the quantum hypothesis and blackbody radiation
Alex
> I haven't seen anyone mention the first ever application of the quantum
> hypothesis: the Planck black body radiation spectrum. Planck assumed
> that the amount of energy per frequency of the EM radiation is
> quantized (i.e. comes only in discrete increments). From this
> assumption, he derived a theoretical prediction for the black body
> radiation spectrum which matched the experiment extremely well.
>From the quantum hypothesis itself you can derive rather Wien's
formula. Planck was trying to explain experimental data and his
radiation law but it was not a theoretical prediction. From
discontinuity of energy you can arrive only at Planck's mean energy
distribution. To get Planck's radiation law you would need additional
presumptions which are quite foreign to the quantum hypothesis.
Alex
> Thermodynamic properties, such as the black body spectrum are
> independent of the way radiation interacts with matter (except for the
> black body idealization itself). Hence the quatization properties may
> be assigned directly to the radiation field.
>
> Igor
Igor Khavkine
> Igor Khavkine wrote:
>
> > I haven't seen anyone mention the first ever application of the quantum
> > hypothesis: the Planck black body radiation spectrum. Planck assumed
> > that the amount of energy per frequency of the EM radiation is
> > quantized (i.e. comes only in discrete increments). From this
> > assumption, he derived a theoretical prediction for the black body
> > radiation spectrum which matched the experiment extremely well.
>
> From the quantum hypothesis itself you can derive rather Wien's
> formula. Planck was trying to explain experimental data and his
> radiation law but it was not a theoretical prediction. From
> discontinuity of energy you can arrive only at Planck's mean energy
> distribution. To get Planck's radiation law you would need additional
> presumptions which are quite foreign to the quantum hypothesis.
Can you explain the distinction that you draw between "Planck's mean
energy distribution" and "Planck's radiation law", and what other
presumptions you think are necessary? If one's goal is to calculate the
energy density per frequency, all that is needed is the assumption that
energy can only be stored in discrete packets of h*nu per mode of
frequency nu (that's the quantum hypothesis) and the mode density per
frequency (derived from Maxwell's equations). This exercise is a
standard one in most statistical mechanics texbooks. Granted, this is
probably not how Planck got his formula in the first place, but a
perfectly valid derivation nonetheless.
> > Thermodynamic properties, such as the black body spectrum are
> > independent of the way radiation interacts with matter (except for the
> > black body idealization itself). Hence the quatization properties may
> > be assigned directly to the radiation field.
I'm tempted to say a few words about the "black body idealization".
Personally, I was never really happy with the standard textbook
introduction of an ideal black body. However, recently, I encountered a
really nice explanation in George Gamow's book, _The 30 Years that
Shook Physics_.
As usual, the idea is quite simple. We want to consider the properties
of electromagnetic radiation when it's in equilibrium with its
surroundings, such as some sort of cavity. An important question is how
this equilibrium is established in practice.
If we were considering a gas of particles, then even if its initial
state was far from equilibrium, it would soon equlibrate due to
thermalization. Thermailzation comes about due to the interparticle
collisions which randomly spread the available energy equally between
the particles. One consequence of thermalization is the equipartition
of energy between the degrees of freedom associated to the particles.
On the other hand, when considering radation, it's fundamental degrees
of freedom are the oscillation amplitudes associated to different modes
that come about as solutions of Maxwell's equations. So it is natural
to ask what is the distribution of energy between these modes in
equilibrium at a temperature. However, since the radiation modes are
decoupled, there will be no thermalization to spread the energy between
the radiation modes. In other words, if EM radiation were contained in
a cavity with perfectly reflecting walls, the radiation will never
reach equilibrium.
One solution is to introduce a mediator. The mediator will interact
with all the radiation modes, while the radiation modes themselves are
still decoupled, and will act as a thermalization agent. At first
glance, it may seem that different mediators will produce different
equilibrium states, which will be dictated by the absorption and
emission properties of the mediator. However, experimentally, it is
found that the equilibrium energy distribution between the radiation
modes depends only on temperature and not on the typer of mediator used
(up to small corrections).
Thus, it appears that an idealization of a mediator should be
sufficient to obtain the equilibrium energy distribution for radation.
The only property that are required of this idealized mediator is that
it be capable transfering energy from any one radiation mode to any
other one. In other words, it must be able to both absorb and emit
radation at any frequency. Historically, this idealized mediator has
been called a "black body" and the equilibrium spectrum of EM
radiation, the "black body radiation spectrum". The name "black" refers
to its ability to absorb radiation (or light) at any frequency.
