672 BELL SYSTEM TECHNICAL JOURNAL 



the situation is similar to that which arises through neglecting "edge 

 effects" in calculating the capacity of a parallel plate condenser. 



In Fig. 11 we show also a series of coupled oscillators with boundary 

 conditions corresponding to those prescribed for the atoms. For 

 this case there are six coupled oscillators, which when uncoupled had 

 six independent normal modes of vibration all with the same fre- 

 quency, like that shown for the single oscillator of Fig. Id. After 

 coupling there are six normal modes all having different frequencies; 

 the standing wave patterns corresponding to these are shown in Figs. 

 1 \i to 1 In. A similar splitting of frequencies occurs when the members 

 of a set of electrical circuits are placed in close proximity as indicated 

 in Fig. Wo. For them the situation is more complicated than for the 

 mechanical oscillators; each mechanical oscillator has but a single 

 frequency, whereas each circuit has a fundamental and a sequence of 

 overtones. Each possible frequency for the electrical circuits is 

 split by coupling into a set of six. 



In Figs. \\b to llg are shown the proper electron wave functons 

 which arise from the \s atomic states. These wave functions have 

 different energies. When the atoms were separated there were six \s 

 wave functions for the six atoms and each of these gave two states — 

 one for each spin. After coupling we find six crystal wave functions 

 and twelve crystal quantum states, the same number of states for each 

 spin as before. This illustrates a fundamental theorem concerning 

 wave functions in crystals which holds for two and three dimensions 

 as well as for one and is true no matter how large the number of atoms 

 in the crystal. This theorem, which we shall refer to as the "conserva- 

 tion of states," may be stated as follows: consider a set of N similar 

 isolated quantum mechanical systems; they may be single atoms or 

 molecules. Any particular quantum state is then repeated N times 

 over, once for each system. Now bring the systems together so that 

 the energy levels have split up. Then for each AT^-times-repeated 

 quantum state of the isolated systems, we find a set of N crystal quan- 

 tum states. In other words, putting the systems together may change 

 the energies and wave functions of the quantum states but no states 

 are gained or lost in the process. 



In Fig. 12 we indicate how the energy levels of the states depend 

 upon the lattice constant. Each energy level in the figure corresponds 

 to two states, one for each spin. For simplicity only two atomic 

 levels are shown here. Higher energy levels split appreciably at 

 larger lattice constants because of the greater spatial extension of their 

 wave functions. For any particular lattice constant the energy levels 

 arising from a given atomic state lie in a certain band of energy. The 



