Quantum Mechanics in a Nutshell - Rilegato

Mahan, Gerald D.

Rilegato

ISBN 10: 0691137137 ISBN 13: 9780691137131

Casa editrice: Princeton Univ Pr, 2009

Riassunto

Covering the fundamentals as well as many special topics of current interest, this is the most concise, up-to-date, and accessible graduate-level textbook on quantum mechanics available. Written by Gerald Mahan, a distinguished research physicist and author of an acclaimed textbook on many-particle physics, Quantum Mechanics in a Nutshell is the distillation of many years' teaching experience.


Emphasizing the use of quantum mechanics to describe actual quantum systems such as atoms and solids, and rich with interesting applications, the book proceeds from solving for the properties of a single particle in potential; to solving for two particles (the helium atom); to addressing many-particle systems. Applications include electron gas, magnetism, and Bose-Einstein Condensation; examples are carefully chosen and worked; and each chapter has numerous homework problems, many of them original.



Quantum Mechanics in a Nutshell expertly addresses traditional and modern topics, including perturbation theory, WKBJ, variational methods, angular momentum, the Dirac equation, many-particle wave functions, Casimir Force, and Bell's Theorem. And it treats many topics--such as the interactions between photons and electrons, scattering theory, and density functional theory--in exceptional depth.


A valuable addition to the teaching literature, Quantum Mechanics in a Nutshell is ideally suited for a two-semester course.

• The most concise, up-to-date, and accessible graduate textbook on the subject


• Contains the ideal amount of material for a two-semester course


• Focuses on the description of actual quantum systems, including a range of applications


• Covers traditional topics, as well as those at the frontiers of research


• Treats in unprecedented detail topics such as photon-electron interaction, scattering theory, and density functional theory


• Includes numerous homework problems at the end of each chapter

Su questo libro

Informazioni sull?autore

Gerald D. Mahan is Distinguished Professor of Physics at Pennsylvania State University. He is the author of two previous physics textbooks, Many-Particle Physics and Applied Mathematics.

Dalla quarta di copertina

"This is an excellent textbook, written in a very readable style, and it should be perfectly accessible to beginning and intermediate physics graduate students. Gerald Mahan, the author of an acclaimed textbook on many-particle theory, has taught quantum mechanics extensively, and his thorough knowledge and deep understanding of the material is evident in every chapter ofQuantum Mechanics in a Nutshell. Its examples are excellently worked out and it has many interesting homework problems."--Uwe C. Tauber, Virginia Tech

"This book compares well with other graduate textbooks on quantum mechanics, and I will seriously consider adopting it the next time I teach the subject. The choice of material is very good. Gerald Mahan has included both the usual standard topics and a large number of special topics, including some of current research interest. The book is rich in interesting applications, and each chapter has lots of well-chosen problems. If a student can master this book, he or she will have gained an excellent foundation in quantum mechanics."--David G. Stroud, Ohio State University

Dal risvolto di copertina interno

"This is an excellent textbook, written in a very readable style, and it should be perfectly accessible to beginning and intermediate physics graduate students. Gerald Mahan, the author of an acclaimed textbook on many-particle theory, has taught quantum mechanics extensively, and his thorough knowledge and deep understanding of the material is evident in every chapter ofQuantum Mechanics in a Nutshell. Its examples are excellently worked out and it has many interesting homework problems."--Uwe C. Tauber, Virginia Tech

"This book compares well with other graduate textbooks on quantum mechanics, and I will seriously consider adopting it the next time I teach the subject. The choice of material is very good. Gerald Mahan has included both the usual standard topics and a large number of special topics, including some of current research interest. The book is rich in interesting applications, and each chapter has lots of well-chosen problems. If a student can master this book, he or she will have gained an excellent foundation in quantum mechanics."--David G. Stroud, Ohio State University

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Quantum Mechanics in a Nutshell

By Gerald D. Mahan

PRINCETON UNIVERSITY PRESS

Copyright © 2009 Princeton University Press
All right reserved.
ISBN: 978-0-691-13713-1

Contents

Preface......................................................xi1 Introduction...............................................12 One Dimension..............................................143 Approximate Methods........................................624 Spin and Angular Momentum..................................875 Two and Three Dimensions...................................1086 Matrix Methods and Perturbation Theory.....................1577 Time-Dependent Perturbations...............................2138 Electromagnetic Radiation..................................2449 Many-Particle Systems......................................28810 Scattering Theory.........................................32011 Relativistic Quantum Mechanics............................352Index........................................................397

Chapter One

1 Introduction

1.1 Introduction

Quantum mechanics is a mathematical description of how elementary particles move and interact in nature. It is based on the wave–particle dual description formulated by Bohr, Einstein, Heisenberg, Schrödinger, and others. The basic units of nature are indeed particles, but the description of their motion involves wave mechanics.

