Any object which is warm radiates thermal energy. Think of it as coming from vibrations of atoms and electrons in the material, if you like. Hot metal is red, then hotter => yellow, then white. Not just metal, though. Coal too. And the sun (gases). Anything. We will consider an ideal object, which absorbs all radiations touching it. To be in equilibrium, this object will also emit at the same rate. This is called a blackbody.
An example is a hollow cavity, and it really doesn't matter what the cavity is made of. The hole into the cavity acts as the "black body".
Now, analyze the light coming back OUT, and you will see:
This is called the Blackbody Spectrum. People were working very hard to understand it in the late 19th century. Rayleigh-Jeans did the calculation (correctly, mind you, but using only classical E+M)
We won't go into the details, but the primary ingredient is that the radiation in a closed box has only certain possible wavelengths, called "modes", or "standing waves". Each standing wave will have an average energy associated with it, given by kT. (k is Boltzman's constant, known from thermodynamics) This is a result of purely classical thermodynamics.
Rayleigh-Jeans result was that the
*total energy*/volume, contained in modes with frequency near nu, is given by
(To get this, you just add up all the possible standing waves available, that have a frequency near nu, and then multiply by the energy/mode.)
But this result is totally crazy. Because, the total energy in the cavity, found by integrating the above formula over all frequencies, gives infinity, i.e. the integral diverges. The formula predicts that there's too much power at high frequencies (the UV end of the spectrum), and was called the "Ultraviolet Catastrophe".
Planck began by using what he called "lucky guesswork", and got
A one parameter FIT to the experimental curve, not a derivation. It's called the Planck radiation formula. The parameter, h, obtained from the data was 6.6 *10^-34 J sec. Notice that as the frequency gets low, one has
which is Rayleigh-Jeans prediction again. But now, as the frequency gets large, the exponential kills you, and there's no catastrophe. This formula fits the data great, and it still does today. ( COBE data for the temperature of the universe as a whole fits this formula spectacularly)
But why? How do we understand this formula? After "the most strenuous work of
my life", Planck discovered that he could derive the formula. But he needed a
very strange assumption, different from Rayleigh-Jeans. He had to assume that
the oscillators in the walls do *not* have continuous distributions of
energies. Rather, he postulated that E =
,
where n is an integer. The oscillators can jump from one energy to the next,
emitting radiation of frequency nu. So, they can only dump energy bundles of
size (h nu) into the cavity. (The average energy associated with a mode is NOT
kT any more) Planck had realized that each standing wave must have some
restrictions on its energy, a very non-classical idea.
Note how small h is. For a tuning fork, (nu = 660 Hz),
h nu = 6.63*10^-34 J sec * (660 /s) = 4.4*10^-31 J.
But, a tuning fork has a total vibrational energy on the order of dozens of milli Joules. So, you could never hope to notice that there are "steps" in the allowed vibrational energy.
On the other hand, an atom emitting orange light has
h nu = (6.6*10^-34 J sec) *(5 *10^14 Hz) = 3*10^-19 J = 2 eV
Now, the total energy of ionization of H is 13.6 eV, so 2 is a reasonably big step in comparison, so the quantization of energy is quite important in atoms.
Classically, there's no reason at all why oscillators in a wall can't dump any amount of energy they feel like. Planck saw this hypothesis as an "act of desperation", and refused to take it too seriously. It was just a trick. But as we've seen, Einstein did take it seriously, and we now have many examples of experimental evidence that E = h nu for photons.
There are many more examples than we've mentioned. For example, photon goes to e+ e- (or annihilation of matter and antimatter to photons) have billiard-ball like kinematics. The gravitational red shift of photons (responsible for black holes, among other things) can be understood as a photon having a gravitational "mass" of m=p/c = E/c^2 = h nu / c^2, and then as it climbs, h nu = h nu' + mgh. And on and on...
So once again, we ask - is light a wave or a particle? It has a dual character. Wave theory and quantum theory complement each other, each is only part of the story. Think again of the double slit experiment:
This shows interference, so light must be a wave. But think of the photoelectric effect, where the light must be a particle.
In this double slit experiment, think of the intensity in the two different pictures:
Wave picture:
= average electric field
Quantum picture: I = N h nu, with N = (# photons/sec/area)
Setting these equal, it appears that N
.
If N is huge, you see an ordinary interference pattern. If N is small (e.g.,
one photon at a time) you would see individual flashes - which doesn't seem
very wavelike. But keep track of the flashes, and the pattern they form will be
an interference pattern! (More flashes occur where the interference pattern is
large) So, it seems like E^2 is telling you the probability of finding a
photon at some place and time. I.e., the intensity of the photon wave tells you
the likelihood that a photon arrives there. This is a way of thinking about the
duality that we'll talk more about soon!
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