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Sources of Quantum Mechanics
Seventeen seminal papers, dating from the years 1917-26, in which the quantum theory as we now know it was developed and formulated. Among the scientists represented: Einstein, Ehrenfest, Bohr, Born, Van Vleck, Heisenberg, Dirac, Pauli and Jordan. All 17 papers translated into English.
430 pages, Paperback
First published January 1, 1967
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About the author
B.L. van der Waerden
43 books6 followersBartel Leendert van der Waerden (Dutch: [vɑn dər ˈʋaːrdən]; February 2, 1903 – January 12, 1996) was a Dutch mathematician and historian of mathematics.
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September 23, 2020
When, long ago, this reviewer took quantum mechanics as an undergraduate at Princeton, his professor Sam Treiman led off his first lecture with the contentious claim that the advent of quantum mechanics represents mankind’s greatest intellectual achievement of all time. At the time, it seemed off-putting, but over the years the justice of such a hyperbolic assertion has begun to seem less easy to dismiss. For, the classical mechanics of the seventeenth century rests upon what is, after all, a fairly straightforward empirical and theoretical foundation. Indeed, one could say that the great physicists of that era enjoyed beginner’s luck; the elementary principles of inertia and impressed force, once hit upon, lend themselves almost as a matter of course to a mathematical formalization that seems to yield far greater predictive power than one has any right to expect, in view of the exiguous effort one had to mobilize behind it. The same cannot be said about the discovery of quantum mechanics. Even merely first to arrive at the puzzling empirical phenomena that indicate a breakdown of the classical world-picture took centuries of dedicated labor on the part of the world’s scientists and mathematicians, a collective endeavor that embraced both the handful of the great and untold legions of the obscure. The so-called old quantum theory was counter-intuitive and ill-defined from a theoretical point of view. Nevertheless, its empirical predictions were so good that physicists could be sure that they were on to something. The matrix mechanics advanced by Heisenberg in 1925 laid the groundwork for a new quantum mechanics that not only enjoys exacting empirical support across a vast domain of physical processes, at first atomic and molecular but later subatomic and nuclear as well, but also has been elaborated into a most satisfactory mathematical edifice. The phenomenal degree of success encountered in this endeavor encourages us to suppose that the world is after all founded upon natural laws and to have uncovered these hidden foundations and to have conferred upon them a precise mathematical formulation is indeed a supremely great accomplishment of the human spirit.
If one goes back and reads the original papers on quantum mechanics in the collection by van der Waerden, the subject of the present review, one can gain a perspective on the development of the theory very different from what most everyone nowadays knows. In the textbooks, one merely postulates the Schrödinger equation and derives from it the spectrum of the hydrogen atom, as if that were sufficient justification, and never gets around to discussing the coupling to the radiation field. In the actual historical development, however, the coupling to the radiation field was the crux around which everything revolved. Bohr’s correspondence principle makes sense only in this context. Born was able to guess the form of the matrix elements for radiative transitions from Kramers’ dispersion formula, well before Heisenberg’s breakthrough. Heisenberg’s starting point was the virtual oscillator idea from the paper on the quantized radiation field by Bohr, Kramers and Slater. The physical motivation of quantum mechanics is much easier to appreciate in the historical development than in the textbooks widely used among the present generation of physicists. It should be something of a surprise that this isn’t more commonly recognized.
