of the energy elements is set equal to
h½, where ½ is the frequency of an oscillator, an expression
for the entropy of an oscillator results that leads to eq. (2).
Although Einstein expressed private misgivings about
Planck s approach in 1901, he did not mention Planck or
169
PART FOUR
black-body radiation in his papers until 1904. A study of the
foundations of statistical physics, which he undertook be-
tween 1902 and 1904, provided Einstein with the tools he
needed to analyze Planck s derivation and explore its con-
sequences. At least three elements of Einstein s general
molecular theory of heat were central to his subsequent
work on the quantum hypothesis: (1) the introduction of
the canonical ensemble; (2) the interpretation of probability
as it occurs in Boltzmann s principle; and (3) the study of
energy fluctuations in thermal equilibrium.
1. In an analysis of the canonical ensemble, Einstein
proved that the equipartition theorem (see the Introduc-
tion, p. 16) holds for any system in thermal equilibrium.
In paper 5 he showed that, when applied to an ensem-
ble of charged harmonic oscillators in equilibrium with
thermal radiation, the equipartition theorem leads, via eq.
(1), to a black-body distribution law now known as the
Rayleigh-Jeans law:
8À½2
Á½ = kT (3)
c3
Despite its rigorous foundation in classical physics, eq. (3)
only agrees with the observed energy distribution for small
values of ½/T ; indeed, as Einstein noted, it implies an infi-
nite total radiant energy.
2. In 1906, Einstein posed a question that preoccupied
him and others at the time: How is it that Planck did not
arrive at the same formula [eq. (3)], but at the expression. . .
[eq. (2)]? One answer lies in Planck s definition of W in
Boltzmann s principle, which, as Einstein repeatedly noted,
differs fundamentally from his own definition of probabilities
as time averages. As noted above, Planck interpreted W as
170
EARLY WORK ON QUANTUM HYPOTHESIS
proportional to the number of complexions of a state of the
system. As Einstein pointed out in 1909, such a definition
of W is equivalent to its definition as the average, over a
long period of time, of the fraction of time that the system
spends in this state only if all complexions corresponding
to a given total energy are equally probable. However, if
this is assumed to hold for an ensemble of oscillators in
thermal equilibrium with radiation, the Rayleigh-Jeans law
results. Hence, the validity of Planck s law implies that all
complexions cannot be equally probable. Einstein showed
that, if the energies available to a canonical ensemble of
oscillators are arbitrarily restricted to multiples of the energy [ Pobierz całość w formacie PDF ]
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absorbs all incident radiation) at a given temperature is a
universal function of the temperature of the body and wave-
length of the radiation. He inferred that equilibrium thermal
radiation in a cavity with walls maintained at a certain tem-
perature behaves like radiation emitted by a black body at
the same temperature.
Heinrich Friedrich Weber, Einstein s physics professor at
the Eidgenössische Technische Hochschule (ETH), was one
of those who attempted to determine the universal black-
body radiation function. He made measurements of the en-
ergy spectrum and proposed an empirical formula for the
167
PART FOUR
distribution function. He showed that, as a consequence of
his formula, »m = constant/T (where »m is the wavelength
with the maximum intensity of the distribution), thus an-
ticipating Wilhelm Wien s formulation of the displacement
law for black-body radiation. Weber described his work in
a course at the ETH given during the winter semester of
1898 1899, which Einstein took.
Einstein soon started to think seriously about the problem
of radiation. By the spring of 1901, he was closely following
Planck s work on black-body radiation. Originally, Planck had
hoped to explain the irreversibility of physical processes by
studying electromagnetic radiation; ultimately, he came to
recognize that this could not be done without introducing
statistical elements into the argument. In a series of papers
between 1897 and 1900, Planck utilized Maxwell s electro-
dynamics to develop a theory of thermal radiation in inter-
action with one or more identical, charged harmonic oscil-
lators within a cavity. He was only able to account for the
radiation s irreversible approach to thermal equilibrium by
employing methods analogous to those that Ludwig Boltz-
mann had used in kinetic theory. Planck introduced the no-
tion of natural (that is, maximally disordered) radiation,
which he defined by analogy with Boltzmann s definition of
molecular chaos. Using Maxwell s theory, Planck derived a
relation between the average energy of a charged oscil-
lator of frequency ½ in equilibrium with thermal radiation
and the energy density per unit frequency interval Á½ of the
radiation at the same frequency:
c3
½ = Á½ (1)
8À½2
where c is the speed of light.
