"Greg Egan - Foundations 4 - Quantum Mechanics" - читать интересную книгу автора (Egan Greg)

naked eye at room temperature. The cavity of a furnace containing nothing but the
thermal radiation from its heated walls, with a tiny hole through which radiation can
escape to be observed, serves as a good approximation to a black body, both theoretically
and experimentally, so black body thermal radiation is also known as cavity radiation.
Maxwell's theory suggested that the electromagnetic field inside a cavity should
be treated as something akin to the three-dimensional equivalent of a piano string being
bashed at random, simultaneously vibrating with every possible harmonic. A piano
string has evenly spaced harmonics, say 500 Hz, 1000 Hz, 1500 Hz, and so on, which
occur when an exact number of half-wavelengths fit the length of the string; the fact that
the ends of the string are fixed prevents other frequencies being produced. An
electromagnetic field in a three-dimensional cavity is subject to similar boundary
 Egan: "Foundations 4"/p.3


conditions, but unlike a piano string the field's vibrations are free to point in different
directions. For example, the field in a cubical cavity might vibrate in such a way that 5, 7
and 4 half-wavelengths span the cavity's width, breadth and height respectively, because
of the way the waves are oriented with respect to the walls. But waves of exactly the
same frequency, oriented differently, would fit just as well with 4, 5 and 7 half-
wavelengths spanning the same three dimensions.




This makes the situation more complicated than it is for a piano string, but it's still
not too hard to count the modes available to the field: the number of distinct ways in
which it can vibrate. Figure 2 isn't a drawing of a furnace cavity; rather, each point here
represents a different mode, with the x, y and z coordinates of the point giving the
number of half-wavelengths that fit across the width, breadth, and height of the cavity.
The more tightly packed the waves are, the shorter their wavelength and the greater their
frequency. The exact frequency of any mode is proportional to its distance from the
centre of the diagram тАФ that's just a matter of Pythagoras's theorem, and the relationship
between frequency and wavelength. So the number of points between the two spherical
shells counts the number of modes in the frequency range тИЖF. For small values of тИЖF,
this is proportional to the surface area of the inner sphere, which is proportional to F2.
Because the walls of the cavity are assumed not to favour any particular
frequency, every possible mode of the electromagnetic field should have, on average, an
equal share of the total energy. The trouble is, the field has an infinite number of modes
тАФ at ever higher frequencies, you just keep finding more of them. If the energy from the
furnace really was free to spread itself between them, giving them all an equal share, that
would be a never ending process, like gas escaping into an infinite vacuum. The average
 Egan: "Foundations 4"/p.4


frequency of the radiation in the cavity would wander off towards the ultraviolet and
beyond, never stabilising at any fixed spectrum.