![The text on this page is estimated to be only 28.42%
accurate
THE LIFE OF SPINOZA liii offered to Spinoza an annuity of
500 florins, but Spinoza declined to take more than 300 florins. Jan
Rieuwertsz was a bookseller at Amsterdam, and some fifteen years
older than Spinoza. He was a Collegiant, and very liberal in his
views. His shop enjoyed the evil reputation of being the seat of
scoffers. He published and stocked the works of many authors of
unorthodox repute, including those of Descartes, Balling, Jelles, and
Spinoza. His son also was a devoted admirer of Spinoza. Such were
some of the men with whom Spinoza stood in friendly relationship
during his last years in Amsterdam. Further details are wanting.
Possibly he had given private tuition to Simon de Vries (who speaks
of him as master ), Balling, and others ; or he may have held
some kind of seminar or class for the informal discussion of religious
and philosophical questions. If so, the substance of his Meta physical
Thoughts (which were subsequently appended to his version of
Descartes' Principia) and of his Short Treatise on God, Man and his
Well-being must have been elaborated during these years, and for
these purposes. This would also account for the continuation or
revival of similar meetings for the discussion of Spinoza's views, as
reported in the letter of Simon de Vries. Little as is known of these
years, there can be no doubt that they were years of storm and
stress in the mental history of Spinoza. This much may be gathered
from the impres sive pages with which he opens his Treatise on the
Improve ment of the Understanding. After experience had taught
me [so he writes] that all things which are ordinarily encountered in
common life are vain and futile, and when I saw that all things which
occasioned me any anxiety or fear had in themselves nothing of
good or evil, except in so far as the mind was moved by them ; I at
length determined to inquire if there were anything which was a true
good capable of im parting itself, and by which alone the mind could
be affected to the](https://image.slidesharecdn.com/59723-250721085745-52ecaa70/85/Quantum-mechanics-classical-results-modern-systems-and-visualized-examples-2nd-ed-Edition-Richard-Robinett-49-320.jpg)
![The text on this page is estimated to be only 21.91%
accurate
[Conclusion of Spinoza's letter to Oldenburg. October 1661.
S« p. cxxiii.]](https://image.slidesharecdn.com/59723-250721085745-52ecaa70/85/Quantum-mechanics-classical-results-modern-systems-and-visualized-examples-2nd-ed-Edition-Richard-Robinett-56-320.jpg)
![The text on this page is estimated to be only 28.63%
accurate
THE LIFE OF SPINOZA Ixi As the character of the age in
which we live [Spinoza adds] is not unknown to you, I would beg of
you most earnestly to be very careful about the communication of
these things to others. I do not want to say that you should
absolutely keep them to yourselves, but only that if ever you wish to
communicate them to others, then you shall have no other object in
view except only the happiness of your neighbour ; being at the
same time clearly assured that the reward of your labour will not
disappoint you therein. Having finished the first draft of his Short
Treatise Spinoza felt that he had attacked all the great problems of
religion and of philosophy, without any preliminary account of the
requirements of philosophic method, without any adequate
justification of his own mode of treatment. To this problem,
accordingly, he turned his attention next, and began his Treatise on
the Improvement of the Under standing. In a letter dated October
1661, in reply to some questions of Henry Oldenburg, Spinoza states
that he had written a complete little treatise on the origin of things,
and their relation to the first cause, and also on the improvement of
the understanding, and that he was actually busy just then copying
and correcting it. It would appear from this that Spinoza's intention
at that time may have been to combine the Short Treatise and the
Treatise on the Improve ment of the Understanding. What actually
happened, how ever, is not quite certain. The editors of Spinoza's
posthumous works only had a fragment of the Treatise on the
Improvement of the Understanding, and apparently nothing of the
Short Treatise, of which we only possess at present two Dutch
versions, discovered about 1860. The editors of the Opera Posthuma
say that the Treatise on the Improvement of the Understanding was
one of Spinoza's earliest works, and that he had never finished it,
but they appear to be uncertain whether it was only want of time or
the inherent difficulties of the subject which prevented him from
finishing it.](https://image.slidesharecdn.com/59723-250721085745-52ecaa70/85/Quantum-mechanics-classical-results-modern-systems-and-visualized-examples-2nd-ed-Edition-Richard-Robinett-57-320.jpg)
![The text on this page is estimated to be only 29.04%
accurate
Ixiv INTRODUCTION sion which Spinoza made on
Oldenburg may be gathered from the following letter which
Oldenburg wrote to Spinoza immediately after his return to London,
in August 1661. It was with such reluctance [he writes] that I tore
myself away from your side, when I recently visited you in your
retreat at Rijnsburg, that no sooner am I back in England than I
already try to join you again, at least as far as this can be effected
by means of correspondence. Solid learn ing combined with
kindliness and refinement (wherewith Nature and Study have most
richly endowed you) have such an attraction that they win the love
of all noble and liberally educated men. Let us, therefore, most
excellent sir, give each other the right hand of unfeigned friendship,
and cultivate it diligently by every kind of attention and service.
Whatever service my humble powers can render, consider as yours.
And permit me to claim a part of those intellectual gifts which you
possess, if I may do so without detriment to you. Our conversation
at Rijnsburg turned on God, infinite Extension and Thought, on the
difference and the agreement between these attributes, on the
nature of the union of the human soul with the body ; and further,
on the Principles of the Cartesian and the Baconian Philo sophy. But
as we then discussed themes of such moment only at a distance, as
it were, and cursorily, and as all those things have since then been
lying heavily on my mind, I now venture to claim the right of our
new friendship to ask you affectionately to explain to me somewhat
more fully your views on the above-mentioned subjects, and not to
mind enlightening me, more especially on these two points, namely,
first, what do you consider to be the true distinc tion between
Extension and Thought ; secondly, what defects do you observe in
the Philosophy of Descartes and of Bacon, and how, do you think,
might they be eliminated, and replaced by something more sound ?
The more freely you write to me about these and the like, the more
closely](https://image.slidesharecdn.com/59723-250721085745-52ecaa70/85/Quantum-mechanics-classical-results-modern-systems-and-visualized-examples-2nd-ed-Edition-Richard-Robinett-60-320.jpg)
Quantum mechanics classical results modern systems and visualized examples 2nd ed Edition Richard Robinett
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- 6. Quantum Mechanics Classical Results, Modern Systems, and Visualized Examples
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- 8. Quantum Mechanics Classical Results, Modern Systems, and Visualized Examples Second Edition Richard W. Robinett Pennsylvania State University 1
- 9. 3 Great Clarendon Street, Oxford OX2 6DP Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide in Oxford NewYork Auckland Cape Town Dar es Salaam Hong Kong Karachi Kuala Lumpur Madrid Melbourne Mexico City Nairobi New Delhi Shanghai Taipei Toronto With offices in Argentina Austria Brazil Chile Czech Republic France Greece Guatemala Hungary Italy Japan Poland Portugal Singapore South Korea Switzerland Thailand Turkey Ukraine Vietnam Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries Published in the United States by Oxford University Press Inc., New York © Oxford University Press 2006 The moral rights of the author have been asserted Database right Oxford University Press (maker) First published 2006 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this book in any other binding or cover and you must impose the same condition on any acquirer British Library Cataloguing in Publication Data Data available Library of Congress Cataloging in Publication Data Robinett, Richard W. (Richard Wallace) Quantum mechanics : classical results, modern systems, and visualized examples / Richard W. Robinett.—2nd ed. p. cm. ISBN-13: 978–0–19–853097–8 (alk. paper) ISBN-10: 0–19–853097–8 (alk. paper) 1. Quantum theory. I. Title. QC174.12.R6 2006 530.12—dc22 2006000424 Typeset by Newgen Imaging Systems (P) Ltd., Chennai, India Printed in Great Britain on acid-free paper by Biddles Ltd, King’s Lynn, Norfolk ISBN 0–19–853097–8 978–0–19–853097–8 10 9 8 7 6 5 4 3 2 1
- 10. Preface to the Second Edition One of the hallmarks of science is the continual quest to refine and expand one’s understanding and vision of the universe, seeking not only new answers to old questions, but also proactively searching out new avenues of inquiry based on past experience. In much the same way, teachers of science (including textbook authors) can and should explore the pedagogy of their disciplines in a scientific way, maintaining and streamlining what has been documented to work, but also improving, updating, and expanding their educational materials in response to new knowledge in their fields,in basic,applied,and educational research. For that reason,I am very pleased to have been given the opportunity to produce a Second Edition of this textbook on quantum mechanics at the advanced undergraduate level. The First Edition of Quantum Mechanics had a number of novel features, so it may be useful to first review some aspects of that work, in the context of this Second Edition. The descriptive subtitle of the text, Classical Results, Modern Systems, and Visualized Examples, was, and still is, intended to suggest a number of the inter-related approaches to the teaching and learning of quantum mechanics which have been adopted here. • Many of the expected familiar topics and examples (the Classical Results) found in standard quantum texts are indeed present in both editions, but we also continue to focus extensively on the classical–quantum connection as one of the best ways to help students learn the subject. Topics such as momentum- space probability distributions,time-dependent wave packet solutions,and the correspondenceprinciplelimitof largequantumnumberscanallhelpstudents use their existing intuition to make contact with new quantum ideas; classical wave physics continues to be emphasized as well,with its own separate chapter, for the same reason. Additional examples of quantum wave packet solutions have been included in this new Edition, as well as a self-contained discussion of the Wigner quasi-probability (phase-space) distribution, designed to help make contact with related ideas in statistical mechanics, classical mechanics, and even quantum optics. • An even larger number of examples of the application of quantum mech- anics to Modern Systems is provided, including discussions of experimental realizations of quantum phenomena which have only appeared since the First Edition. Advances in such areas as materials science and laser trapping/cooling
- 11. vi PREFACE TO THE SECOND EDITION have meant a large number of quantum systems which have historically been only considered as“textbook”examples have become physically realizable. For example, the “quantum bouncer”, once discussed only in pedagogical journ- als, has been explored experimentally in the Quantum states of neutrons in the Earth’s gravitational field.É The production of atomic wave packets which exhibit the classical periodicity of Keplerian orbitsÊ is another example of a Classical Result which has become a Modern System. The ability to manipulate nature at the extremes of small distance (nano- and even atomic-level) and low temperatures (as with Bose–Einstein con- densates) implies that a knowledge of quantum mechanics is increasingly important in modern physical science, and a number of new discussions of applications have been added to both the text and to the Problems, including ones on such topics as expanding/interfering Bose–Einstein condensates, the quantum Hall effect, and quantum wave packet revivals, all in the context of familiar textbook level examples. • We continue to emphasize the use of Visualized Examples (with 200 figures included) to reinforce students’ conceptual understanding of the basic ideas and to enhance their mathematical facility in solving problems. This includes not only pictorial representations of stationary state wavefunctions and time- dependent wave packets, but also real data. The graphical representation of such information often provides the map of the meeting ground of the some- times arcane formalism of a theorist, the observations of an experimentalist, and the rest of the scientific community; the ability to “follow such maps” is an important part of a physics education. Motivated in this Edition (even more than before) by results appearing from Physics Education Research (PER), we still stress concepts which PER stud- ies have indicated can pose difficulties for many students, such as notions of probability, reading potential energy diagrams, and the time-development of eigenstates and wave packets. As with any textbook revision, the opportunity to streamline the presentation and pedagogy, based on feedback from actual classroom use, is one of the most important aspects of a new Edition, and I have taken this opportunity to remove some topics (moving them, however, to an accompanying Web site) and adding new ones. New sections on TheWigner Quasi-Probability Distribution (and many related problems), an Infinite Array of δ-functions: Periodic Potentials and the Dirac Comb,Time-Dependent Perturbation Theory,and Timescales in Bound State É The title of a paper by V. V. Nesvizhevsky et al. (2002). Nature 415, 297. Ê See Yeazell et al. (1989).