Let us comment on the selection of papers in this reprinting. In order to fit them into one manageable volume, one has to be severely economical. Not only are Sommerfeld’s and Schwarzschild’s substantial contributions to the old quantum theory skipped over (as well as Einstein’s on the photoelectric effect, the specific heat of solids and what we would now call integrability versus chaos), even Bohr’s revolutionary paper of 1913 on the spectrum of the hydrogen atom fails to make the cut. Only those papers from the old quantum school that can be put into a direct line leading to the new quantum mechanics gain inclusion. Bohr’s 1918 paper on line spectra contains his statement of the correspondence principle while Ehrenfest’s 1916 paper advances the adiabatic hypothesis, which is really necessary in order for the Bohr-Sommerfeld quantization conditions to have relevance in any physically plausible system as opposed a mere toy model. Einstein’s 1917 paper on the radiation field matters because, via a clever indirect argument from statistical fluctuations, it supports the idea that the radiation involved in elementary atomic and molecular emission or absorption processes comes in the form of directed bundles (or photons, as we would now say) and that the outgoing radiation in the form of spherical waves expected from Maxwellian electrodynamics simply does not exist. Ladenburg’s 1921 paper delivers an empirically verified connection between anomalous dispersion in the neighborhood of absorption lines and the strength of absorption, which supports the postulated validity of the quantum-theoretical expression for all quantum numbers and not only in the limit of large ones, where it must hold if the correspondence principle does. So far, we have reached an impasse. For the old quantum theory finds actual quantitative empirical support only in the simplest case of the hydrogen atom, yet we know that if it is to be general, it has to apply to atomic and molecular systems with more electrons and thus, correspondingly more degrees of freedom. But all attempts to extend the old quantum mechanics to these systems met with failure due to the intractable nature of the calculations. It was Slater who figured out the way past this conundrum. For, in retrospect, implicit in Ladenburg’s paper is the conceit that, as far as its interaction with the radiation field goes, the atom behaves as if it were a collection of harmonic oscillators each having its own characteristic frequency. Now, Slater’s idea was to apply the same concept to the radiation field, viz., to represent it too as a collection of virtual oscillators in communication with the atom. Bohr, Kramers and Slater then wrote their influential 1924 paper on the quantized radiation field, in which they speculate that energy conservation holds only in the statistical average sense and that emission and absorption processes can be treated as statistically independent. This paper contains no actual equations or derivations, but was central to consolidating physicists’ conceptual understanding of how the sought-for quantum theory ought to fit together. The real significance of Kramers’ dispersion theory is not that it yields an empirically reasonable formula for absorption, based upon Drude’s classical theory, but that he realizes that consistency with the correspondence principle demands the inclusion of an additional term of opposite sign in order to account for the possibility of emission. If this be true, then Slater’s idea of virtual oscillators begins to look more self-consistent, since, in principle, a virtual oscillator ought to be able to give off as well as to receive energy (unless in its ground state). Three papers by Kramers had to be included, as only in the second does he outline a derivation and complete it in the third in collaboration with Heisenberg. Van Vleck’s 1924 paper applies the correspondence principle more sharply than had been done before in order to derive Einstein’s formulae for spontaneous and induced radiative transitions, which cleared up Bohr’s long-standing skepticism. Lastly, Kuhn’s sum rule of 1925 can be seen in light of later developments as giving empirical support to Heisenberg’s canonical commutation relations. Meanwhile, Born’s sophisticated 1924 paper appeals to classical perturbation theory to obtain the quantum-mechanical expression for the perturbed Hamiltonian by means of the all-important device of replacing differential coefficients with difference quotients. The same process applied to the classical dispersion formula yields Kramers’ quantum dispersion formula.
Hence, Part I concludes. Part II reproduces just six papers that were essential to the birth of the new quantum mechanics. Heisenberg’s breakthrough paper of 1925 is motivated by two considerations: first, classical radiation theory is inapplicable because it would imply resonance with the frequency of orbital motion, rather than with differences of frequencies, and second, the naïve picture in the old quantum mechanics of a classical phase space subject to ad hoc quantization rules has to be abandoned. He writes the position variable as a Fourier series and substitutes into the equation of motion of the anharmonic oscillator in order to obtain relations among the Fourier coefficients, which are to be viewed as the appropriate quantities in terms of which to describe the kinematics. Remarkably, he obtains thereby quantum-theoretical rules that satisfy the correspondence principle and reproduce Kramers’ dispersion theory. Born and Jordan’s paper recasts these inchoate suggestions into the beginnings of a consistent theory; its key ideas are the identification of Heisenberg’s strange symbolic multiplication as an instance of matrix multiplication, the introduction of the canonical commutation relations and proof of energy conservation and Bohr’s frequency condition. The next paper, the outcome of a collaboration among Heisenberg, Born and Jordan (known as the ‘Dreimännerarbeit’), consolidates the nascent matrix mechanics into a coherent theory and includes, among other things, the first sketch of a perturbation theory and canonical transformations, which leads to the all-important concept of the transformation to principal axes (although an appreciation of the full significance of the eigenvalue problem had to await Schrödinger). So far, the new quantum mechanics has been developed formally. Based on what was already known from the papers in Part I, everyone could be confident that the consistent formalism they eventually arrive at corresponded to a genial idea, but it had yet to be put to the crucial test of agreement with experiment; its first application to physics was worked out shortly thereafter by Pauli, who computes the spectrum of the hydrogen atom from matrix mechanics. Pauli’s work first convinced physicists that the proposed quantum mechanics must be true. In addition to these four basic papers, van der Waerden reprints two by Dirac, whose contribution was to frame the problem of quantization in terms of a passage from the commutative algebra of functions on classical phase space (what he calls c-numbers) to a non-commutative algebra (q-numbers). The Poisson bracket goes over into the commutator and this provides a theoretical justification for the canonical commutation relations (which, strictly speaking, Jordan had motivated but not proved). Dirac shows that in the case of discrete spectrum, his non-commutative algebra can be represented by the infinite-dimensional matrices of Heisenberg, Born and Jordan.
