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    Introduction to Quantum Mechanics with Applications to Chemistry
    Introduction to Quantum Mechanics with Applications to Chemistry
    Introduction to Quantum Mechanics with Applications to Chemistry
    Ebook740 pages8 hours

    Introduction to Quantum Mechanics with Applications to Chemistry

    By Linus Pauling and E. Bright Wilson

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    When this classic text was first published in 1935, it fulfilled the goal of its authors "to produce a textbook of practical quantum mechanics for the chemist, the experimental physicist, and the beginning student of theoretical physics." Although many who are teachers today once worked with the book as students, the text is still as valuable for the same undergraduate audience.
    Two-time Nobel Prize winner Linus Pauling, Research Professor at the Linus Pauling Institute of Science and Medicine, Palo Alto, California, and E. Bright Wilson, Jr., Professor Emeritus of Chemistry at Harvard University, provide a readily understandable study of "wave mechanics," discussing the Schrodinger wave equation and the problems which can be solved with it. Extensive knowledge of mathematics is not required, although the student must have a grasp of elementary mathematics through the calculus. Pauling and Wilson begin with a survey of classical mechanics, including Newton's equations of motion in the Lagrangian form, and then move on to the "old" quantum theory, developed through the work of Planck, Einstein and Bohr. This analysis leads to the heart of the book ― an explanation of quantum mechanics which, as Schrodinger formulated it, "involves the renunciation of the hope of describing in exact detail the behavior of a system." Physics had created a new realm in which classical, Newtonian certainties were replaced by probabilities ― a change which Heisenberg's uncertainty principle (described in this book) subsequently reinforced.
    With clarity and precision, the authors guide the student from topic to topic, covering such subjects as the wave functions for the hydrogen atom, perturbation theory, the Pauli exclusion principle, the structure of simple and complex molecules, Van der Waals forces, and systems in thermodynamic equilibrium. To insure that the student can follow the mathematical derivations, Pauling and Wilson avoid the "temptation to condense the various discussions into shorter and perhaps more elegant forms" appropriate for a more advanced audience. Introduction to Quantum Mechanics is a perfect vehicle for demonstrating the practical application of quantum mechanics to a broad spectrum of chemical and physical problems.

    LanguageEnglish
    PublisherDover Publications
    Release dateJun 8, 2012
    ISBN9780486134932
    Introduction to Quantum Mechanics with Applications to Chemistry

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    Introduction to Quantum Mechanics with Applications to Chemistry - Linus Pauling

    INTRODUCTION TO QUANTUM MECHANICS

    With Applications to Chemistry

    BY

    LINUS PAULING

    Research Professor, Linus Pauling

    Institute of Science and Medicine,

    Palo Alto, California

    AND

    E. BRIGHT WILSON, JR.

    Theodore William Richards

    Professor of Chemistry,

    Emeritus, Harvard University

    DOVER PUBLICATIONS, INC.

    NEW YORK

    Copyright

    Copyright © 1935, 1963 by Linus Pauling and E. Bright Wilson, Jr.

    All rights reserved.

    Bibliographical Note

    This Dover edition, first published in 1985, is an unabridged and unaltered republication of the work first published by The McGraw-Hill Book Co., New York, in 1935.

    Library of Congress Cataloging-in-Publication Data

    Pauling, Linus, 1901–

    Introduction to quantum mechanics.

    Reprint. Originally published: New York; London: McGraw-Hill, 1935.

    Bibliography: p.

    Includes index.

    ISBN-13: 978-0-486-64871-2

    ISBN-10: 0-486-64871-0

    1. Quantum theory. 2. Wave mechanics. 3. Chemistry, Physical and theoretical. I. Wilson, E. Bright (Edgar Bright), 1908– II. Title.

    [QCI74.12.P391985]530.1′284-25919

    Manufactured in the United States by Courier Corporation

    64871013

    www.doverpublications.com

    PREFACE

    In writing this book we have attempted to produce a textbook of practical quantum mechanics for the chemist, the experimental physicist, and the beginning student of theoretical physics. The book is not intended to provide a critical discussion of quantum mechanics, nor even to present a thorough survey of the subject. We hope that it does give a lucid and easily understandable introduction to a limited portion of quantum-mechanical theory; namely, that portion usually suggested by the name wave mechanics, consisting of the discussion of the Schrödinger wave equation and the problems which can be treated by means of it. The effort has been made to provide for the reader a means of equipping himself with a practical grasp of this subject, so that he can apply quantum mechanics to most of the chemical and physical problems which may confront him.

