Quantum Mechanics by MARC CAHAY and SUPERIVO BANDYOPADHYAY; SOLUTION PROBLEMS; from basic to real world application for material scientists, applied physicists and device engineers.

 

Over the last two decades, there has been a dramatic increase in the study of physical and biological systems at the nanoscale. In fact, this millenium has been referred to as the “nanomillenium.” The fields of nanoscience and nanoengineering have been fuelled by recent spectacular discoveries in mesoscopic physics, a new understanding

of DNA sequencing, the advent of the field of quantum computing, tremendous progress in molecular biology, and other related fields.

 All future devices for semiconductor chip applications are likely to be strongly affected by the laws of quantum mechanics, and an understanding of these laws and tenets must be added to the repertoire of a device engineer and scientist.

Another challenge is to understand the quantum mechanical laws that will govern device operation when the projected density of 1013 transistors per cm2,am anticipated by 2017, is finally reached. Density increase, however, comes with a cost: if energy dissipation does not scale down concomitantly with device dimensions there will be thermal runaway, resulting in chip meltdown. This doomsday scenario has been dubbed the “red brick wall” by the International Technology Roadmapfm for Semiconductors . The foremost challenge is to find alternatives to the current semiconductor technology that would lead to a drastic reduction in energy dissipation during device operation. Such a technology, if and when it emerges, willvery likely draw heavily on quantum mechanics as opposed to classical physics.

Alternatives based on semiconductor heterostructures employing AlGaAs/GaAs or other III–V or II–VI materials have been investigated for several decades and have led to myriad quantum mechanical devices and architectures exploiting the special properties of quantum wells, wire and dots.

Chapters

Chapter 1: General Properties of the Schr¨odinger Equation

 This chapter describes some general properties of the time-independent effective mass Schr¨odinger equation (EMSE), which governs the steady-state behavior of an electron in a solid with spatially varying potential profile. The solid may consist of one or more materials (e.g., a heterostructure or superlattice); hence the effective mass of the electron may vary in space. The EMSE is widely used in studying the electronic and optical properties of solids.

Chapter 2: Operators 

All quantum mechanical operators describing physical variables are Hermitian. This chapter derives several useful identities involving operators. This includes derivations of the shift operator, the Glauber identity, the Baker–Hausdorff formula, the hypervirial theorem, Ehrenfest’s theorem, and various quantum mechanical sum rules. The concept of unitary transformation is also introduced and illustrated through a calculation of the polarizability of the one-dimensional harmonic oscillator. Usage of the operator identities and theorems derived in this chapter is illustrated in other chapters. Some general definitions and properties of operators are reviewed in Appendix C, which the reader should consult before trying out the problems in this chapter.

Chapter 3: Bound States 

The problems in this chapter deal with one-dimensional bound state calculations, which can be performed analytically or via the numerical solution of a transcendental equation. These problems give some insight into more complicated three-dimensional bound state problems whose solutions typically require numerically intensive approaches.

Chapter 4: Heisenberg Principle

 This chapter starts with three different proofs of the generalized Heisenberg uncertainty relations followed by illustrations of their application to the study of some bound state and scattering problems, including diffraction from a slit in a screen and quantum mechanical tunneling through a potential barrier.

 

 Chapter 5: Current and Energy Flux Densities

  This set of problems introduces the current density operator, which is applied to the study of various tunneling problems, including the case of a general one-dimensional heterostructure under bias (i.e., subjected to an electric field), the tunneling of an electron through an absorbing one-dimensional delta scatterer and potential well, and the calculation of the dwell time above a quantum well (QW). The dwell time is the time that an electron traversing a QW potential, with energy above the well’s barrier, lingers within the well region. 

 Chapter 6: Density of States 

 This chapter introduces the important concept of density of states (DOS) going from bulk to quantum confined structures.

 Chapter 7: Transfer Matrix 

 In this chapter, the use of the transfer matrix approach to solving the time-independent Schr¨odinger equation is illustrated for simple examples such as tunneling through a one-dimensional delta scatterer and through a square potential barrier. 

 Chapter 8: Scattering Matrix 

 The concept of a scattering matrix to solve tunneling problems is first described, including their cascading rule. Explicit analytical expression of the scattering matrix through a one-dimensional delta scatterer, two delta scatterers in series separated by a distance L, a simple potential step, a square barrier, and a double barrier resonant tunneling diode are then derived.

 Chapter 9: Perturbation Theory 

 This chapter starts with a brief introduction to first-order time-independent perturbation theory and applies it to the study of an electro-optic modulator and calculation of band structure in a crystal. It then introduces Fermi’s Golden Rule, which is a well-known result of time-dependent perturbation theory, and applies it to calculate the scattering rate of electrons interacting with impurities in a solid.

Chapter 10: Variational Approach 

Another important approach to finding approximate solutions to the Schr¨odinger equation is based on the Rayleigh–Ritz variational principle. For a specific problem.

 Electron in a Magnetic Field Many important phenomena in condensed matter physics, such as the quantum Hall effect, require an understanding of the quantum mechanical behavior of electrons in a magnetic field.

 Chapter 11: Electron in a Magnetic Field 

 Many important phenomena in condensed matter physics, such as the quantum Hall effect, require an understanding of the quantum mechanical behavior of electrons in a magnetic field. In this chapter, we introduce the concept of a vector potential and gauge to incorporate magnetic fields in the Hamiltonian of an electron.

 Chapter 12: Electron in an Electromagnetic Field and Optical Properties of Nanostructures

 This chapter deals with the interaction between an electron and an electromagnetic field. We derive the electron–photon interaction Hamiltonian and apply it to calculate absorption coefficients.

Chapter 13: Time-Dependent Schr¨odinger Equation 

This chapter examines several properties of one-dimensional Gaussian wave packets, including a calculation of the spatio-temporal dependence of their probability current and energy flux densities and a proof that their average kinetic energy is a constant of motion. An algorithm to study the time evolution of wave packets based on the Crank Nicholson scheme is discussed for the cases of totally reflecting and absorbing boundary conditions at the ends of the simulation domain.

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