- 12. PREFACE TO THE SECOND EDITION vii Systems: Classical Period and Quantum Revival Times reflect suggestions from various sources on (hopefully) useful new additions. A number of new in-text Examples and end-of-chapter Problems have been added for similar reasons, as wellasanexpandedsetof Appendices,ondimensionsandmathematicalmethods. An exciting new feature of the Second Edition is the development of a Web siteË in support of the textbook, for use by both students and instructors, linked from the Oxford University PressÌ web page for this text. Students will find many additional (extended) homework problems in the form of Worksheets on both formal and applied topics, such as“slow light”, femtosecond chemistry, and quantum wave packet revivals. Additional material in the form of Supplementary Chapters on such topics as neutrino oscillations, quantum Monte Carlo approx- imation methods, supersymmetry in quantum mechanics, periodic orbit theory of quantum billiards, and quantum chaos are available. For instructors, copies of a complete Solutions Manual for the textbook, as well as Worksheet Solutions, will be provided on a more secure portion of the site, in addition to copies of the Transparencies for the figures in the text. An 85-page Guide to the Pedagogical Literature on Quantum Mechanics is also available there, surveying articles from The American Journal of Physics, The European Journal of Physics, and The Journal of Chemical Education from their earliest issues, through the publication date of this text (with periodic updates planned.) In addition, a quantum mechanics assessment test (the so-called Quantum Mech- anics Visualization Instrument or QMVI) is available at the Instructors site,along with detailed information on its development and sample results from earlier educational studies. Given my long-term interest in the science, as well as the pedagogy, of quantum mechanics, I trust that this site will continually grow in both size and coverage as new and updated materials are added. Information on accessing the Instructors area of the Web site is available through the publisher at the Oxford University Press Web site describing this text. I am very grateful to all those from whom I have had help in learning quantum mechanics over the years, including faculty and fellow students in my under- graduate, graduate, and postdoctoral days, current faculty colleagues (here at Penn State and elsewhere), my own undergraduate students over the years, and numerous authors of textbooks and both research and pedagogical articles,many of whom I have never met, but to whom I owe much. I would like to thank all those who helped very directly in the production of the Second Edition of this text, specifically including those who provided useful suggestions for improve- ment or who found corrections, namely, J. Banavar, A. Bernacchi, B. Chasan, Ë See robinett.phys.psu.edu/qm Ì See www.oup.co.uk
- 13. viii PREFACE TO THE SECOND EDITION J. Edmonds, M. Cole, C. Patton, and J. Yeazell. I have truly enjoyed recent col- laborations with both M. Belloni and M. A. Doncheski on pedagogical issues related to quantum theory, and some of our recent work has found its way into the Second Edition (including the cover) and I thank them for their insights, and patience. No work done in a professional context can be separated from one’s personal life (nor should it be) and so I want to thank my family for all of their help and understanding over my entire career, including during the production of this new Edition. The First Edition of this text was thoroughly proof-read by my mother-in-law (Nancy Malone) who graciously tried to teach me the proper use of the English language; her recent passing has saddened us all. My own mother (Betty Robinett) has been, and continues to be, the single most important role model in my life—both personal and professional—and I am deeply indebted to her far more than I can ever convey. Finally, to my wife (Sarah) and children (James and Katherine), I give thanks everyday for the richness and joy they bring to my life. Richard Robinett December, 2005 State College, PA
- 14. Contents Part I The Quantum Paradigm 1 1 A First Look at Quantum Physics 3 1.1 How this Book Approaches Quantum Mechanics 3 1.2 Essential Relativity 8 1.3 Quantum Physics: as a Fundamental Constant 10 1.4 Semiclassical Model of the Hydrogen Atom 17 1.5 Dimensional Analysis 21 1.6 Questions and Problems 23 2 Classical Waves 34 2.1 The Classical Wave Equation 34 2.2 Wave Packets and Periodic Solutions 36 2.2.1 General Wave Packet Solutions 36 2.2.2 Fourier Series 38 2.3 Fourier Transforms 43 2.4 Inverting the Fourier transform: the Dirac δ-function 46 2.5 Dispersion and Tunneling 51 2.5.1 Velocities for Wave Packets 51 2.5.2 Dispersion 53 2.5.3 Tunneling 56 2.6 Questions and Problems 57 3 The Schrödinger Wave Equation 65 3.1 The Schrödinger Equation 65 3.2 Plane Waves and Wave Packet Solutions 67 3.2.1 Plane Waves and Wave Packets 67 3.2.2 The Gaussian Wave Packet 70 3.3 “Bouncing” Wave Packets 75 3.4 Numerical Calculation of Wave Packets 77 3.5 Questions and Problems 79
- 15. x CONTENTS 4 Interpreting the Schrödinger Equation 84 4.1 Introduction to Probability 84 4.1.1 Discrete Probability Distributions 84 4.1.2 Continuous Probability Distributions 87 4.2 Probability Interpretation of the Schrödinger Wavefunction 91 4.3 Average Values 96 4.3.1 Average Values of Position 96 4.3.2 Average Values of Momentum 98 4.3.3 Average Values of Other Operators 100 4.4 Real Average Values and Hermitian Operators 102 4.5 The Physical Interpretation of φ(p) 104 4.6 Energy Eigenstates, Stationary States, and the Hamiltonian Operator 107 4.7 The Schrödinger Equation in Momentum Space 111 4.7.1 Transforming the Schrödinger Equation Into Momentum Space 111 4.7.2 Uniformly Accelerating Particle 114 4.8 Commutators 116 4.9 The Wigner Quasi-Probability Distribution 118 4.10 Questions and Problems 121 5 The Infinite Well: Physical Aspects 134 5.1 The Infinite Well in Classical Mechanics: Classical Probability Distributions 134 5.2 Stationary States for the Infinite Well 137 5.2.1 Position-Space Wavefunctions for the Standard Infinite Well 137 5.2.2 Expectation Values and Momentum-Space Wavefunctions for the Standard Infinite Well 140 5.2.3 The Symmetric Infinite Well 144 5.3 The Asymmetric Infinite Well 146 5.4 Time-Dependence of General Solutions 151 5.4.1 Two-State Systems 151 5.4.2 Wave Packets in the Infinite Well 154 5.4.3 Wave Packets Versus Stationary States 157 5.5 Questions and Problems 157 6 The Infinite Well: Formal Aspects 166 6.1 Dirac Bracket Notation 166 6.2 Eigenvalues of Hermitian Operators 167 6.3 Orthogonality of Energy Eigenfunctions 168 6.4 Expansions in Eigenstates 171
- 16. CONTENTS xi 6.5 Expansion Postulate and Time-Dependence 175 6.6 Parity 181 6.7 Simultaneous Eigenfunctions 183 6.8 Questions and Problems 185 7 Many Particles in the Infinite Well: The Role of Spin and Indistinguishability 192 7.1 The Exclusion Principle 192 7.2 One-Dimensional Systems 193 7.3 Three-Dimensional Infinite Well 195 7.4 Applications 198 7.4.1 Conduction Electrons in a Metal 198 7.4.2 Neutrons and Protons in Atomic Nuclei 200 7.4.3 White Dwarf and Neutron Stars 200 7.5 Questions and Problems 206 8 Other One-Dimensional Potentials 210 8.1 Singular Potentials 210 8.1.1 Continuity of ψ(x) 210 8.1.2 Single δ-function Potential 212 8.1.3 Twin δ-function Potential 213 8.1.4 Infinite Array of δ-functions: Periodic Potentials and the Dirac Comb 216 8.2 The Finite Well 221 8.2.1 Formal Solutions 221 8.2.2 Physical Implications and the Large x Behavior of Wavefunctions 225 8.3 Applications to Three-Dimensional Problems 230 8.3.1 The Schrödinger Equation in Three Dimensions 230 8.3.2 Model of the Deuteron 231 8.4 Questions and Problems 234 9 The Harmonic Oscillator 239 9.1 The Importance of the Simple Harmonic Oscillator 239 9.2 Solutions for the SHO 243 9.2.1 Differential Equation Approach 243 9.2.2 Properties of the Solutions 247 9.3 Experimental Realizations of the SHO 249 9.4 Classical Limits and Probability Distributions 251
- 17. xii CONTENTS 9.5 Unstable Equilibrium: Classical and Quantum Distributions 254 9.6 Questions and Problems 255 10 Alternative Methods of Solution and Approximation Methods 260 10.1 Numerical Integration 261 10.2 The Variational or Rayleigh–Ritz Method 266 10.3 The WKB method 273 10.3.1 WKB Wavefunctions 274 10.3.2 WKB Quantized Energy Levels 277 10.4 Matrix Methods 278 10.5 Perturbation Theory 286 10.5.1 Nondegenerate States 286 10.5.2 Degenerate Perturbation Theory 293 10.5.3 Time-Dependent Perturbation Theory 295 10.6 Questions and Problems 299 11 Scattering 307 11.1 Scattering in One-Dimensional Systems 307 11.1.1 Bound and Unbound States 307 11.1.2 Plane Wave Solutions 310 11.2 Scattering from a Step Potential 310 11.3 Scattering from the Finite Square Well 315 11.3.1 Attractive Well 315 11.3.2 Repulsive Barrier 319 11.4 Applications of Quantum Tunneling 321 11.4.1 Field Emission 321 11.4.2 Scanning Tunneling Microscopy 324 11.4.3 α-Particle Decay of Nuclei 325 11.4.4 Nuclear Fusion Reactions 328 11.5 Questions and Problems 330 12 More Formal Topics 333 12.1 Hermitian Operators 333 12.2 Quantum Mechanics, Linear Algebra, and Vector Spaces 337 12.3 Commutators 341 12.4 Uncertainty Principles 343 12.5 Time-Dependence and Conservation Laws in Quantum Mechanics 346 12.6 Propagators 352 12.6.1 General Case and Free Particles 352 12.6.2 Propagator and Wave Packets for the Harmonic Oscillator 353
