THE QUANTUM PHYSICS OF SOLIDS 653 



in a generalized form the restrictions imposed by Pauli's principle. 

 As is too frequently the case in mathematical physics, it is much easier 

 to state the problem than to solve it; the solutions of Schroedinger's 

 equation are, in fact, so difficult to obtain that exact solutions have 

 been found for atomic systems only of the simplest type, namely those 

 consisting each of a single nucleus and a single electron. For this 

 case, the quantum states and their energies are all exactly known. 

 For other cases approximations of varying degrees of exactness must 

 be used. The difficulty arises from the interactions between the elec- 

 trons. If it were not for these interactions, one could obtain exact 

 solutions for atoms having many electrons. The difficulty is that the 

 interactions — they are merely electrostatic repulsions — prevent each 

 electron from being independently in a definite quantum state. The 

 interaction of each electron with another is in general small compared 

 to its interaction with the nucleus. To a first approximation, then, 

 the electrons are treated as not interacting and then corrections are 

 applied to this over-simplified picture. (In this first approximation, 

 the generalized mathematical statement of Pauli's principle reduces 

 to the one we gave in the last section — only one electron may occupy 

 a given quantum state.) As a result of this procedure of over-simplifi- 

 cation followed by corrections, our exposition will commence with a 

 discussion of the quantum states of an electron in an atom as if these 

 quantum states were private possessions of the electron and not influ- 

 enced or disturbed in any way by the other electrons. We shall then 

 correct this picture to some extent by considering how the energy of 

 a given electron depends upon the behavior of the other electrons. 

 One correction term which we shall introduce in this way is the impor- 

 tant "exchange energy" discussed below. Thus atomic theory repre- 

 sents a field of endeavor in which further progress is made largely by 

 improvements and refinements. It should be emphasized, however, 

 that the corrections and refinements are not additional assumptions, 

 which are added to the theory, but that they represent instead only 

 steps forward in improving the wave mechanical solutions. 



The last paragraph mentions that an approximate treatment of 

 Schroedinger's equation leads to a set of possible quantum states 

 for an electron in the atom. We shall discuss Schroedinger's equa- 

 tion and the wave functions corresponding to the quantum states 

 in more detail later and at present be concerned only with a descrip- 

 tion of the results. In a neutral atom the electrons arrange them- 

 selves in the quantum states in such a way as to make the energy 

 of the atomic system a minimum. Consistent always with Pauli's 



