Applied Quantum Mechanics

The following sections are included:

• Preface

• Contents

Our goal in this treatment of quantum theory is to provide those aspects of the theory which are most needed for a modem understanding of the world of our experience. Central to this are the many approximations which have turned out to be successful for estimating the properties of matter and predicting the events which occur on a microscopic scale. In that sense the goal is entirely pragmatic, but any study of quantum theory cannot help but raise deep philosophical questions. Perhaps the only real understanding of the theory is through a familiarity with how it works out for many different problems. It seems unlikely that understanding comes from thinking about some complicated extreme case, such as Schroedinger's cat. The approach we take here bypasses much of the question by starting with a single premise of wave-particle duality, from which follows all of quantum theory and its interpretation. Any remaining problems are made moot by the quantum-mechanical view that questions are meaningful only if there is some conceivable experimental way to test the answers.

In keeping with this pragmatic approach we will not emphasize mathematical derivations, nor carry out the detailed analysis of harmonic-oscillator states, hydrogen wavefunctions, or angular momentum, which are part of almost all texts in quantum theory. We treat the simplest case in detail, state the general results, and see that they are the plausible generalizations. We leave the more detailed analysis to other texts. The one we used for the course at Stanford was Kroemer's Quantum Mechanics for Engineers and Materials Scientists, which served very well.

The following sections are included:

• The Premise

• Schroedinger's Equation

• Light Waves

• New Meaning for Potentials

• Measurement

• Eigenstates

• Boundary Conditions

• Sound Waves

The following sections are included:

• Free Electrons in One Dimension

• Free Electrons in Three Dimensions

• Quantum Slabs, Wires, and Dots

• Circularly and Spherically-Symmetric Systems

• The Harmonic Oscillator

The following sections are included:

• The Lagrangian

• Hamilton's Equations

• Including the Vector Potential

Perhaps the greatest accomplishment of quantum theory was providing an understanding of atoms, molecules, and solids in terms of the electrons, which had only been discovered in 1897. Classical physics was not even close to describing their behavior, whereas quantum theory did so quantitatively. It provided an essentially exact description of the hydrogen atom, which includes only a single electron, and which we discuss first. Exact treatment when more than one electron is present is impossible, but very quickly a one-electron approximation was introduced, which was extraordinarily successful, and which provides the basis of our modern understanding of the electronic structure and properties not only of atoms, but of molecules and solids.

The following sections are included:

• The Hydrogen Atom

• Many-Electron Atoms

• Pseudopotentials

• Nuclear Structure

The following sections are included:

• The Li₂ Molecule

• The Variational Method

• Molecular Orbitals

• Perturbation Theory

• N₂, CO, and CO₂

The following sections are included:

• The Linear Chain

• Free-Electron Bands and Tight-Binding Parameters

• Metallic, Ionic, and Covalent Solids

Electronic structure is primarily concerned with the lowest-energy state of a system, the ground state. This encompasses many properties, but often we are interested in higher-energy states, and transitions between different states. Up the this point it might appear that the goal of quantum mechanics was to find solutions, or approximate solutions, to the Schroedinger Equation. For the rest of the text we move on to other problems. The parallel in classical mechanics would be first addressing the calculation of normal modes of vibration, and then moving on to trajectories, collisions, and the myriad of other problems an engineer or scientist must deal with. In quantum mechanics the basic premise is still the wave-particle duality, with h' providing the relation between the two, but we now seek the answer to different questions.

The following sections are included:

• A Pair of Coupled States

• Fermi's Golden Rule

• Scattering in One and Three Dimensions

The following sections are included:

• Transmission in a 1-D Chain

• More General Barriers

• Tunneling Systems

• Tunneling Resonance

The following sections are included:

• Second-Order Coupling

• Carrier Emission and Capture

• Time-Dependent Perturbations

• Optical Transitions

• Beta-Ray Emission from Nuclei

To a large extent in our analysis we have been able to treat individual particles or waves, based upon a one-particle or one-electron approximation introduced already in Section 2.2. Occasionally we have introduced collections of particles, as for quantized conductance in Section 2.3. It is appropriate to take time to organize the subject a little more completely, discussing first statistical distributions in equilibrium, then transport theory when systems are out of equilibrium, and finally some aspects of the theory of noise. These are separate subjects from quantum mechanics, but are absolutely necessary if one is to make quantum-mechanical studies of systems involving many particles. They are not always incorporated in undergraduate programs, so it may be necessary to include them in a course on quantum theory. They also lead to quite different results in the context of quantum mechanics. We proceed as we have in other discussions by treating the simplest cases carefully, and seeing how they generalize to more intricate contexts.

