bei48482_FM

and so

u( )

(9.47)

This formula gives the energy density of photons of frequency in equilibrium at the

temperature Twith atoms whose possible energies are Eiand Ej.

Equation (9.47) is consistent with the Planck radiation law of Eq. (9.38) if

BijBji (9.48)

and

 (9.49)

We can draw these conclusions:

1 Stimulated emission does occur and its probability for a transition between two states

is equal to the probability for absorption.

2 The ratio between the probabilities for spontaneous and stimulated emission varies

with ^3 , so the relative likelihood of spontaneous emission increases rapidly with the

energy difference between the two states.

3 All we need to know is one of the probabilities Aji, Bij, Bjito find the others.

Conclusion 3 suggests that the process of spontaneous emission is intimately related

to the processes of absorption and stimulated emission. Absorption and stimulated

emission can be understood classically by considering the interaction between an atom

and an electromagnetic waves, but spontaneous emission can occur in the absence of

any such wave, yet apparently by a comparable interaction. This paradox is removed

by the theory of quantum electrodynamics. As briefly described in Sec. 6.9, this theory

shows that “vacuum fluctuations” in Eand Boccur even when EB0 classically,

and these fluctuations, analogs of the zero-point vibrations of a harmonic oscillator,

stimulate what is apparently spontaneous emission.

9.8 SPECIFIC HEATS OF SOLIDS

Classical physics fails again

Blackbody radiation is not the only familiar phenomenon whose explanation requires

quantum statistical mechanics. Another is the way in which the internal energy of a

solid varies with temperature.

Let us consider the molar specific heat of a solid at constant volume, cV. This is the

energy that must be added to 1 kmol of the solid, whose volume is held fixed, to raise

its temperature by 1 K. The specific heat at constant pressure cpis 3 to 5 percent higher

than cVin solids because it includes the work associated with a volume change as well

as the change in internal energy.

The internal energy of a solid resides in the vibrations of its constituent particles,

which may be atoms, ions, or molecules; we shall refer to them as atoms here for

8 h 

3



c^3

Aji



Bji

AjiBji





B

B

i

j

j

i

eh^ kT 1

320 Chapter Nine

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