    The book is particularly designed for study by men without extensive previous experience with advanced mathematics, such as chemists interested in the subject because of its chemical applications. We have assumed on the part of the reader, in addition to elementary mathematics through the calculus, only some knowledge of complex quantities, ordinary differential equations, and the technique of partial differentiation. It may be desirable that a book written for the reader not adept at mathematics be richer in equations than one intended for the mathematician; for the mathematician can follow a sketchy derivation with ease, whereas if the less adept reader is to be led safely through the usually straightforward but sometimes rather complicated derivations of quantum mechanics a firm guiding hand must be kept on him. Quantum mechanics is essentially mathematical in character, and an understanding of the subject without a thorough knowledge of the mathematical methods involved and the results of their application cannot be obtained. The student not thoroughly trained in the theory of partial differential equations and orthogonal functions must learn something of these subjects as he studies quantum mechanics. In order that he may do so, and that he may follow the discussions given without danger of being deflected from the course of the argument by inability to carry through some minor step, we have avoided the temptation to condense the various discussions into shorter and perhaps more elegant forms.

    After introductory chapters on classical mechanics and the old quantum theory, we have introduced the Schrödinger wave equation and its physical interpretation on a postulatory basis, and have then given in great detail the solution of the wave equation for important systems (harmonic oscillator, hydrogen atom) and the discussion of the wave functions and their properties, omitting none of the mathematical steps except the most elementary. A similarly detailed treatment has been given in the discussion of perturbation theory, the variation method, the structure of simple molecules, and, in general, in every important section of the book.

    In order to limit the size of the book, we have omitted from discussion such advanced topics as transformation theory and general quantum mechanics (aside from brief mention in the last chapter), the Dirac theory of the electron, quantization of the electromagnetic field, etc. We have also omitted several subjects which are ordinarily considered as part of elementary quantum mechanics, but which are of minor importance to the chemist, such as the Zeeman effect and magnetic interactions in general, the dispersion of light and allied phenomena, and most of the theory of aperiodic processes.

    The authors are severally indebted to Professor A. Sommerfeld and Professors E. U. Condon and H. P. Robertson for their own introduction to quantum mechanics. The constant advice of Professor R. C. Tolman is gratefully acknowledged, as well as the aid of Professor P. M. Morse, Dr. L. E. Sutton, Dr. G. W. Wheland, Dr. L. 0. Brockway, Dr. J. Sherman, Dr. S. Weinbaum, Mrs. Emily Buckingham Wilson, and Mrs. Ava Helen Pauling.

    LINUS PAULING.

    E. BRIGHT WILSON, JR.

    PASADENA, CALIF.,

    CAMBRIDGE, MASS.,

    July, 1935.

    CONTENTS

    PREFACE

    CHAPTER I

    SURVEY OF CLASSICAL MECHANICS

    1.Newton's Equations of Motion in the Lagrangian Form

    1a.The Three-dimensional Isotropic Harmonic Oscillator

    1b.Generalized Coordinates

    1c.The Invariance of the Equations of Motion in the Lagrangian Form

    1d.An Example: The Isotropic Harmonic Oscillator in Polar Coordinates

    1e.The Conservation of Angular Momentum

    2.The Equations of Motion in the Hamiltonian Form

    2a.Generalized Momenta

    2b.The Hamiltonian Function and Equations

    2c.The Hamiltonian Function and the Energy

    2d.A General Example

    3.The Emission and Absorption of Radiation

    4.Summary of Chapter I

    CHAPTER II

    THE OLD QUANTUM THEORY

    5.The Origin of the Old Quantum Theory

    5a.The Postulates of Bohr

    5b.The Wilson-Sommerfeld Rules of Quantization

    5c.Selection Rules. The Correspondence Principle

    6.The Quantization of Simple Systems

    6a.The Harmonic Oscillator. Degenerate States

    6b.The Rigid Rotator

    6c.The Oscillating and Rotating Diatomic Molecule

    6d.The Particle in a Box

    6e.Diffraction by a Crystal Lattice

    7.The Hydrogen Atom

    7a.Solution of the Equations of Motion

    7b.Application of the Quantum Rules. The Energy Levels

    7c.Description of the Orbits

    7d.Spatial Quantization

    8.The Decline of the Old Quantum Theory

    CHAPTER III

    THE SCHRÖDINGER WAVE EQUATION WITH THE HARMONIC OSCILLATOR AS AN EXAMPLE

    9.The Schrödinger Wave Equation

    9a.The Wave Equation Including the Time

    9b.The Amplitude Equation

    9c.Wave Functions. Discrete and Continuous Sets of Characteristic Energy Values

    9d.The Complex Conjugate Wave Function ψ*(x, t)

    10.The Physical Interpretation of the Wave Functions

    10a.ψ*(x, t)ψ(x, t) as a Probability Distribution Function

    10b.Stationary States

    10c.Further Physical Interpretation. Average Values of Dynamical Quantities

    11.The Harmonic Oscillator in Wave Mechanics

    11a.Solution of the Wave Equation

    11b.The Wave Functions for the Harmonic Oscillator and their Physical Interpretation