The following sections are included:

• Distribution Functions

• Phonon and Photon Statistics

• Bosons

• Symmetry Under Interchange

• Fermions

The following sections are included:

• Time-Dependent Distributions

• The Boltzmann Equation

• Conductivity, etc.

The following sections are included:

• Classical Noise

• Quantum Noise and van-der-Waals Interaction

• Shot Noise

• Other Sources

We have made applications of quantum mechanics to solids throughout the text, and seen energy bands in simple systems. Crystalline systems are so important, and their understanding so heavily based upon quantum mechanics, that we should present the organization of the subject which is generally used. For many purposes the tight-binding basis is most flexible and easiest to use, as for the tunneling calculation in Chapter 8. However, the nearly-free-electron limit is also useful and provides a good introduction to Brillouin Zones, as well as formulating diffraction of waves in general, and we use it here. Our study of lattice vibrations in Chapter 15 will be closer to an energy-band formulation using tight-binding theory, but the Brillouin Zones are the same.

The following sections are included:

• The Empty-Core Pseudopotential

• A Band Calculation

• Diffraction

• Scattering by Impurities

• Semiconductor Bands

The following sections are included:

• Dynamics of Packets

• Effective Masses and Donor States

• The Dynamics of Holes

The following sections are included:

• The Spectrum

• The Classical-Vibration Hamiltonian

• The Electron-Phonon Interaction

We discussed quantum transitions between electron states due to a classical light field in Section 9.4. Similarly we could calculate electron scattering by classical vibrations of a crystal lattice using the matrix elements in terms of the vibrational amplitudes which we derived in the preceding section. However, it is often important to treat both the electrons and the fields quantum-mechanically. When the fields, or vibration amplitudes, are very large the quantum effects are not so important but they become essential when the fields are small. The treatment of electromagnetic waves and vibrational modes quantum-mechanically is called field theory and we now need to introduce the operators of field theory, both for these waves and modes, and for electrons. This formulation is often called second quantization. Their application in quantum optics is perhaps the most important for us, and the focus for this section of the book. However, it will be best to introduce the operators for electrons first, and to make the first application to phonons, quantized lattice vibrations.

In principle we already have the basic results we need. We have treated the harmonic oscillator quantum-mechanically, and have indicated that all of the findings apply to vibrational modes and to optical excitations. However, in field theory we use only a part of what we developed, the fact that the momentum operator (h'/i)∂/∂x does not commute with the coordinate x (changing the order of the product of two "commuting operators" does not change the product). In this case, (h'/i)∂/∂x (xψ)= (h'/i)ψ + x(h'/i)∂ψ/∂x so interchanging (h'/i)∂/∂x and x leaves a remainder, h'/i. This turns out to be the feature essential to defining field-theoretical annihilation and creation operators for excitations and particles. This is closely related to the point we made in Section 1.1 that it is possible to develop quantum theory without the waves which we have used throughout. Since we can obtain all of the information about the eigenstates of a harmonic oscillator, and its coupling to external fields, from the corresponding commutation relation, we did not really need the waves. Heisenberg represented the observables by matrices, rather than operators on waves. They had the same commutation relations so they led to the same results. For most contemporary physicists and engineers, the procedure with waves is much more comfortable.

The following sections are included:

• Annihilation and Creation Operators for Electrons

• Stepping Operators

• Angular Momentum

The following sections are included:

• Annihilation and Creation Operators for Phonons

• Phonon Emission and Absorption

• Polaron Self-Energy

• Electron-Electron and Nucleon-Nucleon Interactions

The following sections are included:

• Photons and the Electron-Photon Interaction

• Excitation of Atoms

• The Three-Level Laser

• Interband Transitions

The following sections are included:

• Coherence in a Harmonic Oscillator

• A Driven Classical Oscillator

• A Driven Quantum Oscillator

• Coherent Light

• Electromagnetically-Induced Transparency

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.

The following sections are included:

• Coulomb Shifts

• Screening

The following sections are included:

• Localization and Symmetry Breaking

• The Hubbard Hamiltonian

• Peierls Distortions

• Superconductivity

The following sections are included:

• Free Electrons in a Magnetic Field

• Magnetism of Atoms

• Magnetic Susceptibility

• Ferromagnetism

• Spin-Orbit Coupling

The following sections are included:

• Adiabatic and Sudden Approximations

• Vibrational Excitations

• Electronic and Auger Processes

• Inelastic Processes

The following sections are included:

• Epilogue

• Exercises

• References

• Subject Index