Friday, April 30, 2010

Max Planck and Blackbody Radiation

Max Planck is one of the founders of quantum mechanics, and the fundamental constant governing all quantum phenomena bears his name. His historic contribution arose from the study of thermal radiation. Section 14.7 in the 4th edition of Intermediate Physics for Medicine and Biology analyzes thermal radiation (also known as blackbody radiation), but does not tell the fascinating history behind this advance. In fact, Russ Hobbie and I write “we will not discuss the history of these developments, but will simply summarize the properties of the blackbody radiation function that are important to us.” What better place than this blog to fill in the missing history.

Eisberg and Resnick’s textbook Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles is a good place to learn more (I will quote from the first edition having the silver cover, which I used as an undergraduate). In fact, the opening section of their first chapter addresses this very issue.
“At a meeting of the German Physical Society on Dec. 14, 1900, Max Planck read his paper ‘On the Theory of the Energy Distribution Law of the Normal Spectrum.’ This paper, which first attracted little attention, was the start of a revolution in physics. The date of its presentation is considered to be the birthday of quantum physics, although it was not until a quarter century later that modern quantum mechanics, the basis of our present understanding, was developed by Schroedinger and others […] Quantum physics represents a generalization of classical physics that includes the classical laws as special cases. Just as relativity extends the range of application of physical laws to the region of high velocities, so quantum physics extends the range to the region of small dimensions; and, just as a universal constant of fundamental significance, the velocity of light c, characterizes relativity, so a universal constant of fundamental significance, now called Planck’s constant h, characterizes quantum physics. It was while trying to explain the observed properties of thermal radiation that Planck introduced this constant in his 1900 paper…”
Eisberg and Resnick end their first chapter with “A Bit of Quantum History”
“At first Planck was unsure whether his introduction of the constant h was only a mathematical device or a matter of deep physical significance. In a letter to R. W. Wood, Planck called his limited postulate ‘an act of desperation.’ ‘I knew,’ he wrote, ‘that the problem (of the equilibrium of matter and radiation) is of fundamental significance for physics; I knew the formula that reproduces the energy distribution in the normal spectrum; a theoretical interpretation had to be found at any cost, no matter how high.’”
To better understand the mathematics underlying blackbody radiation, try the new homework problem below, based on Eisberg and Resnick’s analysis (you may want to review Sec. 3.7 of our book about the Boltzmann factor before you attempt this problem).

Section 14.7

Problem 22.5 Let us derive the blackbody spectrum, Eq. 14.37.
(a) Assume the energy En of radiation with frequency ν is discrete, En = h ν n, where n=0, 1, 2, … Let the probability Pn of any state be given by the Boltzmann factor, C e-nhν/kT. Normalize this probability distribution (that is, find C by setting the sum of the probabilities over all states equal to one).
(b) Find the average energy Eave for frequency ν by performing the sum Eave = P0 E0 + P1 E1 + … .
(c) The number of frequencies per unit volume in the frequency range from ν to ν+dν is 8πν2dν/c3. Multiply the result from (b) by this quantity, to get the energy density of the radiation.
(d) The spectrum of power per unit area emitted from a blackbody is equal to c/4 times the energy density. Find the power per unit area per unit frequency, Wν(ν,T) (Eq. 14.37).

You may need to use the following two infinite series
1 + x + x2 + x3 + … = 1/(1-x) ,
x + 2x2 + 3x3 + … = x/(1-x)2 .

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