- 18. CONTENTS xiii 12.7 Timescales in Bound State Systems: Classical Period and Quantum Revival Times 357 12.8 Questions and Problems 360 13 Operator and Factorization Methods for the Schrödinger Equation 370 13.1 Factorization Methods 370 13.2 Factorization of the Harmonic Oscillator 371 13.3 Creation and Annihilation Operators 377 13.4 Questions and Problems 380 14 Multiparticle Systems 384 14.1 Generalities 384 14.2 Separable Systems 387 14.3 Two-Body Systems 389 14.3.1 Classical Systems 390 14.3.2 Quantum Case 391 14.4 Spin Wavefunctions 394 14.5 Indistinguishable Particles 396 14.6 Questions and Problems 407 Part II The Quantum World 413 15 Two-Dimensional Quantum Mechanics 415 15.1 2D Cartesian Systems 417 15.1.1 2D Infinite Well 418 15.1.2 2D Harmonic Oscillator 422 15.2 Central Forces and Angular Momentum 423 15.2.1 Classical Case 423 15.2.2 Quantum Angular Momentum in 2D 425 15.3 Quantum Systems with Circular Symmetry 429 15.3.1 Free Particle 429 15.3.2 Circular Infinite Well 432 15.3.3 Isotropic Harmonic Oscillator 435 15.4 Questions and Problems 437 16 The Schrödinger Equation in Three Dimensions 448 16.1 Spherical Coordinates and Angular Momentum 449 16.2 Eigenfunctions of Angular Momentum 454 16.2.1 Methods of Derivation 454
- 19. xiv CONTENTS 16.2.2 Visualization and Applications 463 16.2.3 Classical Limit of Rotational Motion 465 16.3 Diatomic Molecules 467 16.3.1 Rigid Rotators 467 16.3.2 Molecular Energy Levels 469 16.3.3 Selection Rules 472 16.4 Spin and Angular Momentum 475 16.5 Addition of Angular Momentum 482 16.6 Free Particle in Spherical Coordinates 491 16.7 Questions and Problems 492 17 The Hydrogen Atom 501 17.1 Hydrogen Atom Wavefunctions and Energies 501 17.2 The Classical Limit of the Quantum Kepler Problem 507 17.3 Other “Hydrogenic” Atoms 513 17.3.1 Rydberg Atoms 513 17.3.2 Muonic Atoms 515 17.4 Multielectron Atoms 517 17.4.1 Helium-Like Atoms 519 17.4.2 Lithium-Like Atoms 524 17.4.3 The Periodic Table 527 17.5 Questions and Problems 529 18 Gravity and Electromagnetism in Quantum Mechanics 540 18.1 Classical Gravity and Quantum Mechanics 540 18.2 Electromagnetic Fields 543 18.2.1 Classical Electric and Magnetic Fields 543 18.2.2 E and B Fields in Quantum Mechanics 548 18.3 Constant Electric Fields 550 18.4 Atoms in Electric Fields: The Stark Effect 552 18.4.1 Classical Case 552 18.4.2 Quantum Stark Effect 555 18.5 Constant Magnetic Fields 561 18.6 Atoms in Magnetic Fields 564 18.6.1 The Zeeman Effect: External B Fields 564 18.6.2 Spin-Orbit Splittings: Internal B Fields 569 18.6.3 Hyperfine Splittings: Magnetic Dipole–Dipole Interactions 574 18.7 Spins in Magnetic Fields 576 18.7.1 Measuring the Spinor Nature of the Neutron Wavefunction 576 18.7.2 Spin Resonance 578
- 20. CONTENTS xv 18.8 The Aharonov–Bohm Effect 583 18.9 Questions and Problems 586 19 Scattering in Three Dimensions 596 19.1 Classical Trajectories and Cross-Sections 597 19.2 Quantum Scattering 603 19.2.1 Cross-Section and Flux 603 19.2.2 Wave Equation for Scattering and the Born Approximation 606 19.3 Electromagnetic Scattering 612 19.4 Partial Wave Expansions 619 19.5 Scattering of Particles 624 19.5.1 Frames of Reference 625 19.5.2 Identical Particle Effects 631 19.6 Questions and Problems 635 A Dimensions and MKS-type Units for Mechanics, Electricity and Magnetism, and Thermal Physics 641 A.1 Problems 642 B Physical Constants, Gaussian Integrals, and the Greek Alphabet 644 B.1 Physical Constants 644 B.2 The Greek Alphabet 646 B.3 Gaussian Probability Distribution 646 B.4 Problems 648 C Complex Numbers and Functions 649 C.1 Problems 651 D Integrals, Summations, and Calculus Results 653 D.1 Integrals 653 D.2 Summations and Series Expansions 658 D.3 Assorted Calculus Results 661 D.4 Real Integrals by Contour Integration 661 D.5 Plotting 664 D.6 Problems 665 E Special Functions 666 E.1 Trigonometric and Exponential Functions 666 E.2 Airy Functions 667
- 21. xvi CONTENTS E.3 Hermite Polynomials 668 E.4 Cylindrical Bessel Functions 669 E.5 Spherical Bessel Functions 669 E.6 Legendre Polynomials 669 E.7 Generalized Laguerre Polynomials 670 E.8 The Dirac δ-Function 671 E.9 The Euler Gamma Function 672 E.10 Problems 672 F Vectors, Matrices, and Group Theory 674 F.1 Vectors and Matrices 674 F.2 Group Theory 679 F.3 Problems 679 G Hamiltonian Formulation of Classical Mechanics 680 G.1 Problems 685 REFERENCES 687 INDEX 695
- 22. PART I The Quantum Paradigm
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- 24. ONE A First Look at Quantum Physics 1.1 How This Book Approaches Quantum Mechanics It can easily be argued that a fully mature and complete knowledge of quantum mechanics should include historical, axiomatic, formal mathematical, and even philosophical background to the subject. However, for a student approaching quantum theory for the first time in a serious way, it can be the case that an approach utilizing his or her existing knowledge of, and intuition for, classical physics (including mechanics, wave physics, and electricity and magnetism) as well as emphasizing connections to experimental results can be the most pro- ductive. That, at least, is the point of view adopted in this text and can be illustrated by a focus on the following general topics: (1) The incorporation of a wave property description of matter into a consistent wave equation, via the Schrödinger equation; (2) The statistical interpretation of the Schrödinger wavefunction in terms of a probability density (in both position- and momentum-space); (3) The study of single-particle solutions of the Schrödinger equation, for both time-independent energy eigenstates as well as time-dependent systems, for many model systems, in a variety of spatial dimensions, and finally; (4) The influence of both quantum mechanical effects and the constraints arising from the indistinguishability of particles (and how that depends on their spin) on the properties of multiparticle systems, and the resulting implications for the structure of different forms of matter. By way of example of our approach, we first note that Fig. 1.1 illustrates an example of a precision measurement of the wave properties of ultracold neutrons, exhibiting a Fresnel diffraction pattern arising from scattering from
- 25. 4 CHAPTER 1 A FIRST LOOK AT QUANTUM PHYSICS Scanning slit position 100 µm 500 1000 1500 Figure 1.1. Fresnel diffraction pattern obtained from scattering at a sharp edge, obtained using ultracold neutrons by Gähler and Zeilinger (1991). a sharp edge, nicely explained by classical optical analogies. We devote Chapter 2 to a discussion of classical wave physics and Chapter 3 to the description of such wave effects for material particles, via the Schrödinger equation. Figure 1.2 demonstrates an interference pattern using electron beams, built up“electron by electron,” with the obvious fringes resulting only from a large number of indi- vidual measurements. The important statistical aspect of quantum mechanics, simply illustrated by this experiment, is discussed in Chapter 4 and beyond. Itcanbearguedthatmuchof theearlysuccessof quantumtheorycanbetraced to the fact that many exactly soluble quantum models are surprisingly coincid- ent with naturally occurring physical systems, such as the hydrogen atom and the rotational/vibrational states of molecules and such systems are, of course, discussed here. The standing wave patterns obtained from scanning tunnel- ing microscopy of “electron waves” in a circular corral geometry constructed from arrays of iron atoms on a copper surface, seen in Fig. 1.3, reminds us of the continuing progress in such areas as materials science and atom trapping in developing artificial systems (and devices) for which quantum mechanics is applicable. In that context, many exemplary quantum mechanical models, which have historically been considered as only textbook idealizations, have also recently found experimental realizations. Examples include “designer” potential wells approximating square and parabolic shapes made using molecular beam techniques, as well as magnetic or optical traps. The solution of the Schrödinger equation, in a wide variety of standard (and not-so-standard) one-, two-, and three-dimensional applications, is therefore emphasized here, in Chapters 5, 8, 9,