    11c.Mathematical Properties of the Harmonic Oscillator Wave Functions

    CHAPTER IV

    THE WAVE EQUATION FOR A SYSTEM OF POINT PARTICLES IN THREE DIMENSIONS

    12.The Wave Equation for a System of Point Particles

    12a.The Wave Equation Including the Time

    12b.The Amplitude Equation

    12c.The Complex Conjugate Wave Function ψ*(x1 ….zn, t)

    12d.The Physical Interpretation of the Wave Functions

    13.The Free Particle

    14.The Particle in a Box

    15.The Three-dimensional Harmonic Oscillator in Cartesian Coordinates

    16.Curvilinear Coordinates

    17.The Three-dimensional Harmonic Oscillator in Cylindrical Coordinates

    CHAPTER V

    THE HYDROGEN ATOM

    18.The Solution of the Wave Equation by the Polynomial Method and the Determination of the Energy Levels

    18a.The Separation of the Wave Equation. The Translational Motion

    18b.The Solution of the φ Equation

    18c.The Solution of the ϑ Equation

    18d.The Solution of the r Equation

    18e.The Energy Levels

    19.Legendre Functions and Surface Harmonics

    19a.The Legendre Functions or Legendre Polynomials

    19b.The Associated Legendre Functions

    20.The Laguerre Polynomials and Associated Laguerre Functions

    20a.The Laguerre Polynomials

    20b.The Associated Laguerre Polynomials and Functions

    21.The Wave Functions for the Hydrogen Atom

    21a.Hydrogen-like Wave Functions

    21b.The Normal State of the Hydrogen Atom

    21c.Discussion of the Hydrogen-like Radial Wave Functions

    21d.Discussion of the Dependence of the Wave Functions on the Angles ϑ and φ

    CHAPTER VI

    PERTURBATION THEORY

    22.Expansions in Series of Orthogonal Functions

    23.First-order Perturbation Theory for a Non-degenerate Level

    23a.A Simple Example: The Perturbed Harmonic Oscillator

    23b.An Example: The Normal Helium Atom

    24.First-order Perturbation Theory for a Degenerate Level

    24a.An Example: Application of a Perturbation to a Hydrogen Atom

    25.Second-order Perturbation Theory

    25a.An Example: The Stark Effect of the Plane Rotator

    CHAPTER VII

    THE VARIATION METHOD AND OTHER APPROXIMATE METHODS

    26.The Variation Method

    26a.The Variational Integral and its Properties

    26b.An Example: The Normal State of the Helium Atom

    26c.Application of the Variation Method to Other States

    26d.Linear Variation Functions

    26e.A More General Variation Method

    27.Other Approximate Methods

    27a.A Generalized Perturbation Theory

    27b.The Wentzel-Kramers-Brillouin Method

    27c.Numerical Integration

    27d.Approximation by the Use of Difference Equations

    27e.An Approximate Second-order Perturbation Treatment

    CHAPTER VIII

    THE SPINNING ELECTRON AND THE PAULI EXCLUSION PRINCIPLE, WITH A DISCUSSION OF THE HELIUM ATOM

    28.The Spinning Electron

    29.The Helium Atom. The Pauli Exclusion Principle

    29a.The Configurations ls2s and ls2p

    29b.The Consideration of Electron Spin. The Pauli Exclusion Principle

    29c.The Accurate Treatment of the Normal Helium Atom

    29d.Excited States of the Helium Atom

    29e.The Polarizability of the Normal Helium Atom

    CHAPTER IX

    MANY-ELECTRON ATOMS

    30.Slater's Treatment of Complex Atoms

    30a.Exchange Degeneracy

    30b.Spatial Degeneracy

    30c.Factorization and Solution of the Secular Equation

    30d.Evaluation of Integrals

    30e.Empirical Evaluation of Integrals. Applications

    31.Variation Treatments for Simple Atoms

    31a.The Lithium Atom and Three-electron Ions

    31b.Variation Treatments of Other Atoms

    32.The Method of the Self-consistent Field

    32a.Principle of the Method

    32b.Relation of the Self-consistent Field Method to the Variation Principle

    32c.Results of the Self-consistent Field Method

    33.Other Methods for Many-electron Atoms

    33a.Semi-empirical Sets of Screening Constants

    33b.The Thomas-Fermi Statistical Atom

    CHAPTER X

    THE ROTATION AND VIBRATION OF MOLECULES

    34.The Separation of Electronic and Nuclear Motion

    35.The Rotation and Vibration of Diatomic Molecules

    35a.The Separation of Variables and Solution of the Angular Equations

    35b.The Nature of the Electronic Energy Function

    35c.A Simple Potential Function for Diatomic Molecules

    35d.A More Accurate Treatment. The Morse Function

    36.The Rotation of Polyatomic Molecules

    36a.The Rotation of Symmetrical-top Molecules

    36b.The Rotation of Unsymmetrical-top Molecules

    37.The Vibration of Polyatomic Molecules

    37a.Normal Coordinates in Classical Mechanics

    37b.Normal Coordinates in Quantum Mechanics

    38.The Rotation of Molecules in Crystals

    CHAPTER XI

    PERTURBATION THEORY INVOLVING THE TIME, THE EMISSION AND ABSORPTION OF RADIATION, AND THE RESONANCE PHENOMENON

    39.The Treatment of a Time-dependent Perturbation by the Method of Variation of Constants

