The Wave Properties of Electrons

Rolf E. Hummel

Ayahiko Ichimiya, Philip I. Cohen

Wlodek Zawadzki

Herein, relations between particle and wave properties for charge carriers in periodic potentials of crystalline metals and semiconductors are derived. The particle aspects of electrons in periodic potentials are considered using properties of quasimomentum (QM), while the wave aspects are described using wave packets of Bloch waves. The two aspects are combined in the derivation of QM–wavelength relations for energy bands of arbitrary nonparabolicity and nonsphericity. An effective mass relating electron QM to its average velocity for spherical energy bands is defined and used to calculate energy dependences of wavelengths for electrons in narrow‐gap semiconductors, graphene, and surface states of topological insulators. An uncertainty relation between electron QM and its spatial coordinate in periodic potentials is derived. It is emphasized that the described properties apply to the average (not instantaneous) electron behavior. Analogies and differences between the wave and uncertainty properties of electrons in crystalline solids and those in vacuum are traced. An interference experiment with semiconductor nanostructures is proposed to verify the theoretical predictions and demonstrate to what extent the wave properties of electrons in solids are distinct from those in vacuum.

Vadim A. Bednyakov

Abstract An attempt is made to substantiate the absence of wave properties for a single electron. What is commonly called the corpuscular-wave duality or the wave properties of matter, which appear in the form of a diffraction pattern on the target of certain kind, is a result of the obligatory involvement of a large number of electrons in the diffraction process. In the context of quantum mechanics, a form of the diffraction pattern can be predicted using the rule of adding probability amplitudes, which, in the simplest case, are solutions to the Schrödinger equation for free particles. Therefore, the answer to the question posed in the title sounds something like this: a single electron has no wave properties; however, they become apparent whenever there are many electrons and a suitable target.

Clinton J. Davisson

The experiments by the author and L. H. Germer, by G. P. Thomson and by others from which the wave properties of electrons are adduced are briefly described. The agreement between the results of these experiments and the prediction of L. de Broglie is pointed out. The wave and corpuscular properties of electrons are compared with the similar properties of light quanta.

George Paget Thomson