- 26. 1.1 HOW THIS BOOK APPROACHES QUANTUM MECHANICS 5 Figure 1.2. Interference patterns obtained by using an electron microscope showing the fringes being “built up” from an increasingly large number of measurements of individual events. From Merli, Missiroli, and Pozzi (1976). (Photo reproduced by permission of the American Institute of Physics.) and 15–17. In parallel to these examples,more formal aspects of quantum theory are outlined in Chapters 7, 10, 12, 13, and 14. The quantum in quantum mechanics is often associated with the discrete energy levels observed in bound-state systems,most famously for atomic systems such as the hydrogen atom,which we discuss in Chapter 17,emphasizing that this is the quantum version of the classical Kepler problem. We also show, in Fig. 1.4, experimental measurements leading to a map of the momentum-space probab- ility density for the 1S state of hydrogen and the emphasis on momentum-space methods suggested by this result is stressed throughout the text. The influence of additional“real-life”effects,suchasgravityandelectromagnetism,onatomicand other systems are then discussed in Chapter 18. We note that the data in Fig. 1.4
- 27. 6 CHAPTER 1 A FIRST LOOK AT QUANTUM PHYSICS Figure 1.3. Standing wave patterns obtained using scanning tunneling microscopy from a circular “corral” of radius ∼70 Å, constructed from 48 iron atoms on a copper surface. (Photo courtesy of IBM Almaden.) 0.2 0.2 0 0.4 0.6 Differential cross section 0.8 1.0 0.4 0.5 Momentum q (a.u.) 0.8 1.0 H(1s) 1200 eV 800 eV 400 eV (1+q2 )–4 1.2 1.4 Figure 1.4. Electronprobabilitydensityobtainedbyscatteringwiththreedifferentenergyprobes,compared with the theoretically calculated momentum-space probability density for the hydrogen-atom ground state, from Lohmann and Weigold (1981). The data are plotted again the scaled momentum in atomic units (a.u.), q = a0p/. was obtained via scattering processes, and the importance of scattering methods in quantum mechanics is emphasized in both one-dimension (Chapter 11) and three-dimensions (Chapter 19). The fact that spin-1/2 particles must satisfy the Pauli principle has profound implications for the way that matter can arrange
- 28. 1.1 HOW THIS BOOK APPROACHES QUANTUM MECHANICS 7 Ionizational potential (eV) (solid) Nuclear charge (Z) He Ne Ar Kr Xe Rn 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 Polarizability (10 –24 cm 3 ) (dashed) 0 20 40 60 Figure 1.5. Plots of the ionization energy (solid) and atomic polarizability (dashed) versus nuclear charge, showing the shell structure characterized by the noble gas atoms, arising from the filling of atomic energy levels as mandated by the Pauli principle for spin-1/2 electrons. itself, as shown in the highly correlated values of physical parameters shown in Fig. 1.5 for atoms of increasing size and complexity. While it is illustrated here in a numerical way, this should also be reminiscent of the familiar periodic table of the elements, and the Pauli principle has similar implications for nuclear struc- ture. We discuss the role of spin in multiparticle systems described by quantum mechanics in Chapters 7, 14, and 17. We remind the reader that similar dramatic manifestations of quantum phe- nomena (including all of the effects mentioned above) are still being discovered, as illustrated in Fig. 1.6. In a justly famous experiment,É two highly localized and well-separated samples of sodium atoms are cooled to sufficiently low tem- peratures so that they are in the ground states of their respective potential wells (produced by laser trapping.) The trapping potential is removed and the res- ulting coherent Bose–Einstein condensates are allowed to expand and overlap, exhibiting the quantum interference shown in Fig. 1.6 (the solid curve, showing É From the paper entitled Observation of interference between two Bose condensates by Andrews et al. (1997).
- 29. 8 CHAPTER 1 A FIRST LOOK AT QUANTUM PHYSICS 200 Position (µm) 400 0 0 30 60 Absorpition (%) Figure 1.6. Data (from Andrews et al. (1997)) illustrating the interference of two Bose condensates as they expand and overlap (solid curve), compared to a single expanding Bose condensate (dotted curve). regular absorption variations across the central overlap region), while no such interference is observed for a single expanding quantum sample (dotted data.) Many of the salient features of this experiment can be understood using relatively simple ideas outlined in Chapters 3, 4, and 9. The ability to use the concepts and mathematical techniques of quantum mechanics to confront the wide array of experimental realizations that have come to characterize modern physical science will be one of the focuses of this text. Before proceeding, however, we reserve the remainder of this chapter for brief reviews of some of the essential aspects of both relativity and standard results from quantum theory. 1.2 Essential Relativity While we will consider nonrelativistic quantum mechanics almost exclusively, it is useful to briefly review some of the rudiments of special relativity and the fundamental role played by the speed of light, c. For a free particle of rest mass m moving at speed v, the total energy (E), momentum (p), and kinetic energy (T) can be written in the relativistically correct forms E = γ mc2 , p = γ mv, and T ≡ E − mc2 = (γ − 1)mc2 (1.1) where γ ≡ 1 1 − v2/c2 = 1 − v2 c2 −1/2 (1.2)
- 30. 1.2 ESSENTIAL RELATIVITY 9 The nonrelativistic limit corresponds to v/c 1, in which case we can use the series expansion (1 + x)n = 1 + nx + n(n − 1) 2! x3 + n(n − 1)(n − 2) 3! x3 + · · · (1.3) for x = v2/c2 small to show that p ≈ mv and T ≈ 1 + 1 2 v2 c2 + · · · − 1 mc2 ≈ 1 2 mv2 (1.4) which are the familiar nonrelativistic results for motion at speeds slow compared to the speed of light. In quantum mechanics the momentum is a more natural variable than v, and a useful relation can be obtained from Eqn. (1.1), namely E2 = (pc)2 + (mc2 )2 (1.5) This form stresses the fact that E, pc, and mc2 all have the same dimensions (namely energy), and we will often use these forms when convenient. As an example, the rest energies of various atomic particles will often be quoted in energy units; for the electron and proton we have mec2 = 0.511 MeV and mpc2 = 938.3 MeV (1.6) Recall that the electron volt or eV is defined by 1 eV = the energy gained by a fundamental charge e which has been accelerated through 1 V = (1.6 × 10−19 C)(1 V ) = 1.6 × 10−19 J (1.7) Atomic “masses” are often quoted in unified atomic mass units (formerly amu) which are given by 1 u = 931.5 MeV. The nonrelativistic limit of Eqn. (1.5), where pc mc2, is easily seen to be E = mc2 1 + pc mc2 2 1/2 = mc2 + p2 2m − p4 8m3c2 + · · · (1.8) Since the rest energy is “just along for the ride” in most of the problems we consider, we will ignore its contribution to the total energy; thus a phrase such as “. . . a 2 eV electron . . .” should be taken to mean that the electron has a kinetic energy T = E − mc2 ≈ p2/2m ≈ 2 eV. We will often write pc = 2(mc2)T in this limit.
- 31. 10 CHAPTER 1 A FIRST LOOK AT QUANTUM PHYSICS At the other extreme, in the ultrarelativistic limit when E mc2 (or v c), we can write E = pc 1 + mc2 pc 2 1/2 ≈ pc + 1 2 (mc2)2 pc + · · · (1.9) which is also seen to be consistent with the energy–momentum relation for truly massless particles (such as photons), namely E = pc. We list below several typical quantum mechanical systems and the order-of- magnitudes of the energies involved: • Electrons in atoms: For the inner shell electrons of an atom with nuclear charge +Ze, the kinetic energy is of order T ≈ Z2 13.6 eV. We can say, somewhat arbitrarily, that relativistic effects become nonnegligible when T 0.05 mc2 (i.e. a 5% effect). This condition is satisfied when Z 43, implying that the effects of relativity must certainly be considered for heavy atoms. • Deuteron: The simplest nuclear system is the bound state of a proton and neutron where the typical kinetic energies are T ≈ 2 MeV; this is to be com- pared with mpc2 ≈ mnc2 ≈ 939 MeV so that the deuteron can be considered as a nonrelativistic system to first approximation. • Quarks in the proton and pion: The constituent quark model of element- ary particles postulates that three quarks of effective mass roughly mqc2 ≈ 350 MeV form the proton; this implies binding energies and kinetic energies of the order of 1 − 10 MeV which is consistent with“nonrelativity.” The pion, on the other hand, is considered a bound state of two such quarks, but has rest energy mπ c2 ≈ 140 MeV, so that binding energies (and hence kinetic energies) of order several hundred MeV are required and relativistic effects dominate.Ê • Compact objects in astrophysics: The electrons in white dwarf stars and neut- rons in neutron stars have kinetic energies Te ≈ 0.08 MeV and Tn ≈ 140 MeV respectively, so these objects are “barely” nonrelativistic. 1.3 Quantum Physics: as a Fundamental Constant Justasthespeedof light,c,setsthescaleforwhenrelativisticeffectsareimportant, quantum physics also has an associated fundamental, dimensionful parameter, Ê The pion is really a quark–antiquark system. Bound states of heavier quarks and antiquarks, which are more slowly moving, can be more successfully described using nonrelativistic quantum mechanics.