The important parameter in quantum mechanics is Planck's constant h = 6.626 x 10^-34 J s. It is common to divide it by 2p, and to put a slash through the symbol: h = 1.054 x 10^-34 J s. Classical physics treated electromagnetic radiation as waves. It is particles, called photons, whose quantum of energy is h? where ? is the classical angular frequency. For particles with a mass, such as an electron, the classical momentum mv = p = hk, where the wave vector k gives the wavelength k = 2p/? of the particle. Every particle is actually a wave, and some waves are actually particles.

The wave function ?(r, t) is the fundamental function for a single particle. The position of the particle at any time t is described by the function |?(r, t)|, which is the probability that the particle is at position r at time t. The probability is normalized to one by integrating over all positions:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)

In classical mechanics, it is assumed that one can know exactly where a particle is located. Classical mechanics takes this probability to be

|?(r, t)|² = d³(r - vt) (1:2)

The three-dimensional delta-function has an argument that includes the particle velocity v. In quantum mechanics, we never know precisely where to locate a particle. There is always an uncertainty in the position, the momentum, or both. This uncertainty can be summarized by the Heisenberg uncertainty principle:

?x?p_x = h (1.3)

where ?x is the uncertainty in position along one axis, ?p_x is the uncertainty in momentum along the same axis, and h" is Planck's constant h divided by 2p(h = h/2p), and has the value h = 1.05 x 10^-34 joules-second. Table 1.1 lists some fundamental constants.

1.2 Schrdinger's Equation

The exact form of the wave function ?(r, t) depends on the kind of particle, and its environment. Schrödinger's equation is the fundamental nonrelativistic equation used in quantum mechanics for describing microscopic particle motions. For a system of particles, Schrödinger's equation is written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.4)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.5)

The particles have positions r_i, momentum p_j, and spin s_j. They interact with a potential U(r_j, s_j) and with each other through the pair interaction V(r_i - r_j). The quantity H is the Hamiltonian, and the wave function for a system of many particles is ?(r₁, r₂, r₃, · , r_N; s₁, s₂, ..., s_N).

The specific forms for H depends on the particular problem. The relativistic form of the Hamiltonian is different than the nonrelativistic one. The relativistic Hamiltonian is discussed in chapter 11. The Hamiltonian can be used to treat a single particle, a collection of identical particles, or different kinds of elementary particles. Many-particle systems are solved in chapter 9.

No effort is made here to justify the correctness of Schrödinger's equation. It is assumed that the student has had an earlier course in the introduction to modern physics and quantum mechanics. A fundamental equation such as eqn. (1.4) cannot be derived from any postulate-free starting point. The only justification for its correctness is that its predictions agree with experiment. The object of this textbook is to teach the student how to solve Schrödinger's equation and to make these predictions. The students will be able to provide their own comparisons to experiment.

Schrödinger's equation for a single nonrelativistic particle of mass m, in the absence of a magnetic field, is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1:6)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.7)

The potential energy of the particle is V(r). This potential is usually independent of the spin variable for nonrelativistic motions in the absence of a magnetic field. Problems involving spin are treated in later chapters. When spin is unimportant in solving Schrödinger's equation, its presense is usually ignored in the notation: the wave function is written as ?(r).

In quantum mechanics, the particle momentum is replaced by the derivative operator:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.8)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.9)

Schrödinger's equation (1.4) is a partial differential equation in the variables (r, t). Solving Schrödinger's equation for a single particle is an exercise in ordinary differential equations. The solutions are not just mathematical exercises, since the initial and boundary conditions are always related to a physical problem.

Schrödinger's equation for a single particle is always an artificial problem. An equation with V (r) does not ever describe an actual physical situation. The potential V(r) must be provided by some other particles or by a collection of particles. According to Newton's third law, there is an equal and opposite force acting on these other particles, which are also reacting to this mutual force. The only situation in which one particle is by itself has V = 0, which is a dull example. Any potential must be provided by another particle, so Schrödinger's equation is always a many-particle problem. Nevertheless, there are two reasons why it is useful to solve the one-particle problem using classical potentials. The first is that one has to learn using simple problems as a stepping stone to solving the more realistic examples. Secondly, there are cases where the one-particle Schrödinger's equation is an accurate solution to a many-particle problem: i.e., it describes the relative motion of a two-particle system.

1.3 Eigenfunctions

In solving the time-dependent behavior, for the one-particle Schrödinger's equation (1.8), an important subsidiary problem is to find the eigenvalues e_n and eigenfunctions f_n(r) of the time-independent Hamiltonian:

Hf_n(r) = e_nf_n(r) (1.10)

There is a silly convention of treating "eigenfunction" and "eigenvalue" as single words, while "wave function" is two words. The name wave function is usually reserved for the time-dependent solution, while eigenfunction are the solutions of the time-independent equation. The wave function may be a single eigenfunction or a linear combination of eigenfunctions.

The eigenfunctions have important properties that are a direct result of their being solutions to an operator equation. Here we list some important results from linear algebra: The Hamiltonian operator is always Hermitian: H = H.

? Eigenvalues of Hermitian operators are always real.