    39a.A Simple Example

    40.The Emission and Absorption of Radiation

    40a.The Einstein Transition Probabilities

    40b.The Calculation of the Einstein Transition Probabilities by Perturbation Theory

    40c.Selection Rules and Intensities for the Harmonic Oscillator

    40d.Selection Rules and Intensities for Surface-harmonic Wave Functions

    40e.Selection Rules and Intensities for the Diatomic Molecule. The Franck-Condon Principle

    40f.Selection Rules and Intensities for the Hydrogen Atom

    40g.Even and Odd Electronic States and their Selection Rules.

    41.The Resonance Phenomenon

    41a.Resonance in Classical Mechanics

    41b.Resonance in Quantum Mechanics

    41c.A Further Discussion of Resonance

    CHAPTER XII

    THE STRUCTURE OF SIMPLE MOLECULES

    42.The Hydrogen Molecule-ion

    42a.A Very Simple Discussion

    42b.Other Simple Variation Treatments

    42c.The Separation and Solution of the Wave Equation

    42d.Excited States of the Hydrogen Molecule-ion

    43.The Hydrogen Molecule

    43a.The Treatment of Heitler and London

    43b.Other Simple Variation Treatments

    43c.The Treatment of James and Coolidge

    43d.Comparison with Experiment

    43e.Excited States of the Hydrogen Molecule

    43f.Oscillation and Rotation of the Molecule. Ortho and Para Hydrogen

    44.The Helium Molecule-ion HeJ and the Interaction of Two Normal Helium Atoms

    44a.The Helium Molecule-ion He+2

    44b.The Interaction of Two Normal Helium Atoms

    45.The One-electron Bond, the Electron-pair Bond, and the Three-electron Bond

    CHAPTER XIII

    THE STRUCTURE OF COMPLEX MOLECULES

    46.Slater's Treatment of Complex Molecules

    46a.Approximate Wave Functions for the System of Three Hydrogen Atoms

    46b.Factoring the Secular Equation

    46c.Reduction of Integrals

    46d.Limiting Cases for the System of Three Hydrogen Atoms

    46e.Generalization of the Method of Valence-bond Wave Functions

    46f.Resonance among Two or More Valence-bond Structures

    46g.The Meaning of Chemical Valence Formulas

    46h.The Method of Molecular Orbitals

    CHAPTER XIV

    MISCELLANEOUS APPLICATIONS OF QUANTUM MECHANICS

    47.Van der Waals Forces

    47a.Van der Waals Forces for Hydrogen Atoms

    47b.Van der Waals Forces for Helium

    47c.The Estimation of Van der Waals Forces from Molecular Polarizabilities

    48.The Symmetry Properties of Molecular Wave Functions

    48a.Even and Odd Electronic Wave Functions. Selection Rules

    48b.The Nuclear Symmetry Character of the Electronic Wave Function

    48c.Summary of Results Regarding Symmetrical Diatomic Molecules

    49.Statistical Quantum Mechanics. Systems in Thermodynamic Equilibrium

    49a.The Fundamental Theorem of Statistical Quantum Mechanics

    49b.A Simple Application

    49c.The Boltzmann Distribution Law

    49d.Fermi-Dirac and Bose-Einstein Statistics

    49e.The Rotational and Vibrational Energy of Molecules

    49f.The Dielectric Constant of a Diatomic Dipole Gas

    50.The Energy of Activation of Chemical Reactions

    CHAPTER XV

    GENERAL THEORY OF QUANTUM MECHANICS

    51.Matrix Mechanics

    51a.Matrices and their Relation to Wave Functions. The Rules of Matrix Algebra

    51b.Diagonal Matrices and Their Physical Interpretation

    52.The Properties of Angular Momentum

    53.The Uncertainty Principle

    54.Transformation Theory

    APPENDICES

    I.Values of Physical Constants

    II.Proof that the Orbit of a Particle Moving in a Central Field Lies in a Plane

    III.Proof of Orthogonality of Wave Functions Corresponding to Different Energy Levels

    IV.Orthogonal Curvilinear Coordinate Systems

    V.The Evaluation of the Mutual Electrostatic Energy of Two Spherically Symmetrical Distributions of Electricity with Exponential Density Functions