- 32. 1.3 QUANTUM PHYSICS: AS A FUNDAMENTAL CONSTANT 11 namely Planck’s constant. Its first applications came in the understanding of some of the quantum aspects of the electromagnetic (EM) field and the particle nature of EM radiation. • In his investigations of the black body spectrum emitted from heated objects (so-called cavity radiation), Planck found that he could only fit the observed intensity distribution if he made the (then radical) assumption that the EM energy of a given frequency f was quantized and given by En(f ) = nhf where n = 0, 1, 2, 3 . . . (1.10) The constant of proportionality,h,was derived from a“fit”to the experimental data, and has been found to be h = 6.626 × 10−34 J · s (1.11) and is called Planck’s constant; we will far more often use the related form ≡ h 2π = 1.054 × 10−34 J · s = 6.582 × 10−16 eV · s = 6.582 × 10−22 MeV · s (1.12) which is to be read as “h-bar”. • Einstein assumed the energy quantization of Eqn. (1.10) was a more gen- eral characteristic of light, and proposed that EM radiation was composed of photonsË or“bundles”of discrete energy Eγ = hf . He used the photon concept to explain the photoelectric effect, and predicted that the kinetic energy of elec- trons emitted from the surface of metals after being irradiated should be given by 1 2mv2 max = Eγ − W = hf − W (1.13) where W is called the work function of the metal in question. Subsequent experiments were able to confirm this relation, as well as providing another, complementary measurement of h (P1.5) which agreed with the value obtained by Planck. • The relativistic connection between energy and momentum for a massless particle such as the photon could be used to show that it has a momentum given by pγ c = Eγ = hf = hc λ or pγ = h λ (1.14) Ë We use the notation γ (for gamma ray) to indicate a property corresponding to a photon of any energy or frequency.
- 33. 12 CHAPTER 1 A FIRST LOOK AT QUANTUM PHYSICS Figure 1.7. Geometry for Compton scattering. The incident photon scatters from an electron, initially at rest, at an angle θγ . ug Eg Ee ⬘ Eg ⬘ where λ is the wavelength. Arthur Compton noted that the scattering of X- rays by free electrons at rest could be considered as a collision process where the incident photon has an energy and momentum given by Eqn. (1.14), as shown in Fig. 1.7. Conservation of energy and momentum (P1.6) can then be applied to show that the wavelength of the scattered photon, λ, is given by the Compton scattering formula λ − λ = h mec (1 − cos(θγ )) (1.15) where θγ is the angle between the incident and scattered photon directions; X-ray scattering experiments confirmed the validity of Eqn. (1.15). The connection of Planck’s constant to the properties of material particles, such as electrons, came later: • Using yet another experimental “fit” to spectroscopic data, in this case the Balmer–Ritz formula for the frequencies in the spectrum for hydrogen, Bohr used semiclassical arguments to deduce that the angular momentum of the electron was quantized as L = n h 2π = n with n = 1, 2, 3 . . . (1.16) • Motivated by the dual wave-particle nature exhibited by light, for example, in Compton scattering, de Broglie suggested that matter, specifically electrons, would exhibit wave properties. He postulated that the relation λdB = h p (1.17)
- 34. 1.3 QUANTUM PHYSICS: AS A FUNDAMENTAL CONSTANT 13 apply to material particles as well as to photons, thereby defining the de Broglie wavelength. He could show that Eqn. (1.17) reproduced the Bohr condition of Eqn. (1.16), and thus explain the hydrogen atom spectrum. Example 1.1. de Broglie Wavelength of a Truck? Over the roughly 80 years since the Davisson–Germer experiment4 directly demonstrated the wave nature of electrons by the observation of the diffraction of electron beams from nickel crystals, with a wavelength consistent5 with Eqn. (1.17), the quantum mechanical wave-particle duality of objects of increasing size and complexity has been observed. Only 3 years after the prediction by de Broglie, Davisson and Germer accelerated electrons through voltages of order V ∼ 50 V to speeds given by 1 2 mev2 = eV −→ v = 2eV mec2 c = 100 eV 0.51 MeV c ≈ 0.015 c (1.18) which is still nonrelativistic and gives a de Broglie wavelength of λ = h/mv ≈ 1.7 Å, which nicely matched the atomic spacings in their sample (already determined by X-ray scattering experiments). It is sometimes useful to compare the quantum mechanical wavelength of a particle to other physical dimensions,including its own size.While many particles which play a crucial role in determining the structure of matter have finite and measurable sizes, all ultrahigh energy scattering experiments involving electrons (which therefore probe ultra-small distance scales) are so far consistent with the electrons having no internal structure; various experiments can be interpreted as putting upper limits on an electron “size” of order 10−10 Å = 10−5 F or roughly 50, 000 times smaller than a proton or neutron. This justifies the assumption of a “point-like” electron. Sixty years after the Davisson and Germer experiments with electrons, single- and double- slit diffraction of slow neutrons was observed, giving the “most precise realization hitherto for matter waves.”6 In this case,the neutrons have a physical size measured (in other experiments) to be of order 1 F = 10−5 Å and ultracold neutrons with λ = 15 − 30 Å were utilized, so that the spatial extent of the particle is still orders-of-magnitude less than its quantum mechanical wave length. In the last decade or so, however, advances in atom interferometry have led to the observation of interference or diffraction phenomena for small atoms (helium, He), larger atoms (atomic sodium, Na), diatomic molecules (sodium dimer or Na2), small clusters of molecules (of H, He2, and D2), and most recently C60 molecules (buckeyballs), all of atomic dimensions, and with increasingly small de Broglie wavelengths. Representative data (and references) are collected below. 4 See Davisson and Germer (1927). 5 Their exact words are “The equivalent wave-lengths of the electron beams may be calculated from the diffraction data in the usual way. These turn out to be in acceptable agreement with the values of h/mv of the undulatory mechanics.” 6 Zeilinger et al. (1988).
- 35. Exploring the Variety of Random Documents with Different Content
- 36. The text on this page is estimated to be only 28.95% accurate xlii INTRODUCTION or Unitarians, many of whom consequently went over to the Collegiants. After all, then, the decree of toleration embodied in the Union of Utrecht did not secure very much in the way of real toleration. Non-Calvinist Christians were allowed to live in the Netherlands without suffering in person or pro perty on account of their nonconformity. For those days even that was a great deal ; but the right of public worship was quite another matter. And if the Union of Utrecht did not secure real toleration for all Christian sects, much less did it guarantee anything to the Jews, who had not been contemplated in it at all, who had not even been formally admitted into the Netherlands, but whose presence had been more or less connived at. Even in 1619, when the Jewish question was definitely raised in the Netherlands, it was decided to allow each city to please itself whether it would permit Jews to live there or not. Their position was precarious indeed. They had to take care not to give offence to the religious susceptibilities of their neighbours. And their troubles commenced soon enough. About the year 1618 there had arrived in Amsterdam a Marano refugee from Portugal whose name was Gabriel da Costa. Both he and his late father had held office in the Catholic Church, but seized by a sudden longing to return to the religion of his ancestors, Gabriel fled to Amsterdam, where he embraced Judaism and changed his name from Gabriel to Uriel. His ideas about Judaism had been derived chiefly from reading the Old Testament, and his contact with actual Rabbinic Judaism somewhat disappointed him. He thereupon commenced to speak contemptuously of the Jews as Pharisees, and aired his views very freely against the belief in the immortality of the soul, and the inspiration of the Bible. These views were, of course, as much opposed to Christianity as to Judaism. The Jewish physician, de Silva, as already stated, tried to controvert these heretical
- 37. The text on this page is estimated to be only 28.40% accurate THE LIFE OF SPINOZA xliii views in a book published in 1623. Da Costa replied, in 1624, with a treatise which was very confused, and which, while accusing de Silva of slander against the author, actually reiterated those heresies. Partly from fear that an outcry might be raised against the Jews as promulgators of heresy, the Jewish authorities excommunicated Uriel da Costa, and as a kind of official repudiation of all responsi bility for him, they communicated the facts to the civil authorities, who thereupon imprisoned him, fined him, and ordered his book to be burned. Shunned by Jews and Christians alike, da Costa found his existence very lonely and intolerable, and in 1633 he made up his mind, as he said, to become an ape among apes/' and made his peace with the Synagogue. But he soon got quite reckless again, and was excommunicated a second time. Again he grew weary of his isolation, and once more he approached the Synagogue authorities for the removal of the ban. Deter mined not to be duped again, yet reluctant to repel him absolutely, they imposed hard conditions on him. He sub mitted to the conditions — he recanted his sins publicly in the Synagogue, received thirty-nine lashes, and lay pros trate on the threshold of the Synagogue while the congrega tion stepped over him as they passed out. It was a cruel degradation. And so heavily did his humiliation weigh on his mind that he committed suicide soon afterwards. This happened in 1640, and Spinoza must have remembered the scandal. If the Jewish community in Amsterdam felt it necessary to repudiate, in such drastic manner, their responsibility for Uriel da Costa' s heresies, so as to avoid giving offence to their Christian neighbours, there was every reason why they should feel even greater discomfort on account of Spinoza's heresies in 1656. It was a critical period in the annals of Jewish history. During the Muscovite and Cossack inva sion of Poland (1654-1656) entire Jewish communities 'were
- 38. The text on this page is estimated to be only 28.63% accurate xliv INTRODUCTION massacred by the invaders ; nor did the Poles behave much better towards the Jews during the war. Naturally, whoso ever could tried to escape from the scene of slaughter. There was consequently a considerable influx of Polish Jews into Amsterdam. Now, even in the twentieth century, when countless missionaries are sent to spread the Gospel from China to Peru, Jewish refugees have been shown but scant Christian charity under similar circumstances, so we have every reason to suppose that the condition of the Amsterdam Jewish community did not gain in security through this influx of destitute refugees. Then more than ever was it necessary to be circumspect, and avoid giving offence to the people of the land, especially in the matter of the most delicate of all things — religion.* They did not want another scandal. One da Costa affair was enough, and more than enough. Yet they must not incur the responsibility for Spinoza's heresies. So at first they tried to bribe Spinoza. They promised him a considerable annuity if he would only keep quiet, and show some amount of outward conformity to his religion. They must have known well enough that silence and partial outward conformity do not alter a man's views ; they were surely shrewd enough to realise that a heretic does not cease to be a heretic by becoming also a hypocrite. If their sole object had been to suppress heresy in their midst, that was not the way to gain their end. Heresy would not languish through becoming profitable. The real motive that prompted them must have been that just indicated — though it is very likely that they did not realise it so explicitly. If they had done so, and if they had urged these points on Spinoza, he would, undoubtedly, have appreciated the need for caution and silence. But they evidently did not understand him, they evidently misconceived his character entirely, and the * That their apprehensions were not unfounded is clear from the fact that even some twenty years afterwards various Synods of the Reformed Church tried to induce the civil powers to pass strong measures for the forcible ebnversion of the Jews.