? Eigenfunctions with different eigenvalues are orthogonal:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.11)

which is usually written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.12)

These two statements are not actually identical. The confusing case is where there are several different states with the same eigenvalue. They do not have to obey eqn. (1.12), but they can be constructed to obey this relation. We assume that is the case.

? The eigenfunctions form a complete set:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.13)

These properties are used often. Orthogonality is important since it implies that each eigenfunction f_n(r) is linearly independent of the others. Completeness is important, since any function f(r) can be uniquely and exactly expanded in terms of these eigenfunctions:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.14)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.15)

The function of most interest is the wave function ?(r, t). It can be expanded exactly as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.16)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.17)

The coefficients a_n depend on the initial conditions. They depend on neither r nor t. One example is when the system is in thermal equilibrium. If the particles obey MaxwellBoltzmann statistics, the coefficients are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.18)

and O is the grand canonical potential. Another example occurs during an experiment, where the system is prepared in a particular state, such as an atomic beam. Then the coefficients a_n are determined by the apparatus, and not by thermodynamics.

The wave function has a simple physical interpretation. The probability P(r, t) that the particle is at the position r at the time t is given by the square of the absolute magnitude of the wave function:

P(r, t) = |?(r, t)|² (1.19)

In quantum mechanics, there is no certainty regarding the position of the particle. Instead, the particle has a nonzero probability of being many different places in space. The likelihood of any of these possibilities is given by P(r, t). The particle is at only one place at a time.

The normalization of the wave function is determined by the interpretation of P(r, t) as a probability function. The particle must be someplace, so the total probability should be unity when integrated over all space:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.20)

The normalization also applies to the wave function. The eigenfunctions are also orthogonal, so that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.21)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.22)

The summation of the expansion coefficients |a_n|² must be unity. These coefficients

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.23)

are interpreted as the probability that the particle is in the eigenstate f_n(r).

The average value of any function f (r) is obtained by taking the integral of this function over all of space, weighted by the probability P(r, t). The bracket notation denotes the average of a quantity:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.24)

For example, the average potential energy <V> and the average position <r> are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.25)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.26)

A similar average can be taken for any other function of position.

There is no way to take an average of the particle velocity v = r. Since P(r, t) does not depend on r, there is no way to average this quantity. So does not exist. Instead, the average velocity is found by taking the time derivative of the average of r, such as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.27)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.28)

Now use Schrödinger's equation and its complex conjugate, to find

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.29)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.30)

which is used in d/dt:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.31)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.32)

The terms containing the potential energy V canceled. The equivalence of the last two expressions is found by just taking the derivative in the last expression. Each term in brackets generates a factor of ([??]?*) ·([??]?), which cancels.

An integration by parts gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.33)

If A is the quantity in brackets, then (A · [??])r = A, so the final expression is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.34)

The integrand is just the definition of the particle current:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.35)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.36)

The function j(r, t) is the particle current, which has the units of number of particles per second per unit area. If it is multiplied by the charge on the particle, it becomes the electrical current density J = ej, with units of amperes per area.

In the integral (1.34), integrate by parts on the second term in brackets. It then equals the first term:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.37)

The momentum operator is p = h]??]/i, and the integral is the expectation value of the momentum. In quantum mechanics, the average value of the velocity is the average value of the momentum divided by the particle mass.

The expectation value of any derivative operator should be evaluated as is done for the momentum: the operator is sandwiched between ?* and ? under the integral:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.38)

Other examples are the Hamiltonian and the z-component of angular momentum:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.39)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.40)

Once the wave function is known, it can be used to calculate the average value of many quantities that can be measured.

The last relationship to be proved in this section is the equation of continuity:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.41)

where ?(r, t) [equivalent to] P(r, t) is the particle density and j(r, t) is the particle current. The continuity equation is proved by taking the same steps to evaluate the velocity:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.42)

Use the above expressions (1.29, 1.30) for the time derivative of the wave functions. Again the potential energy terms cancel:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.43)

which proves the equation of continuity. Schrödinger's equation has been shown to be consistent with the equation of continuity as long as the density of particles is interpreted to be ?(r, t) = |?(r, t)|², and the current is eqn. (1.35).

1.4 Measurement

Making a measurement on a particle in a quantum system is a major disruption of the probability distribution. Suppose we have N identical particles, for example, atoms, in a large box. They will be in a variety of energy states. It is not possible to say which atom is in which state. If f/_n(r) is an exact eigenstate for an atom in this box, the wave function of an atom is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.44)

(Continues...)

Excerpted from Quantum Mechanics in a Nutshellby Gerald D. Mahan Copyright © 2009 by Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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Dettagli dell?edizione

• Editore: Princeton Univ Pr
• Data di pubblicazione: 2009
• Lingua: Inglese
• ISBN 10: 0691137137
• ISBN 13: 9780691137131
• Rilegatura: Copertina rigida
• Numero di pagine: 399
• Contatto del produttore: non disponibile