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Physica Scripta

Number of electrons in a homogeneous field using delta function normalization

D. Terasawa

Published Nov 8, 2024• 0 citations
Publisher

Abstract

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.

DOI
10.1088/1402-4896/ad908d

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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.
Abstract We obtain accurate eigenvalues of the one-dimensional Schrödinger equation with a Hamiltonian of the form H g = H + gδ(x), where δ(x) is the Dirac delta function. We show that the well known Rayleigh–Ritz variational method is a suitable approach provided that the basis set takes into account the effect of the Dirac delta on the wavefunction. Present analysis may be suitable for an introductory course on quantum mechanics to illustrate the application of the Rayleigh–Ritz variational method to a problem where the boundary conditions play a relevant role and have to be introduced carefully into the trial function. Besides, the examples are suitable for motivating the students to resort to any computer-algebra software in order to calculate the required integrals and solve the secular equations.
The normalization of scattering states is more than a rote step necessary to calculate expectation values. This normalization actually contains important information regarding the density of the scattering spectrum (along with useful details on the bound states). For many applications, this information is more useful than the wavefunctions themselves. In this paper, we show that this correspondence between scattering state normalization and the density of states is a consequence of the completeness relation, and we present formulas for calculating the density of states, which are applicable to certain potentials. We then apply these formulas to the delta function potential and the square well. We then illustrate how the density of states can be used to calculate the partition function for a system of two particles with a point-like (delta potential) interaction.
Usually, expected values for various physical quantities, such as the number of electrons occupying certain states or the Coulomb interaction between different states of electrons, can be expressed in terms of integrals. In contrast, our method, based on differential forms, shows that expected values can be obtained by averaging over time. To confirm the validity of our method, we prepare the two cases: one is a very simple case with no many-body interaction, and the other is the case where the many-body term is included (the simplest Anderson Hamiltonian). Regarding the simple case without inclusion of many-body term, we prove analytically that the number of electrons occupying any state derived from our method is equivalent to the analytical one evaluated from the Greenʼs function method. When the many-body term is included, our results show good numerical agreement with the analytical ones derived from the Greenʼs function method. By the two cases, the calculation of expected values based on our method is considered valid.
An application of the quantum wave impedance method for the analytical study of quantum-mechanical systems containing a set of delta-wells and delta-barriers is considered in this study.The algorithm for determining both transmission and reflection coefficients for a set of delta-barriers was proposed.Similarly, the scheme for analytically deriving the expressions for energy levels and wave functions of bound states for a set of delta-wells was developed.In both cases, the solution for the quantum wave impedance function in the region with a constant potential [1] and the matching conditions at the origins of deltafunctions were used.As an illustration of the application of the proposed approach, the quantum-mechanical systems with symmetric double-barrier potential and double-well potential were investigated.In the case of a single delta-well, the eigenenergy value as well as the expression for the eigenfunction were obtained directly using the non-linear differential equation for the quantum wave impedance function.The results demonstrate the effectiveness of the quantum wave impedance method in handling quantum-mechanical systems with deltalike potentials, offering a robust framework for further studies in this area.
We propose a method that employs functionals of the one-electron reduced density matrix (density matrix) to capture long-range effects of electron correlation. The complementary short-range regime is treated with density functionals. In an effort to find approximations for the long-range density-matrix functional, a modified power functional is applied to the homogeneous electron gas with Coulomb interactions replaced by their corresponding long-range counterparts. For the power $\ensuremath{\beta}=1/2$ and the range-separation parameter $\ensuremath{\omega}=1/{r}_{s}$, the functional reproduces the correlation and the kinetic correlation energies with a remarkable accuracy for intermediate and large values of rs{r}_{s}rs​. Analysis of the Euler equation corresponding to this functional reveals correct rs{r}_{s}rs​ expansion of the correlation energy in the limit of large rs{r}_{s}rs​. The first expansion coefficient is in very good agreement with that obtained from the modified Wigner-Seitz model.