THE QUANTUM PHYSICS OF SOLIDS 663 



moment six times as large as the spin magnetic moment of the electron.^ 

 The same exchange effect which causes the Zd quantum states to fill 

 unevenly in the isolated atom causes, in the case of the metal, an 

 uneven filling of the "energy bands" which arise from these Zd states. 

 We shall return to this topic in the section on ferromagnetism. 



Solving Schroedinge/ s Equation 



The possible quantum states of an atom are obtained by solving 

 Schroedinger's equation for an electron moving in the potential field 

 of the nucleus and the other electrons. In Fig. la we have represented 

 the potential energy of an electron in an atom. If this potential 

 energy (call it U) is known as a function of the position x, y, z, of the 

 electron, then the Schroedinger equation is 



^. + ^.+ a^ + -^(£- W = 0, (1) 



where m is the mass of the electron, h is Planck's constant and £ is 

 an unknown energy and 4^ an unknown wave function, for which a 

 physical interpretation will shortly be given. It is found that this 

 equation possesses proper solutions only for certain values of E\ once 

 these values are known, the equation can be solved for the unknown 

 wave functions. The fact that only certain values of E are possible 

 will probably seem more natural after reading the discussion given 

 below of a mechanical system. The permitted energies and wave 

 functions give the system of quantum states of Fig. 2, 



The wave equation of Schroedinger is similar in form to many of the 

 other wave equations of mathematical physics. In Fig. 1g to 1j we 

 represent a stretched membrane like a rectangular drumhead. If the 

 mass per unit area of the membrane is o- and the surface tension is T, 

 then the wave equation for it is 



5^2 + ai2 +-7-^-0. (2) 



where / is the unknown frequency of vibration and ^ is the unknown 

 vertical displacement. Applied to the membrane, this equation has 

 solutions only for certain values of/; the standing wave patterns corre- 

 sponding to the four lowest frequencies are shown in Figs. 7g to 7j. 



^ For transition elements other than chromium, the motions of the electrons in 

 their wave functions produce magnetic moments that must be considered as well as 

 the spin; for a discussion of this point the reader is again referred to "Spinning Atoms 

 and Spinning Electrons" by K. K. Darrow, Bell System Technical Journal, XVI, p. 

 319 and to texts on atomic physics. 



