# The discretized Schrödinger equation and simple models for semiconductor quantum wells

@article{Boykin2004TheDS, title={The discretized Schr{\"o}dinger equation and simple models for semiconductor quantum wells}, author={Timothy B. Boykin and Gerhard Klimeck}, journal={European Journal of Physics}, year={2004}, volume={25}, pages={503-514} }

The discretized Schrodinger equation is one of the most commonly employed methods for solving one-dimensional quantum mechanics problems on the computer, yet many of its characteristics remain poorly understood. The differences with the continuous Schrodinger equation are generally viewed as shortcomings of the discrete model and are typically described in purely mathematical terms. This is unfortunate since the discretized equation is more productively viewed from the perspective of solid…

## 54 Citations

### The discretized Schrödinger equation for the finite square well and its relationship to solid-state physics

- Physics
- 2005

The discretized Schrödinger equation is most often used to solve one-dimensional quantum mechanics problems numerically. While it has been recognized for some time that this equation is equivalent to…

### The discretized Schr ̈ odinger equation for the finite square well and its relationship to solid-state physics

- Physics
- 2013

The discretized Schrödinger equation is most often used to solve onedimensional quantum mechanics problems numerically. While it has been recognized for some time that this equation is equivalent to…

### Bound states of moving potential wells in discrete wave mechanics

- Physics
- 2017

Discrete wave mechanics describes the evolution of classical or matter waves on a lattice, which is governed by a discretized version of the Schrödinger equation. While for a vanishing lattice…

### Non-Hermiticities even in quantum systems that are closed

- Physics
- 2018

Rarely noted paradoxes in applications of fundamental quantum relations are pointed out, with their resolution leading to emergent non-Hermitian behaviors due to boundary terms – even for closed…

### Current density and continuity in discretized models

- Physics
- 2010

Discrete approaches have long been used in numerical modelling of physical systems in both research and teaching. Discrete versions of the Schrödinger equation employing either one or several basis…

### The Discretized Momentum Operator

- Computer ScienceThe Physics Educator
- 2019

This paper illustrates by using some examples from undergraduate-level one-dimensional quantum mechanics to show how to handle derivatives in a discrete model.

### Quantum dot–ring nanostructure — A comparison of different approaches

- Physics
- 2016

It has been shown recently that a nanostructure composed of a quantum dot (QD) surrounded by a quantum ring (QR) possesses a set of very unique characteristics that make it a good candidate for…

### A simple analytical model for electronic conductance in a one dimensional atomic chain across a defect

- Physics
- 2011

An analytical model is presented for the electronic conductance in a one dimensional atomic chain across an isolated defect. The model system consists of two semi infinite lead atomic chains with the…

## References

SHOWING 1-10 OF 10 REFERENCES

### Tight-binding-like expressions for the continuous-space electromagnetic coupling Hamiltonian

- Physics
- 2001

In quantum mechanics texts one sometimes encounters the unqualified (and generally untrue) assertion that the solution of the Schrodinger equation for a charged particle in the presence of an…

### Single and multiband modeling of quantum electron transport through layered semiconductor devices

- Physics
- 1997

Non-equilibrium Green function theory is formulated to meet the three main challenges of high bias quantum device modeling: self-consistent charging, incoherent and inelastic scattering, and band…

### Generalized eigenproblem method for surface and interface states: The complex bands of GaAs and AlAs.

- Computer SciencePhysical review. B, Condensed matter
- 1996

A method is developed which easily handles those parameter sets at, or parameter sets for, which other approaches fail and is implemented in the second-near neighbor s* model to find the complex bands of GaAs and AlAs.

### Quantitative simulation of a resonant tunneling diode

- Physics
- 1997

Quantitative simulation of an InGaAs/InAlAs resonant tunneling diode is obtained by relaxing three of the most widely employed assumptions in the simulation of quantum devices. These are the single…

### Tight-binding model for GaAs/AlAs resonant-tunneling diodes.

- Physics, MedicinePhysical review. B, Condensed matter
- 1991

On utilise des matrices de transfert pour realiser les calculs et on presente une methode amelioree qui permet de transferer sur des dimensions superieures (>1000 A). Ainsi, on peut inclure les…

### Table of Integrals, Series, and Products

- Mathematics
- 1943

Introduction. Elementary Functions. Indefinite Integrals of Elementary Functions. Definite Integrals of Elementary Functions. Indefinite Integrals of Special Functions. Definite Integrals of Special…

### Tunneling calculations for systems with singular coupling matrices: Results for a simple model.

- PhysicsPhysical review. B, Condensed matter
- 1996

### Valley splitting in strained silicon quantum wells

- Physics
- 2003

A theory based on localized-orbital approaches is developed to describe the valley splitting observed in silicon quantum wells. The theory is appropriate in the limit of low electron density and…

### Quantum transport Heterostructures and Quantum Devices (VLSI Electronics: Microstructure Science vol 24) ed

- W R Frensley and N G Einspruch
- 1994