Applied Quantum Mechanics

Walter A Harrison

• Publisher: WSPC
• Year: 2000
• Print ISBN: 9789810243944
• eBook ISBN: 9789814365628
• Topic: Physical Sciences
• Subtopic: Quantum Theory

Quantum mechanics is widely recognized as the basic law which governs all of nature, including all materials and devices. It has always been essential to the understanding of material properties, and as devices become smaller it is also essential for studying their behavior. Nevertheless, only a small fraction of graduate engineers and materials scientists take a course giving a systematic presentation of the subject. The courses for physics students tend to focus on the fundamentals and formal background, rather than on application, and do not fill the need. This invaluable text has been designed to fill the very apparent gap.

The book covers those parts of quantum theory which may be necessary for a modern engineer. It focuses on the approximations and concepts which allow estimates of the entire range of properties of nuclei, atoms, molecules, and solids, as well as the behavior of lasers and other quantum-optic devices. It may well prove useful also to graduate students in physics, whose courses on quantum theory tend not to include any of these applications. The material has been the basis of a course taught to graduate engineering students for the past four years at Stanford University.

Topics Discussed: Foundations; Simple Systems; Hamiltonian Mechanics; Atoms and Nuclei; Molecules; Crystals; Transitions; Tunneling; Transition Rates; Statistical Mechanics; Transport; Noise; Energy Bands; Electron Dynamics in Solids; Vibrations in Solids; Creation and Annihilation Operators; Phonons; Photons and Lasers; Coherent States; Coulomb Effects; Cooperative Phenomena; Magnetism; Shake-off Excitations; Exercise Problems.

Contents:

• Foundations
• Simple Systems
• Hamiltonian Mechanics
• Atoms and Nuclei
• Molecules
• Crystals
• Transitions
• Tunneling
• Transition Rates
• Statistical Mechanics
• Transport
• Noise
• Energy Bands
• Electron Dynamics in Solids
• Vibrations in Solids
• Creation and Annihilation Operators
• Phonons
• Photons and Lasers
• Coherent States
• Coulomb Effects
• Cooperative Phenomena
• Magnetism
• Shake-off Excitations
• Exercise Problems

Readership: Graduate students in engineering, materials science, chemistry and physics.

VII. Many-Body Effects

The step which made electronic structure understandable was the one-electron approximation, which we introduced in Section 4.2. In looking at the state of one electron, the effects of other electrons were included in an average way by including an averaged potential from those electrons. This one-electron picture provided us with states in terms of which we could discuss transitions and tunneling and optical absorption and emission. They also proved the basis for statistical analysis when many particles were present and could be used to estimate total-energy changes when atoms were rearranged or moved. We turn finally to some cases in which this one-particle picture is inadequate and see how we can proceed to understand such systems.

Chapter 20. Coulomb Effects

The principal interaction between electrons is the Coulomb interaction, and it is the basis of most of the effects we shall discuss. When we discuss superconductivity in Chapter 23, the important interaction between electrons will arise indirectly through the phonons. In particle and nuclear physics the interactions come from fundamentally different sources as we saw in Section 17.4. There are many qualitatively different effects arising from the interaction between particles. If we understand the physical nature of any effect, we can ordinarily frame the problem in terms of that understanding, much as we took variational wavefunctions to correspond to bonding states in molecules, or propagating states in solids. Including many-body effects is not a straight-forward addition of another term to the one-particle Hamiltonian; it is an asking of new questions. We begin with a discussion of Coulomb shifts, which arise because the charge on an individual electron is not infinitesimal.

20.1 Coulomb Shifts

We made a one-electron approximation in constructing electronic states in atoms in Chapter 4. This was a seeking of approximate many-electron states in the form of a product wavefunction, or an antisymmetric combination of product wavefunctions, of the form Ψ₁(r₁)Ψ₂(r₂)…ΨN(r_N) for the N electrons present. This led to a one-electron eigenvalue equation with a potential based upon which states were occupied, and the solution of that equation gave a set of energy eigenstates ε_i, the lowest of which were occupied in the ground state, corresponding for example to a ls²2s²2p² configuration for carbon. We indicated that these eigenvalues were approximately equal to the removal energy of an electron from the corresponding state. It is also true that the energy required to transfer an electron in the atom to an excited state of the atom is given approximately by the difference in the eigenvalues corresponding to the states between which the electron is transferred. For example, changing from a ls²2s²2p² to a Is²2s2p³ configuration for carbon requires approximately ε_2p - ε_2s. The new configuration actually corresponds to a slightly different charge distribution and potential which should be used to obtain new eigenvalues, but we have neglected such small corrections, which are many-body effects.