    VI.Normalization of the Associated Legendre Functions

    VII.Normalization of the Associated Laguerre Functions

    VIII.The Greek Alphabet

    INDEX

    INTRODUCTION TO QUANTUM MECHANICS

    CHAPTER I

    SURVEY OF CLASSICAL MECHANICS

    The subject of quantum mechanics constitutes the most recent step in the very old search for the general laws governing the motion of matter. For a long time investigators confined their efforts to studying the dynamics of bodies of macroscopic dimensions, and while the science of mechanics remained in that stage it was properly considered a branch of physics. Since the development of atomic theory there has been a change of emphasis. It was recognized that the older laws are not correct when applied to atoms and electrons, without considerable modification. Moreover, the success which has been obtained in making the necessary modifications of the older laws has also had the result of depriving physics of sole claim upon them, since it is now realized that the combining power of atoms and, in fact, all the chemical properties of atoms and molecules are explicable in terms of the laws governing the motions of the electrons and nuclei composing them.

    Although it is the modern theory of quantum mechanics in which we are primarily interested because of its applications to chemical problems, it is desirable for us first to discuss briefly the background of classical mechanics from which it was developed. By so doing we not only follow to a certain extent the historical development, but we also introduce in a more familiar form many concepts which are retained in the later theory. We shall also treat certain problems in the first few chapters by the methods of the older theories in preparation for their later treatment by quantum mechanics. It is for this reason that the student is advised to consider the exercises of the first few chapters carefully and to retain for later reference the results which are secured.

    In the first chapter no attempt will be made to give any parts of classical dynamics but those which are useful in the treatment of atomic and molecular problems. With this restriction, we have felt justified in omitting discussion of the dynamics of rigid bodies, non-conservative systems, non-holonomic systems, systems involving impact, etc. Moreover, no use is made of Hamilton's principle or of the Hamilton-Jacobi partial differential equation. By thus limiting the subjects to be discussed, it is possible to give in a short chapter a thorough treatment of Newtonian systems of point particles.

    1. NEWTON'S EQUATIONS OF MOTION IN THE LAGRANGIAN FORM

    The earliest formulation of dynamical laws of wide application is that of Sir Isaac Newton. If we adopt the notation xi, yi, zi for the three Cartesian coordinates of the i′th particle with mass mi, Newton's equations for n point particles are

    where Xi, Yi, Zi are the three components of the force acting on the ith particle. There is a set of such equations for each particle. Dots refer to differentiation with respect to time, so that

    By introducing certain familiar definitions we change Equation 1–1 into a form which will be more useful later. We define as the kinetic energy T (for Cartesian coordinates) the quantity

    If we limit ourselves to a certain class of systems, called conservative systems, it is possible to define another quantity, the potential energy V, which is a function of the coordinates x1y1z1 … xnynzn of all the particles, such that the force components acting on each particle are equal to partial derivatives of the potential energy with respect to the coordinates of the particle (with negative sign); that is,

    It is possible to find a function V which will express in this manner forces of the types usually designated as mechanical, electrostatic, and gravitational. Since other types of forces (such as electro-magnetic) for which such a potential-energy function cannot be set up are not important in chemical applications, we shall not consider them in detail.

    With these definitions, Newton's equations become

    There are three such equations for every particle, as before. These results are definitely restricted to Cartesian coordinates; but by introducing a new function, the Lagrangian function L, defined for Newtonian systems as the difference of the kinetic and potential energy,

    we can throw the equations of motion into a form which we shall later prove to be valid in any system of coordinates (Sec. 1c). In Cartesian coordinates T is a function of the velocities only, and for the systems to which our treatment is restricted V is a function of the coordinates only; hence the equations of motion given in Equation 1–5 on introduction of the function L assume the form

    In the following paragraphs a simple dynamical system is discussed by the use of these equations.

    1a. The Three-dimensional Isotropic Harmonic Oscillator.—As an illustration of the use of the equations of motion in this form, we choose a system which has played a very important part in the development of quantum theory. This is the harmonic oscillator, a particle bound to an equilibrium position by a force which increases in magnitude linearly with its distance r from the point. In the three-dimensional isotropic harmonic oscillator this corresponds to a potential function ½kr², representing a force of magnitude kr acting in a negative direction; i.e., from the position of the particle to the origin. k is called the force constant or Hooke's-law constant. Using Cartesian coordinates we have

    whence

    Multiplication of the first member of Equation 1–9 by gives

    or

    which integrates directly to

    The constant of integration is conveniently expressed as ½kx²0.

    Hence

    or, on introducing the expression 4π²mv²0 in place of the force constant k,

    which on integration becomes

    or

    and similarly

    In these expressions x0, y0, z0, δx, δy, and δz are constants of integration, the values of which determine the motion in any given case. The quantity v0 is related to the constant of the restoring force by the equation

    so that the potential energy may be written as

    As shown by the equations for x, y, and z, v0 is the frequency of the motion. It is seen that the particle may be described as carrying out independent harmonic oscillations along the x, y, and z axes, with different amplitudes x0, y0, and z0 and different phase angles δx, δy, and δz, respectively.

