Re: the quantum hypothesis and blackbody radiation
- From: Igor Khavkine <igor.kh@xxxxxxxxx>
- Date: Sun, 22 Jan 2006 23:53:04 +0000 (UTC)
Alex wrote:
> 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
.
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