However, if we were to remove a second electron from an atom, going from 1s²2s²2p to 1s²2s² for carbon, it is clear that much more energy would be required to remove that second electron than the ε_2p which was required for the first, It would be working against the extra -e²/r from the doubly-charged atom as it was removed. Similarly, adding an electron to a neutral carbon atom, going to a 1s²2s²2p³ configuration would not gain the energy ε_2p. An electron returning to the ionized atom to make it neutral gains ε_2p, but that coming to a neutral atom gains less by an energy equal to the Coulomb interaction U between two p-electrons. It is of order seven electron volts for silicon and the heavier elements but over eleven eV for carbon (estimates are given for all the elements in Harrison (1999), p. 9). This corresponds to e²/r with r of the order of 2Å as expected for charge distributions of atomic size. This is all in accord with the familiar fact that the electron affinity of an atom, the binding energy of the additional electron in a negatively charged atom, is much smaller than the ionization energy of the neutral atom. The difference is this Coulomb interaction U which is also approximately equal to the difference in the first and second ionization energies of the same atom (assuming both removals are from the same state, e. g., a 2p-state).

One might have thought that this Coulomb effect would spoil the prediction of cohesive energy of an alkali halide which we made in Section 6.3. We took the energy gained in forming the solid as the energy gained in adding an electron to the halogen atom, minus the energy required to remove it from the alkali atom. Here we would say that an energy U should be added to the free-atom term value we used for the halogen. That is true, but the energy of that added electron is also lowered by the presence of the six positive alkali ions surrounding it, raised by the twelve nearest halogen ions, etc. The sum over neighbors is called the Madelung energy, equal to -l.8e²d (see, for example, Harrison (1999) 326ff), and approximately cancels the Coulomb U. The cancellation is no accident. The atoms in the ionic crystal select a spacing such that the transfer of electrons between states on different atoms does not greatly change their proximity to the nuclei. Such cancellations have made many of the simplified one-electron estimates meaningful in spite of real Coulomb shifts.

One might also have thought that such Coulomb shifts did not apply to the transfer of an electron from the valence band in a semiconductor to the conduction band in the semiconductor since we think of both states as spread throughout the entire crystal. This would be misguided since the crystal is in fact made of atoms and an atomic description is also meaningful for the crystal. Thus we may think of the transfer of an electron to the conduction band as a transfer from a bonding state (for which the energy eigenvalue applies) to an antibonding state in a site where the bond levels are both occupied. Thus we might expect the eigenvalues - the results of a band calculation - to underestimate the gap by an energy of the order of the U for the constituent atoms. That is true, but we may also see that this enhancement of the gap is reduced by a factor of 1/ε, with ε the dielectric constant equal to 12 for silicon,

This is not because the dielectric medium intervenes between the interacting electrons, but because an extra electron in a bond polarizes the surrounding medium so that the potential is +e²/(εr) and reduced by a factor of 1/ε at the surface of the atom or bond. This enhancement of the gap, of order 7 eV/12 ≈ 0.5 eV for silicon, relative to band calculations, is seen experimentally. It can be calculated more completely by the mathematical methods of many-body theory, as by Hybertsen and Louie (1985), but it is given rather well by Eq. (20.1) for all semiconductors and insulators (Harrison (1999), 207ff).

Adolph, Gavrilenko, Tenelsen, Bechstedt and Del Sol (1996) have made a rather complete study of the effect of this enhancement on various properties. It is found that the enhancements tend to be rather independent of wavenumber in the bands, as suggested by the Eq. (20.1), so that the correction is approxi...

Table of contents

• Cover Page
• Title Page
• Copyright Page
• Preface
• Contents
• I. The Basic Approach
• II. Electronic Structure
• III. Time Dependence
• IV. Statistical Physics
• V. Electrons and Phonons
• VI. Quantum Optics
• VII. Many-Body Effects
• Epilogue
• Exercises
• References
• Subject Index
• Back Cover