THE QUANTUM PHYSICS OF SOLIDS 651 



chanical theory predicts the same result, but with a slightly dififerent 

 value for L. According to the old theory L = Ik^/e"^ = 1.44 X 10~^ 

 volts'/degree^ where k is Boltzmann's constant, while the new gives 

 L = Tr^P/3e^ = 2.45 X 10~^ voltsVdegree^, and the experimental values 

 for several elements are Cu 2.23, Ag 2.31, Au 2.35, Mo 2.61, W 3.04, 

 Fe 2.47 — all times 10"^ volts'/degree^. We see that the constancy 

 of the Lorentz number predicted by both theories is in reasonable 

 agreement with experiment, but that in predicting the numerical 

 value of the constant the new theory is better than the old. 



The fundamental problem of how the electrons and nuclei form 

 stable atoms and crystals was, as we have said above, inexplicable on 

 the older theory. The newer quantum mechanics of Bohr and later 

 that of Schroedinger, Heisenberg, and Dirac were needed. Bohr 

 postulated that out of the infinity of possible motions for the electrons 

 of an atom, only a certain restricted set was permitted. Each per- 

 mitted motion corresponded to a definite energy for the atomic system 

 as a whole. This concept of energy levels for the atom gave a natural 

 interpretation to nature of atomic spectra and explained the meaning of 

 the combination principle. In order to restrict the atomic motions to 

 certain energy levels, Bohr supposed that the laws of atomic dynamics 

 were such that only those modes of motion were permitted for which 

 certain dynamical quantities, called phase integrals, had values equal 

 to multiples of Planck's constant h. For the case of the hydrogen atom 

 these laws led to the now well-known Bohr orbits for the electron and 

 to energy levels which were in good agreement with experiment. For 

 atoms with more electrons it was very difficult to apply Bohr's laws 

 except in a very approximate and unsatisfactory way. However, two 

 very valuable concepts came from his theory which are preserved in the 

 newer wave mechanical theory. These were that the individual elec- 

 trons could be thought of as restricted to certain orbits and that these 

 orbits were specified by giving them certain quantum numbers. It 

 was found that three quantum numbers were needed to specify the 

 orbit. All atoms were found to have the same general scheme of 

 orbits. The number of electrons moving in these orbits varies from 

 atom to atom and for any given atom is equal to the atomic number Z. 

 In order to explain the facts of spectroscopy and the periodic table 

 of the elements, it was necessary to introduce a rule known as Pauli's 

 principle. This principle states simply that no more than two elec- 

 trons may occupy the same orbit in an atom; that is, no more than 

 two electrons of an atom may have the same three quantum numbers. 

 As we shall discuss in the next section, a complete specification of the 

 state of an electron in an atom requires four quantum numbers; two 