Unfortunately, this name omits the important connotation of the
possibility of emission of radiation (light) at all possible
frequencies. I think that this omission has been responsible for why
it's been hard for me to grasp the meaning of a "black body".
I hope this tangent has helped someone else to be less confused.
Igor
Cyberkatru
> possibility of emission of radiation (light) at all possible
> frequencies. I think that this omission has been responsible for why
> it's been hard for me to grasp the meaning of a "black body".
>
> I hope this tangent has helped someone else to be less confused.
>
> Igor
>
Actually that did help me get a little clearer. I was alway a bit befuzzled
by the textbook treatments.
Thanx
Alex
>
> Can you explain the distinction that you draw between "Planck's mean
> energy distribution" and "Planck's radiation law", and what other
> presumptions you think are necessary? If one's goal is to calculate the
> energy density per frequency, all that is needed is the assumption that
> energy can only be stored in discrete packets of h*nu per mode of
> frequency nu (that's the quantum hypothesis) and the mode density per
> frequency (derived from Maxwell's equations). This exercise is a
> standard one in most statistical mechanics texbooks. Granted, this is
> probably not how Planck got his formula in the first place, but a
> perfectly valid derivation nonetheless.
"Planck's radiation law" is his famous formula for blackbody radiation
spectrum. "Planck's mean energy distribution" is what was obtained by
Planck from Boltzmann statistics with quantized energy assumption:
<E> = Eo / (exp[Eo/kT] - 1)
(Instead of "mean energy" it is probably better to talk about average
number of photons.)
>From this semiclassical statistical position you can get (for
3-dimensional system) Wien's formula - it was known (see, for example,
'Atomic Physics' by Max Born) but not mentioned often. To come to
Planck's radiation law, additional (foreign) presumptions were required
(ideas of Maxwell's electrodynamics, an absorbing and emitting
oscillators or something else).
Much later in 1924 Bose stepped away further from classical statistics
(what Bose himself did not recognize at that time). According to Bose,
Planck's (mean energy) distribution represents only a single mode of
thermal radiation in infinite number of modes. Planck's distribution
was "generalized" to become Bose-Einstein distribution. "The standard
exercise in most statistical mechanics textbooks" reproduces Bose
reasoning.
While the understanding of blackbody radiation became more statistical
and less dependent on the additional presumptions, there is an
important drawback (in my opinion) in transition from Planck to Bose
statistics. (Semi)classical statistics can be reduced to simple
deterministic behavior (to "a gas of particles", for example). It
appears that the same cannot be done for Bose statistics (causality is
lost). For more details see http://arxiv.org/abs/cond-mat/0512292
C. M. Heard
>From the quantum hypothesis itself you can derive rather Wien's
>formula. Planck was trying to explain experimental data and his
>radiation law but it was not a theoretical prediction. From
>discontinuity of energy you can arrive only at Planck's mean energy
>distribution. To get Planck's radiation law you would need additional
>presumptions which are quite foreign to the quantum hypothesis.
I believe that you have this backwards. According to Wehr and
Richards, the textbook used in my Modern Physics class some 35
years ago, Wien's law was essentially an empirical curve fit.
Explicitly it reads
dE/dlambda = c1 lambda^-5 / exp(c2 / (lambda * T))
where c1 and c2 are empirically determined constants, lambda is
wavelength, and T is absolute temperature. This provides a good
fit to experimental results at short wavelengths but a poor fit
at long wavelengths, where it should reduce to the classical
Rayleigh-Jeans law. Planck did notice that one could improve
the fit by putting a minus one into the denominator, and this
indeed was another empirical step.
However, Planck also showed that one could rigorously derive
his law from the quantum hypothesis (i.e., that a harmonic
oscillator has allowed energy values of n h_cross omega) _and_
the result from statistical mechanics that the ratio of the
probabalites of occupancys of two states A and B is
exp(-EA/kT) / exp(-EB/kT).
//cmh
Alex
> Alex wrote:
> >From the quantum hypothesis itself you can derive rather Wien's
> >formula. Planck was trying to explain experimental data and his
> >radiation law but it was not a theoretical prediction. From
> >discontinuity of energy you can arrive only at Planck's mean energy
> >distribution. To get Planck's radiation law you would need additional
> >presumptions which are quite foreign to the quantum hypothesis.
>
> I believe that you have this backwards. According to Wehr and
> Richards, the textbook used in my Modern Physics class some 35
> years ago, Wien's law was essentially an empirical curve fit.
