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1925 M. Born, Z. Phys. 34, 858 On Quantum Mechanics M. Born Received 1925 Translation into English: Sources of Quantum Mechanics, Ed. by B. van der...
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Explain, in your own words (without going into mathematical details,

although you may use the odd equation for illustrative purposes), the main features of the Heisenberg-Born-Jordan atom, as described by their new theory of Quantum Mechanics, using quotes from the papers of Born and Jordan and the analysis of that paper by Fedak and Prentis. The explanation should include: what are the new variables describing the states of the atom? What are their fundamental properties? How do they link to observables? The two papers are attached.
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M. Born, Z. Phys. 34, 858 1925 On Quantum Mechanics M. Born Received 1925 — — ± ♦ ± — — Translation into English: Sources of Quantum Mechanics, Ed. by B. L. van der Waerden, North Holland, Amsterdam (1967) 277. — — ± ♦ ± — — The recently published theoretical approach of Heisenberg is here developed into a systematic theory of quantum mechanics (in the first place for systems hav- ing one degree of freedom) with the aid of mathematical matrix methods. After a brief survey of the latter, the mechanical equations of motion are derived from a variational principle and it is shown that using Heisenberg’s quantum condition, the principle of energy conservation and Bohr’s frequency condition follow from the mechanical equations. Using the anharmonic oscillator as example, the question of uniqueness of the solution and of the significance of the phases of the partial vibra- tions is raised. The paper concludes with an attempt to incorporate electromagnetic field laws into the new theory. Introduction The theoretical approach of Heisenberg 1 recently published in this Journal, which aimed at setting up a new kinematical and mechanical formalism in conformity with the basic requirements of quantum theory, appears to us of considerable potential significance. It represents an attempt to render 1 W.Heisenberg, Zs. f. Phys. 33 (1925) 879. 1
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justice to the new facts by selling up a new and really suitable conceptual system instead of adapting the customary conceptions in a more or less ar- tificial and forced manner. The physical reasoning which led Heisenberg to this development has been so clearly described by him that any supple- mentary remarks appear superfluous. But, as he himself indicates, in its formal, mathematical aspects his approach is but in its initial stages. His hypotheses have been applied only to simple examples without being fully carried through to a generalized theory. Having been in an advantageous position to familiarize ourselves with his ideas throughout their formative stages, we now strive (since his investigations have been concluded) to clar- ify the mathematically formal content of his approach and present some of our results here. These indicate that it is in fact possible, starting with the basic premises given by Heisenberg, to build up a closed mathemati- cal theory of quantum mechanics which displays strikingly close analogies with classical mechanics, but at the same time preserves the characteristic features of quantum phenomena. In this we at first confine ourselves, like Heisenberg, to systems hav- ing one degree of freedom and assume these to be – from a classical standpoint – periodic. We shall in the continuation of this publication con- cern ourselves with the generalization of the mathematical theory to sys- tems having ah arbitrary number of degrees of freedom, as also to aperiodic motion. A noteworthy generalization of Heisenberg’s approach lies in our confining ourselves neither to treatment of nonrelativistic mechanics nor to calculations involving Cartesian systems of coordinates. The only restriction which we impose upon the choice of coordinates is to base our considerations upon libration coordinates, which in classical theory are periodic functions of time. Admittedly, in some instances it might be more reasonable to employ other coordinates: for example, in the case of a rotating body to introduce the angle of rotation ϕ , which becomes a linear function of time. Heisenberg also proceeded thus in his treatment of the rotator; however, it remains undecided whether the approach applied there can be justified from the standpoint of a consistent quantum mechanics. The mathematical basis of Heisenberg’s treatment is the law of mul- tiplication of quantum–theoretical quantities, which he derived from an ingenious consideration of correspondence arguments. The development of his formalism, which we give here, is based upon the fact that this rule of multiplication is none other than the well–known mathematical rule of ma- trix multiplication. The infinite square array (with discrete or continuous indices) which appears at the start of the next section, termed a matrix, is a representation of a physical quantity which is given in classical theory as 2