168
EARLY WORK ON QUANTUM HYPOTHESIS
Planck calculated the average energy of an oscillator by
making assumptions about the entropy of the oscillators that
enabled him to derive Wien s law for the energy density
of the black-body spectrum, which originally seemed well
supported by the experimental evidence. But by the turn of
the century new observations showed systematic deviations
from Wien s law for large values of »T .
Planck presented a new energy density distribution for-
mula that agreed closely with observations over the entire
spectrum: 1
8Àh½3 1
Á½ = (2)
c3 eh½/kT - 1
In this expression, now known as Planck s law or Planck s
formula, k = R/N is Boltzmann s constant, where R is the
gas constant, N is Avogadro s (or Loschmidt s) number, and
h is a new constant (later called Planck s constant). To de-
rive this formula, Planck calculated the entropy of the os-
cillators, using what Einstein later called the Boltzmann
principle : S = k ln W where S is the entropy of a macro-
scopic state of the system, the probability of which is W .
Following Boltzmann, Planck took the probability of a state
to be proportional to the number of complexions, or possi-
ble microconfigurations of the system corresponding to that
state. He calculated this number by dividing the total en-
ergy of the state into a finite number of elements of equal
magnitude, and counting the number of possible ways of
distributing these energy elements among the individual os-
cillators. If the size of the energy elements is set equal to
h½, where ½ is the frequency of an oscillator, an expression
for the entropy of an oscillator results that leads to eq. (2).
Although Einstein expressed private misgivings about
Planck s approach in 1901, he did not mention Planck or
169
PART FOUR
black-body radiation in his papers until 1904. A study of the
foundations of statistical physics, which he undertook be-
tween 1902 and 1904, provided Einstein with the tools he
needed to analyze Planck s derivation and explore its con-
sequences. At least three elements of Einstein s general
molecular theory of heat were central to his subsequent
work on the quantum hypothesis: (1) the introduction of
the canonical ensemble; (2) the interpretation of probability
as it occurs in Boltzmann s principle; and (3) the study of
energy fluctuations in thermal equilibrium.
1. In an analysis of the canonical ensemble, Einstein
proved that the equipartition theorem (see the Introduc-
tion, p. 16) holds for any system in thermal equilibrium.
In paper 5 he showed that, when applied to an ensem-
ble of charged harmonic oscillators in equilibrium with
thermal radiation, the equipartition theorem leads, via eq.
(1), to a black-body distribution law now known as the
Rayleigh-Jeans law:
8À½2
Á½ = kT (3)
c3
Despite its rigorous foundation in classical physics, eq. (3)
only agrees with the observed energy distribution for small
values of ½/T ; indeed, as Einstein noted, it implies an infi-
nite total radiant energy.
2. In 1906, Einstein posed a question that preoccupied
him and others at the time: How is it that Planck did not
arrive at the same formula [eq. (3)], but at the expression. . .
[eq. (2)]? One answer lies in Planck s definition of W in
Boltzmann s principle, which, as Einstein repeatedly noted,
differs fundamentally from his own definition of probabilities
as time averages. As noted above, Planck interpreted W as
170
EARLY WORK ON QUANTUM HYPOTHESIS
proportional to the number of complexions of a state of the
system. As Einstein pointed out in 1909, such a definition
of W is equivalent to its definition as the average, over a
long period of time, of the fraction of time that the system
spends in this state only if all complexions corresponding
to a given total energy are equally probable. However, if
this is assumed to hold for an ensemble of oscillators in
thermal equilibrium with radiation, the Rayleigh-Jeans law
results. Hence, the validity of Planck s law implies that all
complexions cannot be equally probable. Einstein showed
that, if the energies available to a canonical ensemble of
oscillators are arbitrarily restricted to multiples of the energy [ Pobierz całość w formacie PDF ]