bei48482_FM

140 Chapter Four

so that

  (4.20)

When p1, the frequency of the radiation is exactly the same as the frequency

of rotation fof the orbital electron given in Eq. (4.19). Multiples of this frequency are

radiated when p2, 3, 4,.... Hence both quantum and classical pictures of the

hydrogen atom make the same predictions in the limit of very large quantum num-

bers. When n2, Eq. (4.19) predicts a radiation frequency that differs from that given

by Eq. (4.20) by almost 300 percent. When n10,000, the discrepancy is only about

0.01 percent.

The requirement that quantum physics give the same results as classical physics in

the limit of large quantum numbers was called by Bohr the correspondence princi-

ple.It has played an important role in the development of the quantum theory of

matter.

Bohr himself used the correspondence principle in reverse, so to speak, to look for

the condition for orbit stability. Starting from Eq. (4.19) he was able to show that stable

orbits must have electron orbital angular momenta of

mr  n1, 2, 3,... (4.21)

Since the de Broglie electron wavelength is hm, Eq. (4.21) is the same as

Eq. (4.12), n 2 r, which states that an electron orbit must contain an integral num-

ber of wavelengths.

4.7 NUCLEAR MOTION

The nuclear mass affects the wavelengths of spectral lines

Thus far we have been assuming that the hydrogen nucleus (a proton) remains

stationary while the orbital electron revolves around it. What must actually happen, of

course, is that both nucleus and electron revolve around their common center of mass,

which is very close to the nucleus because the nuclear mass is much greater than that

of the electron (Fig. 4.17). A system of this kind is equivalent to a single particle of

mass mthat revolves around the position of the heavier particle. (This equivalence is

nh



2 

Condition for

orbital stability

2 p



n^3

E 1



h

Frequency of

photon

Figure 4.17Both the electron and nucleus of a hydrogen atom revolve around a common center of

mass (not to scale !).

Hydrogen

nucleus

Center of mass

Axis

Electron

bei48482_ch04.qxd 1/14/02 12:20 AM Page 140