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##### From Quantum Transport: Atom to Transistor

Our objective for the next few chapters is to learn how the Hamiltonian matrix [ H] for a given device structure (see Fig. 1.6.5) is written down. We start in this chapter with (1) the hydrogen atom (Section 2.1) and how it led scientists to the Schr dinger equation, (2) a simple approach called the finite difference method (Section 2.2) that can be used to convert this differential equation into a matrix equation, and (3) a few numerical examples (Section 2.3) showing how energy levels are calculated by diagonalizing the resulting Hamiltonian matrix.

## 2.1 Hydrogen Atom

Early in the twentieth century scientists were trying to build a model for atoms which were known to consist of negative particles called electrons surrounding a positive nucleus. A simple model pictures the electron (of mass m and charge ? q) as orbiting the nucleus (with charge Zq) at a radius r (Fig. 2.1.1) kept in place by electrostatic attraction, in much the same way that gravitational attraction keeps the planets in orbit around the Sun.

 (2.1.1)

Fig. 2.1.1: Stationary orbits of an electron around a nucleus can be obtained by requiring their circumferences to be integer multiples of the de Broglie wavelength.

A faster electron describes an orbit with a smaller radius. The total energy of the electron is related to the radius of its orbit by the relation

 (2.1.2)

However, it was soon realized that this simple viewpoint was inadequate since, according to classical electrodynamics, an orbiting electron...

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##### Topics of Interest

As we move from the hydrogen atom (one electron only) to multi-electron atoms, we are immediately faced with the issue of electron-electron interactions, which is at the heart of almost all the...

10.1 Why Does an Atom Emit Light? We started this book by noting that the first great success of the Schr dinger equation was to explain the observed optical spectrum of the hydrogen atom. It was...

The hydrogen atom is the Rosetta stone of the early twentieth century atomic physics. The attempt to decipher its structure and properties led to the development of quantum mechanics and the...

6.1 QUANTUM KINETIC EQUATION FOR THE DENSITY MATRIX In a semiclassic approximation in which the quantum fluctuations of the radiation field are ignored, the resonance interaction of the atom (or...

An electron in a hydrogenic atom or ion can occupy any of the n ?sjm j eigen states of the Hamiltonian of the atom or ion. In ions or atoms with more than one electron, the solutions of the time...

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