The foundations of quantum mechanics languish in a sorry state today, everyone being preoccupied by sterile debates over interpretation. The most vigorously defended of these, Bohm’s pilot wave interpretation offers neither any physical principles nor any insight into the solution of actual physical problems; all it amounts to is a strained attempt to inject physical meaning into the expressions that arise under an artificial rewriting of Schrödinger’s equation (it should be clear that a mere change of variables cannot lead to novel physics). The great originators of quantum mechanics who were still around when Bohm came forward with his theory dismissed it derisively – scarcely the nefarious conspiracy to suppress truth that his present-day partisans imagine, but a frank recognition of the evident worthlessness of Bohm’s formalism and accompanying speculations as far as it is a question of getting real physics done. The only physicist of any distinction ever to have been a disciple of Bohm is Bell.
The root of the malaise may be sought in the prevailing tendency to very abstract and recondite theorizing in the absence of any close connection to experiment; and along with this, a disproportionate emphasis on spinning out mathematical consequences of unjustified premises rather than on understanding and evaluating them in physical terms. The solution: let us return to the sources [in French, ressourcement]! If we are to imitate the great founders of the quantum mechanics, we need mentally to dissolve the formalism with which we are all-too familiar (which in its essentials has never advanced beyond the point it had already reached by 1932 at the hands of von Neumann) and to think anew inductively from the sheer raw phenomena, as they did. The requisite mathematics should follow as a matter of course; physical principle is logically prior—in other words, diametrically the opposite of what everyone does these days, just taking the path of least resistance!
Why does van der Waerden pass over Schrödinger? If one also reads Schrödinger’s collected papers on wave mechanics he will be struck by how haphazard is the route by which he arrives at his eponymous equation, by how quickly he is able to flesh out the essentials of the theory by relying on the mathematics already worked out by Courant and Hilbert in their Methoden der mathematischen Physik and in general by how wanting his writing is in terms of style; not a polished edifice! Schrödinger has the stamina to see his way through forbidding calculations (the Stark effect, for instance, which is too advanced to make it into the textbooks), but he lacks the form-giving power of artistic genius (the kind of power that is so evident in Schiller’s mature work). The seasoned reader will be impressed, in this connection, by his signal failure to make any original or interesting observations in his Nature and the Greeks and Science and Humanism (the contrast with Heisenberg is apparent). To be a Kulturmensch means not just to master an arcane discipline but also to have something intelligent to say about how it relates to the rest of knowledge and culture. It would be of interest to pursue Heisenberg’s intriguing suggestion that quantum mechanics involves a return to the Aristotelian concept of potentia. The problem is that Schrödinger’s equation only describes the temporal evolution of the wavefunction but does not offer an account of causation, such as what happens during measurement, when the system makes a transition from the potential to the actual. In some sense, time advances only when a measurement is performed.
Hence, one may concur with van der Waerden’s relegation of Schrödinger to a secondary status in the discovery of quantum-mechanical theory. But the esteemed reader of this review is cordially invited to have recourse to the original literature in order to form his own conclusions!
If one goes back and reads the original papers on quantum mechanics in the collection by van der Waerden, the subject of the present review, one can gain a perspective on the development of the theory very different from what most everyone nowadays knows. In the textbooks, one merely postulates the Schrödinger equation and derives from it the spectrum of the hydrogen atom, as if that were sufficient justification, and never gets around to discussing the coupling to the radiation field. In the actual historical development, however, the coupling to the radiation field was the crux around which everything revolved. Bohr’s correspondence principle makes sense only in this context. Born was able to guess the form of the matrix elements for radiative transitions from Kramers’ dispersion formula, well before Heisenberg’s breakthrough. Heisenberg’s starting point was the virtual oscillator idea from the paper on the quantized radiation field by Bohr, Kramers and Slater. The physical motivation of quantum mechanics is much easier to appreciate in the historical development than in the textbooks widely used among the present generation of physicists. It should be something of a surprise that this isn’t more commonly recognized.