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The 1925 Born and Jordan paper “On quantum mechanics” William A. Fedak a ! and Jeffrey J. Prentis b ! Department of Natural Sciences, University of Michigan–Dearborn, Dearborn, Michigan 48128 s Received 12 September 2007; accepted 9 October 2008 d The 1925 paper “On quantum mechanics” by M. Born and P. Jordan, and the sequel “On quantum mechanics II” by M. Born, W. Heisenberg, and P. Jordan, developed Heisenberg’s pioneering theory into the frst complete Formulation oF quantum mechanics. The Born and Jordan paper is the subject oF the present article. This paper introduced matrices to physicists. We discuss the original postulates oF quantum mechanics, present the two-part discovery oF the law oF commutation, and clariFy the origin oF Heisenberg’s equation. We show how the 1925 prooF oF energy conservation and Bohr’s Frequency condition served as the gold standard with which to measure the validity oF the new quantum mechanics. © 2009 American Association of Physics Teachers. f DOI: 10.1119/1.3009634 g I. INTRODUCTION The name “quantum mechanics” was coined by Max Born. 1 ±or Born and others, quantum mechanics denoted a canonical theory oF atomic and electronic motion oF the same level oF generality and consistency as classical mechanics. The transition From classical mechanics to a true quantum mechanics remained an elusive goal prior to 1925. Heisenberg made the breakthrough in his historic 1925 paper, “Quantum-theoretical reinterpretation oF kinematic and mechanical relations.” 2 Heisenberg’s bold idea was to retain the classical equations oF Newton but to replace the classical position coordinate with a “quantum-theoretical quantity.” The new position quantity contains inFormation about the measurable line spectrum oF an atom rather than the unobservable orbit oF the electron. Born realized that Heisenberg’s kinematical rule For multiplying position quan- tities was equivalent to the mathematical rule For multiplying matrices. The next step was to Formalize Heisenberg’s theory using the language oF matrices. The frst comprehensive exposition on quantum mechanics in matrix Form was written by Born and Jordan, 4 and the sequel was written by Born, Heisenberg, and Jordan. 5 Dirac independently discovered the general equations oF quantum mechanics without using matrix theory. 6 These papers devel- oped a Hamiltonian mechanics oF the atom in a completely new quantum s noncommutative d Format. These papers ush- ered in a new era in theoretical physics where Hermitian matrices, commutators, and eigenvalue problems became the mathematical trademark oF the atomic world. We discuss the frst paper “On quantum mechanics.” 4 This Formulation oF quantum mechanics, now reFerred to as matrix mechanics, 7 marked one oF the most intense peri- ods oF discovery in physics. The ideas and Formalism behind the original matrix mechanics are absent in most textbooks. Recent articles discuss the correspondence between classical harmonics and quantum jumps, 8 the calculational details oF Heisenberg’s paper, 9 and the role oF Born in the creation oF quantum theory. 10 ReFerences 11 19 represent a sampling oF the many sources on the development oF quantum mechan- ics. Given Born and Jordan’s pivotal role in the discovery oF quantum mechanics, it is natural to wonder why there are no equations named aFter them, 20 and why they did not share the Nobel Prize with others. 21 In 1933 Heisenberg wrote Born saying “The Fact that I am to receive the Nobel Prize alone, For work done in Göttingen in collaboration—you, Jordan, and I, this Fact depresses me and I hardly know what to write to you. I am, oF course, glad that our common eFForts are now appreciated, and I enjoy the recollection oF the beautiFul time oF collaboration. I also believe that all good physicists know how great was your and Jordan’s contribution to the structure oF quantum mechanics—and this remains un- changed by a wrong decision From outside. Yet I myselF can do nothing but thank you again For all the fne collaboration and Feel a little ashamed.” 23 Engraved on Max Born’s tombstone is a one-line epitaph: pq ² qp = h / 2 p i . Born composed this elegant equation in early July 1925 and called it “die verschärFte Quantenbedingung” 4 —the sharpened quantum condition. This equation is now known as the law oF commutation and is the hallmark oF quantum algebra. In the contemporary approach to teaching quantum me- chanics, matrix mechanics is usually introduced aFter a thor- ough discussion oF wave mechanics. The Heisenberg picture is viewed as a unitary transFormation oF the Schrödinger picture. 24 How was matrix mechanics Formulated in 1925 when the Schrödinger picture was nowhere in sight? The Born and Jordan paper 4 represents matrix mechanics in its purest Form. II. BACKGROUND TO “ON QUANTUM MECHANICS” Heisenberg’s program, as indicated by the title oF his paper, 2 consisted oF constructing quantum-theoretical rela- tions by reinterpreting the classical relations. To appreciate what Born and Jordan did with Heisenberg’s reinterpreta- tions, we discuss in the Appendix Four key relations From Heisenberg’s paper. 2 Heisenberg wrote the classical and quantum versions oF each relation in parallel—as Formula couplets. Heisenberg has been likened to an “expert decoder who reads a cryptogram.” 25 The correspondence principle 8 , 26 acted as a “code book” For translating a classical relation into its quantum counterpart. Unlike his predecessors who used the correspondence principle to produce specifc relations, Heisenberg produced an entirely new theory—complete with a new representation oF position and a new rule oF multipli- cation, together with an equation oF motion and a quantum condition whose solution determined the atomic observables s energies, Frequencies, and transition amplitudes d . 128 128 Am. J. Phys. 77 s 2 d , ±ebruary 2009 http://aapt.org/ajp © 2009 American Association oF Physics Teachers