    The energy of the system is the sum of the kinetic energy and the potential energy, and is thus equal to

    On evaluation, it is found to be independent of the time, with the value 2π²mv²0(x²0 + y²0 + z²0) determined by the amplitudes of oscillation.

    The one-dimensional harmonic oscillator, restricted to motion along the x axis in accordance with the potential function V = ½kx² = 2π²mv²0x², is seen to carry out harmonic oscillations along this axis as described by Equation 1–14. Its total energy is given by the expression 2π²mv²0x²0.

    1b. Generalized Coordinates.—Instead of Cartesian coordinates x1, y1, z1, …, xn, yn, zn, it is frequently more convenient to use some other set of coordinates to specify the configuration of the system. For example, the isotropic spatial harmonic oscillator already discussed might equally well be described using polar coordinates; again, the treatment of a system composed of two attracting particles in space, which will be considered later, would be very cumbersome if it were necessary to use rectangular coordinates.

    If we choose any set of 3n coordinates, which we shall always assume to be independent and at the same time sufficient in number to specify completely the positions of the particles of the system, then there will in general exist 3n equations, called the equations of transformation, relating the new coordinates qk to the set of Cartesian coordinates xi, yi, zi

    There is such a set of three equations for each particle i. The functions fi, gi, hi may be functions of any or all of the 3n new coordinates qk, so that these new variables do not necessarily split into sets which belong to particular particles. For example, in the case of two particles the six new coordinates may be the three Cartesian coordinates of the center of mass together with the polar coordinates of one particle referred to the other particle as origin.

    As is known from the theory of partial differentiation, it is possible to transform derivatives from one set of independent variables to another, an example of this process being

    This same equation can be put in the much more compact form

    This gives the relation between any Cartesian component of velocity and the time derivatives of the new coordinates. Similar relations, of course, hold for i and i for any particle. The quantities i, by analogy with i are called generalized velocities, even though they do not necessarily have the dimensions of length divided by time (for example, qi may be an angle).

    Since partial derivatives transform in just the same manner, we have

    Since Qi is given by an expression in terms of V and qi which is analogous to that for the force Xi in terms of V and xi, it is called a generalized force.

    In exactly similar fashion, we have

    1c. The Invariance of the Equations of Motion in the Lagrangian Form.—We are now in a position to show that when Newton's equations are written in the form given by Equation 1–7 they are valid for any choice of coordinate system. For this proof we shall apply a transformation of coordinates to Equations 1–5, using the methods of the previous section. Multiplication of Equation 1–5a by of 1–5b by , etc., gives

    with similar equations in y and z. Adding all of these together gives

    where the result of Equation 1–20 has been used. In order to reduce the first sum, we note the following identity, obtained by differentiating a product,

    From Equation 1–19b we obtain directly

    Furthermore, because the order of differentiation is immaterial, we see that

    By introducing Equations 1–26 and 1–25 in 1–24 and using the result in Equation 1–23, we get

    which, in view of the results of the last section, reduces to

    Finally, the introduction of the Lagrangian function L = T – V, with V a function of the coordinates only, gives the more compact form

    (It is important to note that L must be expressed as a function of the coordinates and their first time-derivatives.)

    Since the above derivation could be carried out for any value of j, there are 3n such equations, one for each coordinate qi. They are called the equations of motion in the Lagrangian form and are of great importance. The method by which they were derived shows that they are independent of the coordinate system.

    We have so far rather limited the types of systems considered, but Lagrange's equations are much more general than we have indicated and by a proper choice of the function L nearly all dynamical problems can be treated with their use. These equations are therefore frequently chosen as the fundamental postulates of classical mechanics instead of Newton's laws.

    1d. An Example: The Isotropic Harmonic Oscillator in Polar Coordinates.—The example which we have treated in Section 1a can equally well be solved by the use of polar coordinates r, ϑ, and φ (Fig. 1–1). The equations of transformation corresponding to Equation 1–18 are

    With the use of these we find for the kinetic and potential energies of the isotropic harmonic oscillator the following expressions:

    and

    The equations of motion are

    In Appendix II it is shown that the motion takes place in a plane containing the origin. This conclusion enables us to simplify the problem by making a change of variables. Let us introduce new polar coordinates r, ϑ′, χ such that at the time t = 0 the plane determined by the vectors r and v, the position and velocity vectors of the particle at t = 0, is normal to the new z′ axis. This transformation is known in terms of the old set of coordinates if two parameters ϑ0 and φ0, determining the position of the axis z′ in terms of the old coordinates, are given (Fig. 1–2).

