Number of electrons in a homogeneous field using delta function normalization

Abstract The number of electrons in a homogeneous electric field is derived quantum mechanically for a certain range. Using the delta function normalization method, which is described in detail in Landau–Lifshitz’s Quantum Mechanics, the normalization coefficient of the wavefunctions with continuous energy eigenvalue spectra is obtained. Subsequently, the number of electrons in a one-dimensional homogeneous field in a certain region is derived from this normalization coefficient. We illustrate the derivation of physical quantities using normalization coefficients through selected examples. The derivation provides an interesting application of delta function normalization, and thus can serve as a good textbook example of this normalization method, which used to play a limited role in quantum mechanics.