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Matrices are not explicitly mentioned in Heisenberg’s pa- per. He did not arrange his quantum-theoretical quantities into a table or array. In looking back on his discovery, Heisenberg wrote, “At that time I must confess I did not know what a matrix was and did not know the rules of ma- trix multiplication.” 18 In the last sentence of his paper he wrote “whether this method after all represents far too rough an approach to the physical program of constructing a theo- retical quantum mechanics, an obviously very involved prob- lem at the moment, can be decided only by a more intensive mathematical investigation of the method which has been very superFcially employed here.” 27 Born took up Heisenberg’s challenge to pursue “a more intensive mathematical investigation.” At the time Heisen- berg wrote his paper, he was Born’s assistant at the Univer- sity of Göttingen. Born recalls the moment of inspiration when he realized that position and momentum were matrices: 28 After having sent Heisenberg’s paper to the Zeitschrift für Physik for publication, I began to ponder about his symbolic multiplication, and was soon so involved in it ±or I felt there was some- thing fundamental behind it And one morning, about 10 July 1925, I suddenly saw the light: Heisenberg’s symbolic multiplication was nothing but the matrix calculus, well known to me since my student days from the lectures of Rosanes in Breslau. I found this by just simplifying the notation a little: instead of q s n , n + t d , where n is the quantum num- ber of one state and t the integer indicating the transition, I wrote q s n , m d , and rewriting Heisen- berg’s form of Bohr’s quantum condition, I recog- nized at once its formal signiFcance. It meant that the two matrix products pq and qp are not identi- cal. I was familiar with the fact that matrix multi- plication is not commutative; therefore I was not too much puzzled by this result. Closer inspection showed that Heisenberg’s formula gave only the value of the diagonal elements s m = n d of the ma- trix pq qp ; it said they were all equal and had the value h / 2 p i where h is Planck’s constant and i = Î ²1. But what were the other elements s m Þ n d ? Here my own constructive work began. Repeating Heisenberg’s calculation in matrix notation, I soon convinced myself that the only reasonable value of the nondiagonal elements should be zero, and I wrote the strange equation pq ² qp = h 2 p i 1 , s 1 d where 1 is the unit matrix. But this was only a guess, and all my attempts to prove it failed. On 19 July 1925, Born invited his former assistant Wolf- gang Pauli to collaborate on the matrix program. Pauli de- clined the invitation. 29 The next day, Born asked his student Pascual Jordan to assist him. Jordan accepted the invitation and in a few days proved Born’s conjecture that all nondi- agonal elements of pq ² qp must vanish. The rest of the new quantum mechanics rapidly solidiFed. The Born and Jordan paper was received by the Zeitschrift für Physik on 27 Sep- tember 1925, two months after Heisenberg’s paper was re- ceived by the same journal. All the essentials of matrix me- chanics as we know the subject today Fll the pages of this paper. In the abstract Born and Jordan wrote “The recently pub- lished theoretical approach of Heisenberg is here developed into a systematic theory of quantum mechanics s in the Frst place for systems having one degree of freedom d with the aid of mathematical matrix methods.” 30 In the introduction they go on to write “The physical reasoning which led Heisenberg to this development has been so clearly described by him that any supplementary remarks appear super³uous. But, as he himself indicates, in its formal, mathematical aspects his approach is but in its initial stages. His hypotheses have been applied only to simple examples without being fully carried through to a generalized theory. Having been in an advanta- geous position to familiarize ourselves with his ideas throughout their formative stages, we now strive s since his investigations have been concluded d to clarify the math- ematically formal content of his approach and present some of our results here. These indicate that it is in fact possible, starting with the basic premises given by Heisenberg, to build up a closed mathematical theory of quantum mechanics which displays strikingly close analogies with classical me- chanics, but at the same time preserves the characteristic features of quantum phenomena.” 31 The reader is introduced to the notion of a matrix in the third paragraph of the introduction: “The mathematical basis of Heisenberg’s treatment is the law of multiplication of quantum-theoretical quantities, which he derived from an in- genious consideration of correspondence arguments. The de- velopment of his formalism, which we give here, is based upon the fact that this rule of multiplication is none other than the well-known mathematical rule of matrix multiplica- tion . The inFnite square array which appears at the start of the next section, termed a matrix , is a representation of a physical quantity which is given in classical theory as a func- tion of time. The mathematical method of treatment inherent in the new quantum mechanics is thereby characterized by the employment of matrix analysis in place of the usual number analysis.” The Born-Jordan paper 4 is divided into four chapters. Chapter 1 on “Matrix calculation” introduces the mathemat- ics s algebra and calculus d of matrices to physicists. Chapter 2 on “Dynamics” establishes the fundamental postulates of quantum mechanics, such as the law of commutation, and derives the important theorems, such as the conservation of energy. Chapter 3 on “Investigation of the anharmonic oscil- lator” contains the Frst rigorous s correspondence free d calcu- lation of the energy spectrum of a quantum-mechanical har- monic oscillator. Chapter 4 on “Remarks on electrodynamics” contains a procedure—the Frst of its kind—to quantize the electromagnetic Feld. We focus on the material in Chap. 2 because it contains the essential physics of matrix mechanics. 129 129 Am. J. Phys., Vol. 77, No. 2, ±ebruary 2009 William A. ±edak and Jeffrey J. Prentis
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