Electron Wave Optics 23 



and this could be identified with the space of geometry and 

 electro-magnetism, but two electrons required six dimensions, 

 three electrons nine, and so on. These hyperspaces are obviously 

 only convenient fictions, quasi-geometrical interpretations of the 

 somewhat abstract mathematics of quantum theory. If we admit 

 this, it would be illogical to claim an exceptional position for 

 one electron, to assume that the waves of the first electron occupy 

 space, but that as soon as a second electron appears, one of the 

 two has to go outside with its waves. The reason why in electron 

 optics we can use this strictly speaking inadmissible picture of 

 electron waves in space is the slight degree of interaction of 

 electrons in electron beams. The greatest concentration which 

 is realized near the object is usually less than 10^^ electrons/cm'"^ 

 and in all the rest of the beam it is very much less.* What small 

 interaction there is, can be usually considered as a space charge 

 effect. This means that only one electron is considered as a 

 particle at a time, the others are accounted for by imagining 

 their charges spread out continuously over the space occupied by 

 the beam. In this sense, the problems of electron optics are 

 always one electron problems and Schrodinger's original picture 

 can be used with impunity.f Moreover, in calculating the dif- 

 fraction efifects at apertures, we can apply the well-known 

 methods and results of optics without any modification. Though 

 the momentum of an electron is very much larger than the 

 momentum of a photon of the same energy, it is not sufficient 

 to shake a physical aperture appreciably. Important differences 

 arise only in the case of scattering by single atoms or molecules, 

 to which we shall return later. 



To round off the picture, we must give a somewhat more 

 precise meaning to the intensity of electron waves. If we cal- 

 culate, e.g., the Airy diffraction figure and the densities, which 

 we must expect in it, this does not mean of course that an elec- 



* Electron concentration in metals, in which the quantum mechanical 

 interaction becomes very marked, are of the order lO-'^/cm^. 



t The improvements in the quantum mechanics of electrons due to Dirac 

 can be ignored in electron optics, as the spin of free electrons cannot be 

 observed. 