- 39. The text on this page is estimated to be only 28.25% accurate THE LIFE OF SPINOZA xlv attempt to gag him with a bribe was just the way best cal culated to defeat their end. The only person who might have understood him, and whose intervention might have been successful, was Manasseh ben Israel. But he was in England then, on a mission to Cromwell. So threats were tried next ; but the threat of excommunication had no effect on Spinoza. They had reached the end of their tether. The only course open to them, as they felt, was to put him under the ban. The feeling against him was, no doubt, so strong that a fanatic might have tried to do him some physical violence. And it may be that such an attack gave rise to the story of an attempt to assassinate Spinoza with a dagger, as he was leaving the Synagogue or the theatre. But there is no evidence of this, and the probability is decidedly against it. Some time in June 1656 Spinoza was summoned before the court of Rabbis. Witnesses gave evidence of his here sies. Spinoza did not deny them — he tried to defend them. Thereupon he was excommunicated for a period of thirty days only — in the hope that he might still relent. But he did not. Accordingly, on the 2jth July 1656, the final ban was pronounced upon him publicly in the Synagogue at Amster dam. It was couched in the following terms : The members of the council do you to wit that they have long known of the evil opinions and doings of Baruch de Espinoza, and have tried by divers methods and promises to make him turn from his evil ways. As they have not succeeded in effecting his improve ment, but, on the contrary, have received every day more informa tion about the horrible heresies which he practised and taught, and other enormities which he has committed, and as they had many trustworthy witnesses of this, who have deposed and testified in the presence of the said Spinoza, and have convicted him ; and as all this has been investigated in the presence of the Rabbis, it has been resolved with their consent that the said Espinoza should be anathe matised and cut off from the people of Israel, and now he is
- 40. anathematised with the following anathema : ' With the judgment of the angels and with that of the saints, with
- 41. The text on this page is estimated to be only 28.32% accurate xlvi INTRODUCTION the consent of God, Blessed be He, and of all this holy congrega tion, before these sacred Scrolls of the Law, and the six hundred and thirteen precepts which are prescribed therein, we anathematise, cut off, execrate, and curse Baruch de Espinoza with the anathema wherewith Joshua anathematised Jericho, with the curse wherewith Elishah cursed the youths, and with all the curses which are written in the Law: cursed be he by day, and cursed be he by night ; cursed be he when he lieth down, and cursed be he when he riseth up ; cursed be he when he goeth out, and cursed be he when he cometh in ; the Lord will not pardon him ; the wrath and fury of the Lord will be kindled against this man, and bring down upon him all the curses which are written in the Book of the Law; and the Lord will destroy his name from under the heavens; and, to his undoing, the Lord will cut him off from all the tribes of Israel, with all the curses of the firmament which are written in the Book of the Law ; but ye that cleave unto the Lord your God live all of you this day!' We ordain that no one may communicate with him verbally or in writing, nor show him any favour, nor stay under the same roof with him, nor be within four cubits of him, nor read anything com posed or written by him. This amiable document of the holy congregation is nothing less than a blasphemy. It must be remembered, however, that the actual anathema was a traditional formula, and (unlike the preamble and conclusion) was not specially written for the occasion. No doubt it shows a greater familiarity with the phraseology of the Bible than with its best teaching. But the Jews who excommunicated Spinoza were no worse than their neighbours in this respect. These awful curses were but the common farewells with which the churches took leave of their insubordinate friends. Nor were these the worst forms of leave-taking, by any means. After all, swearing breaks no bones, and burns none alive, as did the rack and the stake which were so common in those days. The Catholic Church excommunicated only
- 42. when it could not torture and kill ; and then its ana themas, though they may have been more polished in diction,
- 43. The text on this page is estimated to be only 28.82% accurate THE LIFE OF SPINOZA xlvii were incomparably more brutal in effect. The ban pronounced upon William the Silent, for instance, contained nothing less than an urgent invitation to cut-throats that they should murder him, in return for which pious deed they would receive absolution for all their crimes, no matter how heinous, and would be raised to noble rank ; and that ban actually accomplished its sinister object! It is, therefore, unjust to single out this ban against Spinoza and judge it by presentday standards. Nor should it be forgotten that if Judaism alone had been concerned, more leniency would have been shown, the whole thing might have been ignored. Elisha ben Abuyah, the Faust of the Talmud, was not persecuted by the Jews, in spite of his heresies. The ban against Spinoza was the due paid to Caesar, rather than to the God of Israel. As in the case of da Costa, and for the same reasons, the Jewish authorities officially communicated the news of Spinoza's excommunication to the civil authorities, who, in order to appease the wrath of the Jewish Rabbinate and the Calvinist clergy, banished Spinoza from Amsterdam, though only for a short period. On the whole there is some reason to suppose that the anathema was not a curse, but a blessing in disguise. It freed him entirely from sectarian and tribal considera tions ; it helped to make him a thinker of no particular sect and of no particular age, but for all men and for all times. However reprehensible his heretical utterances arid un orthodox doings may have been considered by some of his fellow- Jews, yet there can be no doubt that Spinoza did not really desire to sever his connection entirely with them. This is evident from the fact that he did not ignore, as he might have done, the summons to come before the court of Rabbis in order to defend himself against the charge of heresy. It is true that when informed of his final excom munication he is reported to have said : Very well, this
- 44. The text on this page is estimated to be only 28.71% accurate xlviii INTRODUCTION does not force me to do anything which I would not have done of my own accord, had I not been afraid of a scandal. But the last words of this expression of his natural resentment only seem to confirm the suggestion about his previous anxiety to avoid a complete rupture, if he could do so honestly. It was partly perhaps also for this reason that even after his excommunication he addressed to the Synagogue authorities an Apology (written in Spanish) in which he probably sought to defend his heretical views by showing that they had the support of some of the most eminent Rabbis, and to condemn the iniquity of fastening on him horrible practices and other enormities because of his neglect of mere ceremonial observances. Unfortu nately, this document has not yet been recovered, though some of its contents are said to have been subsequently in corporated in his Tractatus Theologico-Politicus. It would throw a flood of light on Spinoza's mental history. How ever, the Apology did not mend matters. Cut off from his community, without kith or kin, he stood alone, but firm and unshaken. Unlike da Costa, he never winced. He seems to have got into touch with Jews again afterwards ; but it was they who had to seek him. §5. THE LAST YEARS OF SPINOZA'S STAY IN AND NEAR AMSTERDAM— 1656-1660 Banished from Amsterdam, Spinoza went to live in Ouwerkerk, a little village to the south of Amsterdam. Possibly he had some Christian friends there who had influence with the civil authorities ; and apparently he meant to return to Amsterdam at the earliest opportunity. Maybe also he was not altogether uninfluenced by the thought that the Jewish cemetery was there, and that his mother, his sister, his father,
- 45. The text on this page is estimated to be only 28.93% accurate THE LIFE OF SPINOZA xlix and others once dear to him, had found their last restingplace in it. For his support he had to rely on the lenses which he made — an art which he had mastered during the years immediately preceding his exile. He made lenses for spec tacles, microscopes, and telescopes, and his friends sold them for him. The work suited his tastes well enough, be cause it kept him in touch with his scientific studies. And he evidently excelled in it, for later on his fame as an optician attracted the notice of Huygens and Leibniz, among others. But it was an unfortunate occupation otherwise. The fine glass-dust which he inhaled during his work must have been very injurious to his health, especially when we bear in mind that he eventually died of consump tion, and that he probably inherited the disease from his mother, who died so young. For the time being, however, it was a congenial occupation, and, with his frugal habits, left him sufficient time to pursue his scientific and philo sophic studies. As already suggested, Spinoza did not stay long in Ouwerkerk, but returned, after a few months, to Amsterdam, where he remained till 1660. Of the events which happened during this period (1656-1660) we possess the most meagre information. Apparently he gave some private lessons in philosophy, and pursued his studies unremittingly. At the end of this period he had already left Descartes behind him, and had thought out the essentials of his own philosophy. From Spinoza's subsequent correspondence, we obtain a glimpse of his friends and associates during this period, while the opening pages of his Improvement of the Under standing at once enlighten and mystify us about his life during those last years in Amsterdam. After leaving Amsterdam in 1660 Spinoza continued a friendly correspondence with several residents in Amster dam, whom he also visited for a short time in 1663. These
- 46. The text on this page is estimated to be only 28.49% accurate 1 INTRODUCTION correspondents must therefore have been known to him during his stay in Amsterdam, and what is known about them helps to throw some light on this obscure period in Spinoza's life-history. They were Pieter Balling, Jarig Jelles, Dirck Kerckrinck, Lodewijk Meyer, Simon Joosten de Vries, and Jan Rieuwertsz. Pieter Balling had acted for some time as the representa tive or agent of various Spanish merchants. And it is just possible that Spinoza's knowledge of Spanish first brought him into touch with him. Balling was a Mennonite, and a pronounced enemy of dogmatism. In 1662 he published a book entitled The Light on the Candlestick, in which he attacked religion based on stereotyped dogmas, and advo cated a religion, partly rationalistic, partly mystical, based on the inward light of the soul. The whole spirit of the book might be summed up in the familiar lines of Matthew Arnold : These hundred doctors try To preach thee to their school. We have the truth, they cry. And yet their oracle, Trumpet it as they will, is but the same as thine. Once read thy own breast right, And thou has done with fears. Man gets no other light, Search he a thousand years. Sink in thyself: there ask what ails thee, at that shrine. In 1664 he translated into Dutch Spinoza's version of Descartes' Principia. In a letter written in the same year, we see Spinoza trying to console Balling on the loss of his child, and dealing tenderly with Balling's premonitions of his impending loss. Jarig Jelles was at one time a spice-merchant in Amster dam, but feeling that knowledge is better than choice gold, that wisdom is better than rubies, and all the things
