Lectures on Quantum Mechanics for Mathematics Students www.com STUDENT MATHEMATICAL LIBRARY Volume 47 Lectures on Quantum Mechanics for Mathematics Students L. Yakubovskii Translated by Harold McFaden AMS American Mathematical Society www.com Editorial Board Gerald B. Osgood (Chair) Robin Forman Michael Starbird The cover graphic was generated by Matt Strassler with help from Peter Skands. Processed through CMS by Albert De Roeck, Christophe Saout and Joanna Weng.
Visualized by Ianna Osborne. 2000 Mathematics Subject Classification. For additional information and updates on this book, visit www.org/bookpages/stml-47 Library of Congress Cataloging-in-Publication Data Faddeev, L. [Lektsii po kvantovoi mekhanike dlia studentov-matematikov.
English] Lectures on quantum mechanics for mathematical students / L. - (Student mathematical library ; v. Iakubovskii, Oleg Aleksandrovich.12-dc22 2008052385 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research.
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The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http:1/wv.com Contents Preface ix Preface to the English Edition xi P. The algebra of observables in classical mechanics 1 §2. Liouville's theorem, and two pictures of motion in classical mechanics 13 §4.
Physical bases of quantum mechanics 15 §5. A finite-dimensional model of quantum mechanics 27 §6. States in quantum mechanics 32 §7. Heisenberg uncertainty relations 36 §8.
Physical meaning of the eigenvalues and eigenvectors of observables 39 §9. Two pictures of motion in quantum mechanics. The Schrodinger equation. Quantum mechanics of real systems.
The Heisenberg commutation relations 49 §11. Coordinate and momentum representations 54 §12. "Eigenfunctions" of the operators Q and P 60 §13. The energy, the angular momentum, and other examples of observables 63 v www.com vi Contents §14.
The interconnection between quantum and classical mechanics. Passage to the limit from quantum mechanics to classical mechanics 69 §15. One-dimensional problems of quantum mechanics. A free one-dimensional particle 77 § 16.
The harmonic oscillator 83 §17. The problem of the oscillator in the coordinate representation 87 § 18. Representation of the states of a one-dimensional particle in the sequence space 12 90 §19. Representation of the states for a one-dimensional particle in the space D of entire analytic functions 94 §20.
The general case of one-dimensional motion 95 §21. Three-dimensional problems in quantum mechanics. A three-dimensional free particle 103 §22. A three-dimensional particle in a potential field 104 §23.
The rotation group 108 §25. Representations of the rotation group 111 §26. Spherically symmetric operators 114 §27. Representation of rotations by 2 x 2 unitary matrices 117 §28.
Representation of the rotation group on a space of entire analytic functions of two complex variables 120 §29. Uniqueness of the representations Dj 123 §30. Representations of the rotation group on the space L2(S2). The radial Schrodinger equation 130 §32.
The hydrogen atom. The alkali metal atoms 136 §33. The variational principle 154 §35. Physical formulation of the problem 157 §36.
Scattering of a one-dimensional particle by a potential barrier 159 www.com Contents vii §37. Physical meaning of the solutions ik, and 02 164 §38. Scattering by a rectangular barrier 167 §39. Scattering by a potential center 169 §40.
Motion of wave packets in a central force field 175 §41. The integral equation of scattering theory 181 §42. Derivation of a formula for the cross-section 183 §43. Abstract scattering theory 188 §44.
Properties of commuting operators 197 §45. Representation of the state space with respect to a complete set of observables 201 §46. Spin of a system of two electrons 208 §48. Systems of many particles.
The identity principle 212 §49. Symmetry of the coordinate wave functions of a system of two electrons. The helium atom 215 §50. Multi-electron atoms.
One-electron approximation 217 §51. The self-consistent field equations 223 §52. Mendeleev's periodic system of the elements 226 Appendix: Lagrangian Formulation of Classical Mechanics 231 www.com Preface This textbook is a detailed survey of a course of lectures given in the Mathematics-Mechanics Department of Leningrad University for mathematics students. The program of the course in quantum me- chanics was developed by the first author, who taught the course from 1968 to 1973.
Subsequently the course was taught by the second au- thor. It has certainly changed somewhat over these years, but its goal remains the same: to give an exposition of quantum mechanics from a point of view closer to that of a mathematics student than is com- mon in the physics literature. We take into account that the students do not study general physics. In a course intended for mathemati- cians, we have naturally aimed for a more rigorous presentation than usual of the mathematical questions in quantum mechanics, but not for full mathematical rigor, since a precise exposition of a number of questions would require a course of substantially greater scope.
In the literature available in Russian, there is only one book pursuing the same goal, and that is the American mathematician G. Mackey's book, Mathematical Foundations of Quantum Me- chanics. The present lectures differ essentially from Mackey's book both in the method of presentation of the bases of quantum mechan- ics and in the selection of material. Moreover, these lectures assume somewhat less in the way of mathematical preparation of the stu- dents.