> Explicitly it reads
>
> dE/dlambda = c1 lambda^-5 / exp(c2 / (lambda * T))
>
> where c1 and c2 are empirically determined constants, lambda is
> wavelength, and T is absolute temperature. This provides a good
> fit to experimental results at short wavelengths but a poor fit
> at long wavelengths, where it should reduce to the classical
> Rayleigh-Jeans law. Planck did notice that one could improve
> the fit by putting a minus one into the denominator, and this
> indeed was another empirical step.
The real history is not so simple and sequential as some Modern Physics
textbooks. You probably would be surprised to find that Rayleigh-Jeans
law has been complete only in 1905 (while Planck's law had been
formulated in 1900). A quick progress at the end of 19th century
brought to a new level the measurements of blackbody radiation
spectrum. Planck had access to the latest data and found the best
formula to describe experimental output. Wien's formula is the
best-known (nowadays) previous candidate.
The only difference between Wien's and Planck's formulas, which affects
the shape of the distribution, is the minus one in the denominator.
Planck has found an explanation to the denominator from ideas of quanta
and classical statistics (not to the whole formula). He arrived at
expression for mean energy of quantum oscillators:
<E> = Eo / (exp[Eo/kT] - 1)
It is interesting that from this classical statistical position you can
come to ... the shape of Wien's distribution (not many textbooks point
to this option).
> However, Planck also showed that one could rigorously derive
> his law from the quantum hypothesis (i.e., that a harmonic
> oscillator has allowed energy values of n h_cross omega) _and_
> the result from statistical mechanics that the ratio of the
> probabalites of occupancys of two states A and B is
> exp(-EA/kT) / exp(-EB/kT).
>
> //cmh
Planck never rigorously derived his law from statistical mechanics (his
reasoning was not pure statistical). Einstein's attempt (1916-17) is
not even mentioned in many statistical mechanics textbooks. Trying to
find such rigorous derivation Bose came to new statistics (without
understanding the novelty of his claims). It happened much later in
1924.
vage...@gmail.com
>of freedom are the oscillation amplitudes associated to different modes
>that come about as solutions of Maxwell's equations. So it is natural
>to ask what is the distribution of energy between these modes in
>equilibrium at a temperature. However, since the radiation modes are
>decoupled, there will be no thermalization to spread the energy between
>the radiation modes. In other words, if EM radiation were contained in
>a cavity with perfectly reflecting walls, the radiation will never
>reach equilibrium.
>One solution is to introduce a mediator. The mediator will interact
>with all the radiation modes, while the radiation modes themselves are
>still decoupled, and will act as a thermalization agent. At first
>glance, it may seem that different mediators will produce different
>equilibrium states, which will be dictated by the absorption and
>emission properties of the mediator. However, experimentally, it is
>found that the equilibrium energy distribution between the radiation
>modes depends only on temperature and not on the typer of mediator used
>(up to small corrections).
This reminds me of a question I had. Ok, if you have a radiation
field you need a mediator to thermalize the radiation modes, but
if you have a medium, lets say a gas, with no radiation field a
priori, then what is the mechanism that produces the thermal
radiation.
Igor Khavkine
> This reminds me of a question I had. Ok, if you have a radiation
> field you need a mediator to thermalize the radiation modes, but
> if you have a medium, lets say a gas, with no radiation field a
> priori, then what is the mechanism that produces the thermal
> radiation.
You'd have to be specific about what you mean by "no radiation field".
Radiation represents the state of the Electro-Magnetic field that's
always there, just like the position and velocity of a particle
characterize its state.
But suppose that we can find a state of the EM field that can be said
to have "no radiation" (which we sometimes can). Then your question may
be rephrased as: what happens to the state of the EM field as the
molecules of a gas start moving around and bouncing off each other?
And the answer would be: the molecules of the gas will "leak" EM
radiation during collisions. You have to remember that the molecules
themselves interact through the EM field and are inthemselves made up
of electrons and nuclei with non-homogeneous charge distributions (even
though they look neutral at common length scales). Whenever
non-homogeneous charge distributions start moving and accelerating (as
they do during collisions), they excite the EM field and produce
radiation.
If you consider the EM field itself as part of the thermodynamic system
including the gas, then it's clear that if the field starts out in some
very "cold" state (no radiation), then heat will from from the
molecules into the field until the two parts of the system reach
equilibrium. Equilibrium is possible since molecules not only leak, but
also absorb radiation. At this point the energy distribution of the EM
field will have the Planck form and the molecules of the gas will have
slightly less energy than when they started.
Hope this helps.
Igor