Blackbody Radiation and the Quantization of Energy

Energy quantization, the noncontinuous nature of energy,was discovered from observing radiant

energy from blackb o dies. A blackbody absorbs electromagnetic EM energy eciently. An ideal

blackb o dy absorbs all electromagnetic energy that b ears up on it. Ecient absorb ers of energy also

are ecient emitters or else they would heat up without limit. Energy radiated from blackb o dies

was studied b ecause the characteristics of the radiation are indep endent of the material constitution

of the blackb o dy. Therefore, conclusions reached by the study of energy radiated from blackb o dies

do not require quali cations related to the material studied.

If you heat up a go o d absorb er emitter of EM energy, it will radiate energy with total p ower

p er unit area prop ortional to the fourth p ower of temp erature in degrees Kelvin. This is the Stefan

- Boltzmann law:

4

R = T ;

8 2 4

where the Stefan constant  =5:6703  10 W/m K . This lawwas rst prop osed by Josef Stefan

in 1879 and studied theoretically by Boltzmann a few years later, so it is named after b oth of them.

The EM energy emitted from a blackb o dy actually dep ends strongly on the wavelength, , of

the energy. Thus, radiantpower is a function of wavelength, R , and total p ower p er unit area is

simply an integral over all wavelengths:

Z

R = R d: 1

The function R  is called the blackbody spectrum. Recall that

! 2

c = f = ! =2f k =

k 

The blackb o dy sp ectrum is di erent for di erent temp eratures, but has the same general shap e as

the solid lines in Figure 1 show. More total p ower is emitted as the temp erature of the blackbody

grows and the p eak of the sp ectrum,  , shifts to shorter wavelengths. In fact,  is inversely

m m

prop ortional to temp erature T such that:

3

 T = constant =2:898  10 mK ;

m

which is known as Wiens' displacement law b ecause it quanti ed the displacement of the sp ectral

peak as temp erature grows.

Without going into any detail, it turns out that classical EM theory predicts that

4

R =2ckT  Rayleigh-Jeans Law ; 2

23 5

where k =1:381  10 J/K = 8:617  10 eV/K is the Boltzmann constant. Note that at ro om

temp erature the quantity kT  1=40 eV. Insp ection of this prediction, shown in Figure 1 as the

dotted line, indicates that it do es not agree with observations of the blackb o dy sp ectrum. In fact,

not only is there disagreement, but substitution of equation 2 into 1 results in the catastrophic

result that the power per unit area radiated by a blackbody is in nite. Because the disagreement

between the observed curves and the curve predicted by the Rayleigh-Jeans formula worsens as

period decreases this is called the ultraviolet catastrophe. Recall that ultraviolet light is shorter

wavelength than visible lightwhich is itself shorter wavelength than infrared, etc. 1 Planck’s Law 1 ) λ 1650 K 0.8

Rayleigh - Jeans Formula 0.6 1450 K

0.4

1250 K

normalized spectral amplitude R( 0.2

0 0 2000 4000 6000 8000

wavelength (nm)

Figure 1: Solid curves are blackb o dy sp ectra predicted by Planck's Law eqn 4 for the three

temp eratures indicated, and the dotted curve is the prediction from the Rayleigh-Jeans Law eqn

2 for a temp erature of 1650 K.

One of the key assumptions in the derivation of the erroneous Rayleigh-Jeans lawisthecontin-

uous nature of EM energy. Planck was able to mo del the shap e of the blackbody sp ectrum much

more accurately by assuming that energy is quantized, that it is not continuous:

E = nhf n =0; 1; 2; 3;::: 3

n

where h is Planck's constant. With this assumption, in 1900 Planck rederived the theoretical shap e

of the blackb o dy sp ectrum and found the following:

2 5

2c h

R = Planck's Law 4

hc=k T

e 1

Fitting this functional form, which is shown for three temp eratures in Figure 1, for the unknown h,

34 15

Planck estimated Planck's constant, currentvalues of which are h =6:626  10 J-s = 4:136  10

eV-s.

This is generally considered the earliest indication of the quantization of energy. Equation 3

is one that we will use throughout our study of quantum mechanics. It will turn out that energy is

quantized by this formula for all forms of EM energy and for matter, as well. 2