    FIG. 1–1.— The relation of polar coordinates r, ϑ, and φ to Cartesian axes.

    FIG. 1–2.—The rotation of axes.

    In terms of the new coordinates, the Lagrangian function L and the equations of motion have the same form as previously, because the first choice of axis direction was quite arbitrary. However, since the coordinates have been chosen so that the plane of the motion is the x′y′ plane, the angle ϑ′ is always equal to a constant, π/2. Inserting this value of ϑ′ in Equation 1–33 and writing it in terms of χ instead of φ we obtain

    which has the solution

    The r equation, Equation 1–35, becomes

    or, using Equation 1–37,

    an equation differing from the related one-dimensional Cartesian coordinate equation by the additional term – p²χ/mr³ which represents the centrifugal force.

    Multiplication by and integration with respect to the time gives

    so that .

    This can be again integrated, to give

    in which x = r², a = –p²χ/m², b is the constant of integration in Equation 1–39, and c = – 4π²v²0. This is a standard integral which yields the equation

    with A given by

    We have thus obtained the dependence of r on the time, and by integrating Equation 1–37 we could obtain χ as a function of the time, completing the solution. Elimination of the time between these two results would give the equation of the orbit, which is an ellipse with center at the origin. It is seen that the constant v0 again occurs as the frequency of the motion.

    1e. The Conservation of Angular Momentum.—The example worked out in the previous section illustrates an important principle of wide applicability, the principle of the conservation of angular momentum.

    Equation 1–37 shows that when is the angular velocity of the particle about a fixed axis z′ and r is the distance of the particle from the axis, the quantity pχ = mr² is a constant of the motion.¹ This quantity is called the angular momentum, of the particle about the axis z′.

    It is not necessary to choose an axis normal to the plane of the motion, as z′ in this example, in order to apply the theorem. Thus Equation 1–33, written for arbitrary direction z, is at once integrable to

    Here r sin ϑ is the distance of the particle from the axis z, so that the left side of this equation is the angular momentum about the axis z.² It is seen to be equal to a constant, pφ.

    FIG. 1–3.— Figure showing the relation between dχ, dϑ, and dφ.

    In order to apply the principle, it is essential that the axis of reference be a fixed axis. Thus the angle ϑ of polar coordinates has associated with it an angular momenta about an axis in the xy plane, but the principle of conservation of angular momentum cannot be applied directly to this quantity because the axis is not, in general, fixed but varies with φ. A simple relation involving pϑ connects the angular momenta pχ and Pφ about different fixed axes, one of which, pχ, relates to the axis normal to the plane of the motion. This is

    an equation easily derived by considering Figure 1–3. The sides of the small triangle have the lengths r sin ϑdφ, rdχ, and rdϑ. Since they form a right triangle, these distances are connected by the relation

    which gives, on introduction of the angular velocities , and and multiplication by m/dt,

    Equation 1–41 follows from this and the definitions of pχ, pϑ, and pφ.

    Conservation of angular momentum may be applied to more general systems than the one described here. It is at once evident that we have not used the special form of the potential-energy expression except for the fact that it is independent of direction, since this function enters into the r equation only. Therefore the above results are true for a particle moving in any spherically symmetric potential field.

    Furthermore, we can extend the theorem to a collection of point particles interacting with each other in any desired way but influenced by external forces only through a spherically symmetric potential function. If we describe such a system by using the polar coordinates of each particle, the Lagrangian function is

    Instead of φ1, φ2, …, φn, we now introduce new angular coordinates α, β, …, k given by the linear equations

    The values given the constants b1, …, kn are unimportant so long as they make the above set of equations mutually independent. α is an angle about the axis z such that if α is increased by ∆α, holding β, …, K constant, the effect is to increase each φi by ∆α, or, in other words, to rotate the whole system of particles about z without changing their mutual positions. By hypothesis the value of V is not changed by such a rotation, so that V is independent of α. We therefore obtain the equation

    Moreover, from Equation 1–42 we derive the relation

    Hence, calling the distance ri sin ϑi of the ith particle from the z axis ρi, we obtain the equation

    This is the more general expression of the principle of the conservation of angular momentum which we were seeking. In such a system of many particles with mutual interactions, as, for example, an atom consisting of a number of electrons and a nucleus, the individual particles do not in general conserve angular momentum but the aggregate does.

    The potential-energy function V need be only cylindrically symmetric about the axis z for the above proof to apply, since the essential feature was the independence of V on the angle α about z. However, in that case z is restricted to a particular direction in space, whereas if V is spherically symmetric the theorem holds for any choice of axis.

    Angular momenta transform like vectors, the directions of the vectors being the directions of the axes about which the angular momenta are determined. It is customary to take the sense of the vectors such as to correspond to the right-hand screw rule.