Let us comment on the selection of papers in this reprinting. In order to fit them into one manageable volume, one has to be severely economical. Not only are Sommerfeld’s and Schwarzschild’s substantial contributions to the old quantum theory skipped over (as well as Einstein’s on the photoelectric effect, the specific heat of solids and what we would now call integrability versus chaos), even Bohr’s revolutionary paper of 1913 on the spectrum of the hydrogen atom fails to make the cut. Only those papers from the old quantum school that can be put into a direct line leading to the new quantum mechanics gain inclusion. Bohr’s 1918 paper on line spectra contains his statement of the correspondence principle while Ehrenfest’s 1916 paper advances the adiabatic hypothesis, which is really necessary in order for the Bohr-Sommerfeld quantization conditions to have relevance in any physically plausible system as opposed a mere toy model. Einstein’s 1917 paper on the radiation field matters because, via a clever indirect argument from statistical fluctuations, it supports the idea that the radiation involved in elementary atomic and molecular emission or absorption processes comes in the form of directed bundles (or photons, as we would now say) and that the outgoing radiation in the form of spherical waves expected from Maxwellian electrodynamics simply does not exist. Ladenburg’s 1921 paper delivers an empirically verified connection between anomalous dispersion in the neighborhood of absorption lines and the strength of absorption, which supports the postulated validity of the quantum-theoretical expression for all quantum numbers and not only in the limit of large ones, where it must hold if the correspondence principle does. So far, we have reached an impasse. For the old quantum theory finds actual quantitative empirical support only in the simplest case of the hydrogen atom, yet we know that if it is to be general, it has to apply to atomic and molecular systems with more electrons and thus, correspondingly more degrees of freedom. But all attempts to extend the old quantum mechanics to these systems met with failure due to the intractable nature of the calculations. It was Slater who figured out the way past this conundrum. For, in retrospect, implicit in Ladenburg’s paper is the conceit that, as far as its interaction with the radiation field goes, the atom behaves as if it were a collection of harmonic oscillators each having its own characteristic frequency. Now, Slater’s idea was to apply the same concept to the radiation field, viz., to represent it too as a collection of virtual oscillators in communication with the atom. Bohr, Kramers and Slater then wrote their influential 1924 paper on the quantized radiation field, in which they speculate that energy conservation holds only in the statistical average sense and that emission and absorption processes can be treated as statistically independent. This paper contains no actual equations or derivations, but was central to consolidating physicists’ conceptual understanding of how the sought-for quantum theory ought to fit together. The real significance of Kramers’ dispersion theory is not that it yields an empirically reasonable formula for absorption, based upon Drude’s classical theory, but that he realizes that consistency with the correspondence principle demands the inclusion of an additional term of opposite sign in order to account for the possibility of emission. If this be true, then Slater’s idea of virtual oscillators begins to look more self-consistent, since, in principle, a virtual oscillator ought to be able to give off as well as to receive energy (unless in its ground state). Three papers by Kramers had to be included, as only in the second does he outline a derivation and complete it in the third in collaboration with Heisenberg. Van Vleck’s 1924 paper applies the correspondence principle more sharply than had been done before in order to derive Einstein’s formulae for spontaneous and induced radiative transitions, which cleared up Bohr’s long-standing skepticism. Lastly, Kuhn’s sum rule of 1925 can be seen in light of later developments as giving empirical support to Heisenberg’s canonical commutation relations. Meanwhile, Born’s sophisticated 1924 paper appeals to classical perturbation theory to obtain the quantum-mechanical expression for the perturbed Hamiltonian by means of the all-important device of replacing differential coefficients with difference quotients. The same process applied to the classical dispersion formula yields Kramers’ quantum dispersion formula.