Nevertheless, we have borrowed much both from Mackey's ix www.com x Preface book and from von Neumann's classical book, Mathematical Founda- tions of Quantum Mechanics. The approach to the construction of quantum mechanics adopted in these lectures is based on the assertion that quantum and classi- cal mechanics are different realizations of one and the same abstract mathematical structure. The features of this structure are explained in the first few sections, which are devoted to classical mechanics. These sections are an integral part of the course and should not be skipped over, all the more so because there is hardly any overlap of the material in them with the material in a course of theoretical me- chanics.
As a logical conclusion of our approach to the construction of quantum mechanics, we have a section devoted to the interconnec- tion of quantum and classical mechanics and to the passage to the limit from quantum mechanics to classical mechanics. In the selection of the material in the sections devoted to appli- cations of quantum mechanics we have tried to single out questions connected with the formulation of interesting mathematical problems. Much attention here is given to problems connected with the theory of group representations and to mathematical questions in the theory of scattering. In other respects the selection of material corresponds to traditional textbooks on general questions in quantum mechanics, for example, the books of V.
The authors are grateful to V. Babich, who read through the manuscript and made a number of valuable comments.com Preface to the English Edition The history and the goals of this book are adequately described in the original Preface (to the Russian edition) and I shall not repeat it here. The idea to translate the book into English came from the numerous requests of my former students, who are now spread over the world. For a long time I kept postponing the translation because I hoped to be able to modify the book making it more informative.
However, the recent book by Leon Takhtajan, Quantum Mechanics for Mathematicians (Graduate Studies in Mathematics, Volume 95, American Mathematical Society, 2008), which contains most of the material I was planning to add, made such modifications unnecessary and I decided that the English translation can now be published. Just when the decision to translate the book was made, my coau- thor Oleg Yakubovskii died. He had taught this course for more than 30 years and was quite devoted to it. He felt compelled to add some physical words to my more formal exposition.
The Russian text, published in 1980, was prepared by him and can be viewed as a combination of my original notes for the course and his experience of teaching it. It is a great regret that he will not see the English translation.com xii Preface to the English Edition Leon Takhtajan prepared a short appendix about the formalism of classical mechanics. It should play the role of introduction for stu- dents who did not take an appropriate course, which was obligatory at St. I want to add that the idea of introducing quantum mechanics as a deformation of classical mechanics has become quite fashionable nowadays.
Of course, whereas the term "deformation" is not used explicitly in the book, the idea of deformation was a guiding principle in the original plan for the lectures. Petersburg, November 2008 www. The algebra of observables in classical mechanics We consider the simplest problem in classical mechanics: the problem of the motion of a material point (a particle) with mass m in a force field V(x), where x(xl, x2i x3) is the radius vector of the particle. The force acting on the particle is F=-gradV=-ax.
The basic physical characteristics of the particle are its coordi- nates x1, x2, x3 and the projections of the velocity vector v(vl, v2, v3). All the remaining characteristics are functions of x and v; for exam- ple, the momentum p = mv, the angular momentum 1 = x x p = mx x v, and the energy E = mv2/2 + V (x). The equations of motion of a material point in the Newton form are dv (1} mdt = - aV , dx dt = v. It will be convenient below to use the momentum p in place of the velocity v as a basic variable.
In the new variables the equations of motion are written as follows: dp aV dx p (2) dt ax ' dt m Noting that m = ap and = aX , where H = + V (x) is the Hamiltonian function for a particle in a potential field, we arrive at the equations in the Hamiltonian form (3) dx_8H dp__aH dt ap ' dt ax It is known from a course in theoretical mechanics that a broad class of mechanical systems, and conservative systems in particular, 1 www. Yakubovskii are described by the Hamiltonian equations aH aH (4) 9t = apt , Pt = - qat , i = 1, 2,. , pn) is the Hamiltonian function, qt and pt are the generalized coordinates and momenta, and n is called the number of degrees of freedom of the system. We recall that for a conservative system, the Hamiltonian function H coincides with the expression for the total energy of the system in the variables qt and pt.
We write the Hamiltonian function for a system of N material points interacting pairwise: (5) H N = E 2m i=1 -2 Pt N N + 1: Vj (xi - xj) + 1] V (xi). t<j i=1 Here the Cartesian coordinates of the particles are taken as the gener- alized coordinates q, the number of degrees of freedom of the system is n = 3N, and V i (xt - xj) is the potential of the interaction of the ith and jth particles. The dependence of Utj only on the difference xt - xj is ensured by Newton's third law. (Indeed, the force acting on ay;; = 9v,; _ -F t.