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Unformatted text preview: Ben Sauerwine Practice for Qualifying Exams Problem Source: CMU August 2003 Qualifying Exam Consider a hollow body at temperature T containing electromagnetic radiation in equilibrium with its walls. The energy spectrum of this so-called blackbody radiation can be measured by observing it through a negligibly small hole in the cavity wall. As argued by Planck, the energy levels of the radiation are quantized and are the same as those of a simple harmonic oscillator of angular frequency ω . (a) What are the quantized energy levels of a radiation mode of angular frequency ω ? Use the Boltzmann distribution to find the thermal average number of photons for such a mode. ( ) ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ − = = − = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ = = = = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + = − − − − − − ∑ ∑ ∑ T k T k n i n T k i i T k i i T k i i T k E n n B B B B B B i e e Z Z N N e e e e Z n E po zero ignore n E ω ω ω ω ω ω ω h h h h h h h 1 1 1 int 2 1 (b) The density of modes in...
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