Hence, Part I concludes. Part II reproduces just six papers that were essential to the birth of the new quantum mechanics. Heisenberg’s breakthrough paper of 1925 is motivated by two considerations: first, classical radiation theory is inapplicable because it would imply resonance with the frequency of orbital motion, rather than with differences of frequencies, and second, the naïve picture in the old quantum mechanics of a classical phase space subject to ad hoc quantization rules has to be abandoned. He writes the position variable as a Fourier series and substitutes into the equation of motion of the anharmonic oscillator in order to obtain relations among the Fourier coefficients, which are to be viewed as the appropriate quantities in terms of which to describe the kinematics. Remarkably, he obtains thereby quantum-theoretical rules that satisfy the correspondence principle and reproduce Kramers’ dispersion theory. Born and Jordan’s paper recasts these inchoate suggestions into the beginnings of a consistent theory; its key ideas are the identification of Heisenberg’s strange symbolic multiplication as an instance of matrix multiplication, the introduction of the canonical commutation relations and proof of energy conservation and Bohr’s frequency condition. The next paper, the outcome of a collaboration among Heisenberg, Born and Jordan (known as the ‘Dreimännerarbeit’), consolidates the nascent matrix mechanics into a coherent theory and includes, among other things, the first sketch of a perturbation theory and canonical transformations, which leads to the all-important concept of the transformation to principal axes (although an appreciation of the full significance of the eigenvalue problem had to await Schrödinger). So far, the new quantum mechanics has been developed formally. Based on what was already known from the papers in Part I, everyone could be confident that the consistent formalism they eventually arrive at corresponded to a genial idea, but it had yet to be put to the crucial test of agreement with experiment; its first application to physics was worked out shortly thereafter by Pauli, who computes the spectrum of the hydrogen atom from matrix mechanics. Pauli’s work first convinced physicists that the proposed quantum mechanics must be true. In addition to these four basic papers, van der Waerden reprints two by Dirac, whose contribution was to frame the problem of quantization in terms of a passage from the commutative algebra of functions on classical phase space (what he calls c-numbers) to a non-commutative algebra (q-numbers). The Poisson bracket goes over into the commutator and this provides a theoretical justification for the canonical commutation relations (which, strictly speaking, Jordan had motivated but not proved). Dirac shows that in the case of discrete spectrum, his non-commutative algebra can be represented by the infinite-dimensional matrices of Heisenberg, Born and Jordan.
The foundations of quantum mechanics languish in a sorry state today, everyone being preoccupied by sterile debates over interpretation. The most vigorously defended of these, Bohm’s pilot wave interpretation offers neither any physical principles nor any insight into the solution of actual physical problems; all it amounts to is a strained attempt to inject physical meaning into the expressions that arise under an artificial rewriting of Schrödinger’s equation (it should be clear that a mere change of variables cannot lead to novel physics). The great originators of quantum mechanics who were still around when Bohm came forward with his theory dismissed it derisively – scarcely the nefarious conspiracy to suppress truth that his present-day partisans imagine, but a frank recognition of the evident worthlessness of Bohm’s formalism and accompanying speculations as far as it is a question of getting real physics done. The only physicist of any distinction ever to have been a disciple of Bohm is Bell.
The root of the malaise may be sought in the prevailing tendency to very abstract and recondite theorizing in the absence of any close connection to experiment; and along with this, a disproportionate emphasis on spinning out mathematical consequences of unjustified premises rather than on understanding and evaluating them in physical terms. The solution: let us return to the sources [in French, ressourcement]! If we are to imitate the great founders of the quantum mechanics, we need mentally to dissolve the formalism with which we are all-too familiar (which in its essentials has never advanced beyond the point it had already reached by 1932 at the hands of von Neumann) and to think anew inductively from the sheer raw phenomena, as they did. The requisite mathematics should follow as a matter of course; physical principle is logically prior—in other words, diametrically the opposite of what everyone does these days, just taking the path of least resistance!
Why does van der Waerden pass over Schrödinger? If one also reads Schrödinger’s collected papers on wave mechanics he will be struck by how haphazard is the route by which he arrives at his eponymous equation, by how quickly he is able to flesh out the essentials of the theory by relying on the mathematics already worked out by Courant and Hilbert in their Methoden der mathematischen Physik and in general by how wanting his writing is in terms of style; not a polished edifice! Schrödinger has the stamina to see his way through forbidding calculations (the Stark effect, for instance, which is too advanced to make it into the textbooks), but he lacks the form-giving power of artistic genius (the kind of power that is so evident in Schiller’s mature work). The seasoned reader will be impressed, in this connection, by his signal failure to make any original or interesting observations in his Nature and the Greeks and Science and Humanism (the contrast with Heisenberg is apparent). To be a Kulturmensch means not just to master an arcane discipline but also to have something intelligent to say about how it relates to the rest of knowledge and culture. It would be of interest to pursue Heisenberg’s intriguing suggestion that quantum mechanics involves a return to the Aristotelian concept of potentia. The problem is that Schrödinger’s equation only describes the temporal evolution of the wavefunction but does not offer an account of causation, such as what happens during measurement, when the system makes a transition from the potential to the actual. In some sense, time advances only when a measurement is performed.
Hence, one may concur with van der Waerden’s relegation of Schrödinger to a secondary status in the discovery of quantum-mechanical theory. But the esteemed reader of this review is cordially invited to have recourse to the original literature in order to form his own conclusions!
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