- 47. The text on this page is estimated to be only 28.90% accurate THE LIFE OF SPINOZA li that may be desired are not to be compared to her/' he left his business in the charge of a manager, and devoted him self to study. He wrote a book to show that Cartesianism did not lead to atheism, but was, on the contrary, quite compatible with the Christian religion. Spinoza seems to have helped him in the composition of this book. Jelles was one of the friends who persuaded Spinoza to publish his version of Descartes' Principia, and even defrayed the cost of its publication. He also took an active share in the publication of Spinoza's posthumous works, the preface to which is so similar in tone to the book of Jelles that he is regarded as its author by some very competent authorities. Dirck Kerckrinck was seven years younger than Spinoza, whom he first met at Van den Enden's school (? 1652-6). He studied medicine, and became the author of various medical treatises. Colerus relates some gossip to the effect that Spinoza and Kerckrinck were rivals for the hand of Clara Maria, the gifted daughter of Van den Enden, and that she accepted Kerckrinck because he was rich, while Spinoza was poor. But as Clara Maria was born in 1644, this very natural attempt to introduce a touch of romance into Spinoza's life of single blessedness is an utter failure. Clara Maria was barely sixteen when Spinoza left Amster dam for good in 1660, and he had ceased to be her father's pupil in 1654 or, at the latest, in 1656. As an inmate in her father's house he may have been fond of her as a mere child, and some expression of endearment uttered in that sense probably gave rise to this pretty tale. It is true, how ever, that Kerckrinck did marry her in 1671, as already mentioned. Spinoza possessed several of the medical works of Kerckrinck, who had, no doubt, sent them to him as an old friend of his. Lodewijk Meyer was a medical practitioner in Amsterdam. He was about two years older than Spinoza, and a man of versatile talents. He had studied not only medicine but
- 48. The text on this page is estimated to be only 28.85% accurate Hi INTRODUCTION also philosophy and theology, made his bid as poet and dramatist, lexicographer and stage -manager, and was the moving spirit in a certain literary society, the name and motto of which was (as we need scarcely be surprised to hear) Nil volentibus arduum. It was he who wrote the interesting preface to Spinoza's version of Descartes' Principia. Simon Joosten de Vries was an Amsterdam merchant. He was only about a year younger than Spinoza, though his attitude towards Spinoza was always that of a humble disciple. He studied medicine under the direction of Spinoza, and his attachment to Spinoza is evident from a letter of his written in 1663, after Spinoza had left Amster dam. For a long time, he writes, I have been longing to be with you ; but the weather and the hard winter have not been propitious to me. Sometimes I complain of my lot in being removed from you by a distance which separates us so much. Happy, most happy, is your companion Casearius, who lives with you under the same roof, and who can converse with you on the most excellent topics during dinner, or supper, or on your walks. But although we are so far apart in the body, yet you have constantly been present to my mind, especially when I take your writings in my hand, and apply myself to them. In the same letter he reports about a philosophical society for the study of Spinoza's philosophy, as communicated to de Vries and others in manuscript form, and asks for further elucidation of some difficult points. The sincerity and extent of his devotion was further shown by his offer of a gift of 2000 florins to Spinoza, which was, however, declined. Later on, Simon de Vries, whose health was even less satisfactory than Spinoza's, feeling that his end was drawing nigh, desired to make Spinoza his heir. Again the philosopher dissuaded him, urging the prior claims of the testator's own kindred. On the death of Simon de Vries his brother
- 49. The text on this page is estimated to be only 28.42% accurate THE LIFE OF SPINOZA liii offered to Spinoza an annuity of 500 florins, but Spinoza declined to take more than 300 florins. Jan Rieuwertsz was a bookseller at Amsterdam, and some fifteen years older than Spinoza. He was a Collegiant, and very liberal in his views. His shop enjoyed the evil reputation of being the seat of scoffers. He published and stocked the works of many authors of unorthodox repute, including those of Descartes, Balling, Jelles, and Spinoza. His son also was a devoted admirer of Spinoza. Such were some of the men with whom Spinoza stood in friendly relationship during his last years in Amsterdam. Further details are wanting. Possibly he had given private tuition to Simon de Vries (who speaks of him as master ), Balling, and others ; or he may have held some kind of seminar or class for the informal discussion of religious and philosophical questions. If so, the substance of his Meta physical Thoughts (which were subsequently appended to his version of Descartes' Principia) and of his Short Treatise on God, Man and his Well-being must have been elaborated during these years, and for these purposes. This would also account for the continuation or revival of similar meetings for the discussion of Spinoza's views, as reported in the letter of Simon de Vries. Little as is known of these years, there can be no doubt that they were years of storm and stress in the mental history of Spinoza. This much may be gathered from the impres sive pages with which he opens his Treatise on the Improve ment of the Understanding. After experience had taught me [so he writes] that all things which are ordinarily encountered in common life are vain and futile, and when I saw that all things which occasioned me any anxiety or fear had in themselves nothing of good or evil, except in so far as the mind was moved by them ; I at length determined to inquire if there were anything which was a true good capable of im parting itself, and by which alone the mind could be affected to the
- 50. The text on this page is estimated to be only 28.66% accurate liv INTRODUCTION exclusion of all else ; whether, indeed, anything existed by the dis covery and acquisition of which I might have continuous and supreme joy to all eternity, I say that / at length determined : for at first sight it seemed unwise to be willing to let go something certain for something that was yet uncertain. I saw, forsooth, the advantages which are derived from honour and riches, and that I should be obliged to abstain from the quest of these if I wished to give serious application to something different and new : and if, perchance, supreme happiness should lie in them, I saw clearly that I should have to do without it ; but if, on the other hand, it did not lie in them, and I applied myself only to them, then I should also have to go without the highest happiness. I, therefore, re volved in my mind whether, perchance, it would not be possible to arrive at the new plan of life, or, at least, some certainty about it, without any change in the order and usual plan of my life, a thing which I have often attempted in vain. Now the things which one mostly meets with in life, and which, so far as one may gather from their actions, men esteem as the highest good, are reducible to these three, namely, riches, honour, and pleasure. By these three the mind is so distracted that it can scarcely think of any other good. . . . When, therefore, I saw that all these things stood in the way of my applying myself to any new plan of life; in fact, that they were so opposed to it that one must necessarily abstain either from the one or from the other, I was forced to inquire which would be the more useful to me; for, as I have already said, I seemed to be willing to let go a sure good for something uncertain. But after brooding a little over this subject I found, in the first place, that if I let go those things and devoted myself to the new plan of life I should be letting go a good uncertain by its very nature ... for one which was uncertain, not in its nature . . . but only as regards its attainment. After unremitting reflection I came to see that, if I could only make up my mind thoroughly, then I should give up sure evils for a sure good. . . . Not with out reason did I use the words, if I could only
- 51. make up my mind thoroughly. For although I saw this so clearly in my mind, yet I could not thus put aside all avarice, sensuous pleasure, and the desire for fame. This one thing I saw, that so long as my mind revolved these thoughts, so long, did it turn away from those things, and consixjer seripusly the new plan of life. This was .a great comfort
- 52. The text on this page is estimated to be only 28.19% accurate THE LIFE OF SPINOZA Iv to me. . . . And although at first these periods were rare and only of very brief duration, yet as the true good gradually became better known to me so these periods grew more frequent and longer. The above confession was written by Spinoza in 1661. The inner struggle between worldly allurements and the beck of the spirit was over then. Indeed already his earlier work, the Short Treatise, which was completed in 1660, bears unmistakable evidence of the peace which crowned that inward conflict. This conflict must therefore be referred to the years immediately preceding 1660. His last years in Amsterdam, when he made his first acquaintance with real life and the struggle for existence, must have brought home to him often enough the desirableness of worldly goods, and the hardships of poverty and obscurity. After all, he was human, and he could scarcely escape the common lot of mortals — the conflict between the two souls which dwell in mortal breast. But Spinoza was not given to speak about himself. He lifts but a corner of the veil, behind which we may well conjecture scenes of storm and stress during the period intervening between his excommunication in 1656 and his departure from Amsterdam in 1660. Early in that year, weary of the whir and the worldliness of that com mercial centre, he went to dwell among unworldly folk with old-world virtues in an out-of-the-world village — Rijnsburg. He withdrew from the madding crowd, but not in disgust or misanthropy. He had caught a glimpse of the highest good of man, and he wanted to strengthen his hold thereon under more favourable conditions. He had discovered that the sorrows of man arise from the love of the transient, while love for an object eternal and infinite feeds the mind with unmixed joy, free from all sorrow.
- 53. The text on this page is estimated to be only 28.33% accurate Ivi INTRODUCTION §6. SPINOZA'S STAY IN RIJNSBURG— 1660-1663 Rijnsburg is a village some six or seven miles north-west of Leyden. Its modest cottages, narrow lanes, quiet water ways, and quaint medieval church still present an old-world appearance very much as in the days of Spinoza — except, of course, for the clumsy, though convenient, steam-trams which pass by on their way to and from Leyden and Katwijk — or Noordwijk-aan-Zee. Within easy walking dis tance from it, on the road to Leyden, is Endgeest, a nice rural little place where Descartes once stayed for a number of years, but now noted chiefly for its lunatic asylum. During the seventeenth century Rijnsburg was the head quarters of the Collegiants. This sect, whose origin has already been explained above, repudiated infant baptism, and insisted on adult baptism by immersion. And Rijnsburg, on the old Rhein, was their place of baptism. That was the reason why the Collegiants were also commonly called the Rijnsburgers. Now Spinoza, as we have seen, numbered several Collegiants among his friends, and it was probably on the suggestion of one of his Collegiant friends that he went to live there. At all events, early in the year 1660 he seems to have taken up his quarters there, probably with a surgeon of the name Hermann Homan, in a newly built little cottage standing in a narrow lane, which has since then come to be known as Spinoza Lane. Some time after wards, apparently, the landlord's pious humanitarianism led him to inscribe or to have inscribed on a stone in the cottage wall the well-meant message expressed in the concluding stanza of Kamphuyzen's May Morning : Alas ! if all men would be wise, And would be good as well, The Earth would be a Paradise, Now it is mostly Hell.