    2. THE EQUATIONS OF MOTION IN THE HAMILTONIAN FORM

    2a. Generalized Momenta.—In Cartesian coordinates the momentum related to the direction xk is mk k, which, since V is restricted to be a function of the coordinates only, can be written as

    Angular momenta can likewise be expressed in this manner. Thus, for one particle in a spherically symmetric potential field, the angular momentum about the z axis was defined in Section 1e by the expression

    Reference to Equation 1–31, which gives the expression for the kinetic energy in polar coordinates, shows that

    Likewise, in the case of a number of particles, the angular momentum conjugate to the coordinate α is

    as shown by the discussion of Equation 1–46. By extending this to other coordinate systems, the generalized momentum pk conjugate to the coordinate qk is defined as

    The form taken by Lagrange's equations (Eq. 1–29) when the definition of Pk is introduced is

    so that Equations 2–5 and 2–6 form a set of 6n first-order differential equations equivalent to the 3n second-order equations of Equation 1–29.

    being in general a function of both the q′s and ′s, the definition of pk given by Equation 2–5 provides 3n relations between the variables qk, k, and pk, permitting the elimination of the 3n velocities k, so that the system can now be described in terms of the 3n coordinates qk and the 3n conjugate momenta pk. Hamilton in 1834 showed that the equations of motion can in this way be thrown into an especially simple form, involving a function H of the pk's and qk's called the Hamiltonian function.

    2b. The Hamiltonian Function and Equations.—For conservative systems¹ we shall show that the function H is the total energy (kinetic plus potential) of the system, expressed in terms of the pk's and qk's. In order to have a definition which holds for more general systems, we introduce H by the relation

    Although this definition involves the velocities k, H may be made a function of the coordinates and momenta only, by eliminating the velocities through the use of Equation 2–5. From the definition we obtain for the total differential of H the equation

    or, using the expressions for pk and k given in Equations 2–5 and 2–6 (equivalent to Lagrange's equations),

    whence, if H is regarded as a function of the qk's and pk's, we obtain the equations

    These are the equations of motion in the Hamiltonian or canonical form.

    2c. The Hamiltonian Function and the Energy.—Let us consider the time dependence of H for a conservative system. We have

    using the same substitutions for pk and k (Eqs. 2–5 and 2–6) as before. H is hence a constant of the motion, which is called the energy of the system. For Newtonian systems, in which we shall be chiefly interested, the Hamiltonian function is the sum of the kinetic energy and the potential energy,

    expressed as a function of the coordinates and momenta. This is proved by considering the expression for T for such systems. For any set of coordinates, T will be a homogeneous quadratic function of the velocities

    where the aij's may be functions of the coordinates.

    Hence

    so that

    2d. A General Example.—The use of the Hamiltonian equations may be illustrated by the example of two point particles with masses m1 and m2, respectively, moving under the influence of a mutual attraction given by the potential energy function V(r), in which r is the distance between the two particles. The hydrogen atom is a special case of such a system, so that the results obtained below will be used in Chapter II. If the coordinates of the first particle are x1, y1, z1 and those of the second x2, y2, z2, the Lagrangian function L is

    The solution of this problem is facilitated by the introduction, in place of x1, y1, z1, x2, y2, z2, of the Cartesian coordinates x, y, z of the center of mass of the system and the polar coordinates r, ϑ, φ of one particle referred to the other as origin.

    The coordinates of the center of gravity are determined by the equations

    with similar equations for y1, y2, y and z1, z2, z. The polar coordinates r, ϑ, φ are given by

    Elimination of x2, y2, and z2 between these two sets of equations leads to the relations

    while elimination of x1, y1, and z1 gives the set

    The substitution of the time derivatives of these quantities in the Lagrangian function L results in the expression

    in which µ, called the reduced mass, is given by

    The momenta conjugate to x, y, z and r, ϑ, φ are

    The Hamiltonian function is therefore

    The equations of motion become

    It is noticed that the last six of these equations (2–26, 2–27) are identical with the equations which define the momenta involved. An inspection of Equations 2–25 indicates that they are closely related to Equations 1–33, 1–34, and 1–35. If in these equations m is replaced by µ and if in Equation 1–35 4π²mv²0r is replaced by we obtain just the equations which result from substituting for pr, pϑ, pφ their expressions in terms of , , and in Equation 2–25. The first three, 2–24, show that the center of gravity of the system moves with a constant velocity, while the next three are the equations of motion of a particle of mass µ bound to a fixed center by a force whose potential energy function is V(r).

    This problem illustrates the fact that in most actual problems the Lagrangian equations are reached in the process of solution of the equations of motion in the Hamiltonian form. The great value of the Hamiltonian equations lies in their particular suitability for general considerations, such as, for example, Liouville's theorem in statistical mechanics, the rules of quantization in the old quantum theory, and the formulation of the Schrödinger wave equation. This usefulness is in part due to the symmetrical or conjugate form of the equations in p and q.

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