- 54. The text on this page is estimated to be only 29.02% accurate THE LIFE OF SPINOZA Ivii And it was by this inscription that, on the authority of an old tradition, the cottage has been identified. It is still in existence, and is still surrounded by open fields rich in garden produce and bulbs. Restored, and equipped with all that diligent search could find and that money could procure in the way of things interesting to students of Spinoza, the cottage is now known as the Spinoza-huis or Spinoza Museum, and serves as a kind of shrine sacred to the memory of the philosopher, and many pilgrims bend their footsteps there to pay homage to a profound mind and lofty character, and feel something of his calm of mind in that haunt of ancient peace. One reason which prompted Spinoza to seek a quiet retreat was probably the desire to write down his thoughts in some systematic form. Dissatisfied with the Scholastic philosophy still in vogue then, he and his friends had turned eagerly to the writings of Descartes. The opposition of the strict Calvinists to the Cartesian philosophy rather tended to recommend it to the Remonstrants (including the Collegiants), and, indeed, to all who had suffered from, or were opposed to, the religious intolerance of the dominant Reformed Church. The cry for impartiality and an open mind in the interpretation of Scripture was felt to have a certain kinship with the Cartesian method of philosophising, his preliminary doubt of whatever could be reasonably disputed. Hence there was a gradual coalition between liberal religion and Cartesian philosophy. Spinoza's friends were mostly Cartesians, and remained such to the end. Whether Spinoza himself was ever a thoroughgoing Cartesian is not known. That Descartes' writings exercised a very potent influence on Spinoza there is no doubt what ever. By 1660, however, Spinoza had already outgrown the fundamentals of Cartesian Metaphysics, though he still con tinued to follow Descartes in his Physics. Now we have already remarked that, during his last years in Amsterdam,
- 55. The text on this page is estimated to be only 28.66% accurate Iviii INTRODUCTION Spinoza seems to have acted as teacher or leader of a small philosophical circle. Its members, including Spinoza him self, were primarily interested in religious questions at first. They approached philosophy from the side of religion, and only in so far as religious problems led up to philosophical considerations. God and His attributes, Nature and Crea tion, Man and his Well-being, the nature of the Human Mind and the Immortality of the Soul — these were the topics which chiefly interested them, and on which, we may assume, Spinoza had accumulated various notes for those informal talks with them. These notes he wanted to elaborate and to systematise. This was the first task which occupied him at Rijnsburg, and it resulted in the Short Treatise on God, Man and his Well-being. But he continued to keep in touch with his Amsterdam friends and sent them the parts of the manuscript as he completed them. Though Cartesian in appearance, and partly also in substance, the Short Treatise, Spinoza's very first philosophical essay, already marks a definite departure from the philosophy of Descartes, in its identification of God with Nature, and its consequent determinism and naturalism. Spinoza himself fully realised the extent of his deviation from Descartes, and the novelty of his views even as compared with the novelties of Cartesianism, which was at that time the new philo sophy par excellence. So he begged his friends not to be impatient with his novel views, but to consider them carefully, remembering that a thing does not therefore cease to be true because it is not accepted by many. He also realised that some of these views were liable to prove rather dangerous to minds more eager for novelty than for truth. He was therefore careful about the kind of people to whom he communicated his views, and also begged his trusted friends to be careful likewise. Caution was also necessary on account of the unremitting vigilance of heretic-hunters.
- 56. The text on this page is estimated to be only 21.91% accurate [Conclusion of Spinoza's letter to Oldenburg. October 1661. S« p. cxxiii.]
- 57. The text on this page is estimated to be only 28.63% accurate THE LIFE OF SPINOZA Ixi As the character of the age in which we live [Spinoza adds] is not unknown to you, I would beg of you most earnestly to be very careful about the communication of these things to others. I do not want to say that you should absolutely keep them to yourselves, but only that if ever you wish to communicate them to others, then you shall have no other object in view except only the happiness of your neighbour ; being at the same time clearly assured that the reward of your labour will not disappoint you therein. Having finished the first draft of his Short Treatise Spinoza felt that he had attacked all the great problems of religion and of philosophy, without any preliminary account of the requirements of philosophic method, without any adequate justification of his own mode of treatment. To this problem, accordingly, he turned his attention next, and began his Treatise on the Improvement of the Under standing. In a letter dated October 1661, in reply to some questions of Henry Oldenburg, Spinoza states that he had written a complete little treatise on the origin of things, and their relation to the first cause, and also on the improvement of the understanding, and that he was actually busy just then copying and correcting it. It would appear from this that Spinoza's intention at that time may have been to combine the Short Treatise and the Treatise on the Improve ment of the Understanding. What actually happened, how ever, is not quite certain. The editors of Spinoza's posthumous works only had a fragment of the Treatise on the Improvement of the Understanding, and apparently nothing of the Short Treatise, of which we only possess at present two Dutch versions, discovered about 1860. The editors of the Opera Posthuma say that the Treatise on the Improvement of the Understanding was one of Spinoza's earliest works, and that he had never finished it, but they appear to be uncertain whether it was only want of time or the inherent difficulties of the subject which prevented him from finishing it.
- 58. The text on this page is estimated to be only 28.66% accurate Ixii INTRODUCTION In the meantime Spinoza seems to have acquired some reputation not only with the Rijnsburgers but even among some of the professors at Leyden. This may have been due to his participation in the Collegiant Conferences held at Rijnsburg. These conferences for the discussion of religious questions were open to all who cared to come. And some of the students from the neighbouring University at Leyden made a practice of attending these meetings and taking part in the debates. Some of them very likely came there for fun, though others, no doubt, had worthier motives. It was in this way that Spinoza came into touch with, among others, Johannes Casearius and the brothers Adriaan and Johannes Koerbagh, of whom more will be said anon. And in this way also Spinoza's name and history may have become known to some of the Leyden professors, among them Johannes Coccejus, professor of theology, famous as the author of the first standard Hebrew dictionary, and even more so as the author of the dictum that an interpreter of the Scriptures should approach his task with a mind free from all dogmatic prejudices — the dictum which helped to bring about a kind of alliance between the Remonstrants and the Cartesians, to which reference has already been made. Now Coccejus was a native of Bremen, and when his countryman Henry Oldenburg visited Leyden in 1661, eager as usual to make the acquaintance of everybody who was remarkable in any way, Coccejus may have suggested to him a visit to Spinoza. Possibly Oldenburg had also heard something about Spinoza from Huygens, who was in correspondence with the English scientists among whom Oldenburg had moved, had actually visited London that very year, and may have met Oldenburg at one of the meetings of the Gresham College, which was soon to blossom into the Royal Society. At all events, in July 1661 Oldenburg visited Spinoza in Rijnsburg.
- 59. The text on this page is estimated to be only 28.36% accurate THE LIFE OF SPINOZA Ixiii Henry Oldenburg, as already remarked, was a native of Bremen, where he was born about 1620. During the war between England and Holland which followed Cromwell's enforcement of the Navigation Act, in 1651, the shipowners of Bremen seem to have suffered. It was therefore decided to send an envoy to make representations to Cromwell con cerning the neutrality of Bremen. Accordingly in 1653 Henry Oldenburg was entrusted with this diplomatic mis sion, which brought him into touch with Milton, who was then Latin Secretary to the Council, and other eminent Englishmen of the time. For some reason he remained in England after the conclusion of his mission, staying in Oxford in 1656, and acting as private tutor to various young gentlemen, including Boyle's nephew, Richard Jones, with whom he travelled in France, Germany, and Italy, during the years 1657-1660, attending the most important academies of science, and becoming acquainted with the great lights of the scientific world. During his stay in Oxford, Olden burg had been associated with the leading spirits of the Invisible College, a society for the discussion of scientific problems. There was a similar society in London, the Gresham College. With the Restoration of Charles II., in 1660, it was decided to apply for a Charter for the formation of a Royal Society to carry on the work of these two societies, and an acting secretary was required to undertake the work of organisation, c. Just then Olden burg returned from his continental tour, and his wide reading and extensive knowledge of men and matters marked him out as just the man for the post, for which he was accordingly nominated. In the following year, 1661, Oldenburg had occasion to visit his native town, Bremen, and on his return journey via Holland, he visited Leyden (among other places), and thence Rijnsburg, where, as already mentioned, he had a long interview with Spinoza. The subject discussed on that occasion and the impres
- 60. The text on this page is estimated to be only 29.04% accurate Ixiv INTRODUCTION sion which Spinoza made on Oldenburg may be gathered from the following letter which Oldenburg wrote to Spinoza immediately after his return to London, in August 1661. It was with such reluctance [he writes] that I tore myself away from your side, when I recently visited you in your retreat at Rijnsburg, that no sooner am I back in England than I already try to join you again, at least as far as this can be effected by means of correspondence. Solid learn ing combined with kindliness and refinement (wherewith Nature and Study have most richly endowed you) have such an attraction that they win the love of all noble and liberally educated men. Let us, therefore, most excellent sir, give each other the right hand of unfeigned friendship, and cultivate it diligently by every kind of attention and service. Whatever service my humble powers can render, consider as yours. And permit me to claim a part of those intellectual gifts which you possess, if I may do so without detriment to you. Our conversation at Rijnsburg turned on God, infinite Extension and Thought, on the difference and the agreement between these attributes, on the nature of the union of the human soul with the body ; and further, on the Principles of the Cartesian and the Baconian Philo sophy. But as we then discussed themes of such moment only at a distance, as it were, and cursorily, and as all those things have since then been lying heavily on my mind, I now venture to claim the right of our new friendship to ask you affectionately to explain to me somewhat more fully your views on the above-mentioned subjects, and not to mind enlightening me, more especially on these two points, namely, first, what do you consider to be the true distinc tion between Extension and Thought ; secondly, what defects do you observe in the Philosophy of Descartes and of Bacon, and how, do you think, might they be eliminated, and replaced by something more sound ? The more freely you write to me about these and the like, the more closely
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