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Quantum Mechanics
This text attempts to present the whole range of quantum mechanics, from the fundamental assumptions to the experimental numbers. The author presents a unified theoretical formulation and includes examples from recent research. Earlier editions of this text have become a standard text and reference work; it has been reprinted and translated. In this third edition, the author has made some minor corrections of the previous edition and has added two new one on quantal phase factors and their consequences (the "Berry phase") and another on the gauge theory of molecular physics.
522 pages, Hardcover
First published January 1, 1979
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May 18, 2022
In the present work, Quantum Mechanics: Foundations and Applications, Arno Bohm largely succeeds in his intention to present the theory from the ground up as a completed edifice, at an unapologetically advanced level. Characteristic of his treatment is an emphasis on the algebraic formalism to the practical exclusion of the more easily picturable approach of Schrödinger – what is not a quirk, but reflects an apparently conscious decision to focus on the logical construction of the theory in terms of observable quantities, represented through their corresponding operators. Thus, the reader can expect Bohm, when investigating a new class of phenomena, to start from a consideration of what is physically observable and to construct an algebra of observables on this basis, rather than merely to write down a model Hamiltonian on analogy with the classical case as anyone else would. In the same vein, he does not satisfy himself with solving formally for the eigenvalues but always discusses how the theoretical results obtained connect to experiment, with all the complications introduced by the departure of the actual system under study from the idealization embodied in the model of it. Another nice feature of Bohm’s exposition is the frequent inclusion of experimental data on the systems under consideration, often going beyond the simplest possible instantiation and interpreted with level diagrams. For instance, for helium or for diatomic molecules. It is conventional to pay lip service to the ideal of physics as an empirically grounded and confirmed discipline, but it is rare for a textbook on quantum mechanics to go so far in demonstrating this. For instance, there are no figures illustrating experimental results in either Dirac or Cohen-Tannoudji, to name two [our reviews here and here].
Chapter four is good on the experimental basis of the rotator and the vibrating rotator – Bohm does not just derive the spectrum in a toy model, but gives energy level diagrams to compare with the experimental results on CO, HCl etc. Chapter five on the Wigner-Eckart theorem, a topic not usually covered even in graduate-level textbooks. He explains what a tensor operator is, then states but does not prove the Wigner-Eckart theorem – important because it allows one to express what one wants to know in terms of a much smaller number of reduced matrix elements (he shows how this works for scalar and vector operators, pp. 178-181). Again the following section §5.4 on parity is exhaustive and hardly ever covered by other authors yet vital if one wants any connection to experiment. Chapter 6 on the hydrogen atom follows Pauli’s original derivation in matrix mechanics via the Lenz vector and Wigner-Eckart theorem – note, the usual q and p are not physical observables; thus, in accord with his pedagogical principles, Bohm will not just write out and solve the Hamiltonian in the position coordinates as is usually done. Chapter seven is devoted to alkali atoms, which played an important role in the development of quantum mechanics (appealed to by Sommerfeld in the old quantum mechanics, and by Pauli when working out his exclusion principle). Again, level diagrams are shown for lithium, sodium and cesium.
In all this, one will appreciate Bohm’s constant emphasis on the physical basis in which system can be prepared versus convenient theoretical basis, leading to Clebsch-Gordon coefficients by which to interrelate the two, with many explicit formulae quoted. The important point to which most everyone else is oblivious: the q resp. p observables represent not just a convenient basis but in many cases a physical one that is good for representing states in which we are interested (the completion including high lying states is unphysical). To draw out the content of this last parenthetical comment: if one were to use any arbitrary and very complicated observable to generate the orthonormal basis, it would be uninterpretable. The question as to why the position and momentum bases exist is very profound and would repay reflection; it has to do with the ineluctability of spatial intuition in making the world known to us. Yet, as Niels Bohr never ceases to point out, there are limits to spatial intuition, especially in the quantum world. How is it to be complemented, and by what? This too marks out a very deep field for reflection and inquiry.
A last thing to note is that Bohm wants to prepare the ground for a far more extensive treatment of scattering theory than is the rule in beginning graduate-level textbooks. Hence, in chapter three he includes a very little on rigged Hilbert spaces, a subject to which Bohm has contributed original research memoirs. Chapter eight presents perturbation theory in a format designed to streamline passage from the discrete (Wigner-Brillouin resp. Rayleigh-Schrödinger) to the continuous case (Lippmann-Schwinger equation), which figures importantly in chapters eighteen to twenty-one. The exposition nevertheless is very compressed, e.g. introduces the principal-value integral without explanation (for which, see Gelfand or Reed-Simon). Chapters 18-20 on resonances, the S-matrix, analyticity and causality conditions get a little detailed and one loses sight of the flow of the derivation, especially in chapter 21 where Gamov vectors are explained in terms of rigged Hilbert spaces, Hardy class functions and the Titchmarsch and Paley-Wiener theorems. Here, the rapid-fire quotation without explanation of major results of real analysis leaves one with, at best, an impressionistic feeling for what is going on: to have a hope of comprehending all this one will have to refer to Walter Thirring, in the third volume on quantum mechanics of atoms and molecules in his A Course in Mathematical Physics or to Reed-Simon, in the third volume on scattering theory of Methods of Modern Mathematical Physics.
Chapter thirteen contains a good analysis of the Stern-Gerlach experiment, far more thorough than is the rule in textbook treatments. One will find here a lucid though terse explanation of the Bell inequalities, not very philosophically oriented but just the derivation of the relevant formulae. Compare Bohm’s derivation of the cross section from the Lippmann-Schwinger equation [p 362ff] with that given, say, by Cohen-Tannoudji. Chapter 22 covers the Berry phase, first off with a reprise of its general derivation but what’s good in Bohm’s account is §22.3 with an explicit computation in the case of a spinning quantum system in a slowly varying magnetic field (an analogue of Dirac’s magnetic monopole).
As Bohm goes on to show in the succeeding section, the Berry phase is an adiabatic approximation to the non-adiabatic exact Aharanov-Anandan phase: he splits the phase factor into geometrical and dynamical parts. Finally chapter 23 investigates this circle of ideas in the context of the Born-Oppenheimer approximation in molecular systems, in a gauge-theoretical formalism that includes the Berry phase which Born and Oppenheimer miss. Bohm describes conceptually what is at stake: the antireductionist spirit mentioned in the introduction [see p. 642, resp. 571]. Compare Bohm’s treatment with the original paper [Zur Quantentheorie der Molekeln, Annalen der Physik 389, 457-484 (1927)]. In line with his overall pedagogical intent, Bohm rounds out his discussion with a concrete derivation of the Berry connection and its curvature in diatomic molecules [p. 645ff] also yielding thereby a satisfying physical realization of a Dirac monopole (not electromagnetic of course).
The epilogue gets a little philosophical. Bohm highlights the two revisions of the quantum mechanical outlook over that of classical mechanics. First of all,
Probability statements in classical physics are always associated with insufficient knowledge, i.e., they are statements about the observer’s knowledge and not about the physical system, which according to the principles of classical physics can be known to unlimited accuracy. In quantum theory one can only say with what probability certain values can be expected, even if one knows the state as well as possible, i.e., even if the system is in a pure state. Thus in quantum theory statements are inherently probabilistic; the occurrence of probability functions is not just a consequence of the observer’s insufficient knowledge, but a property attributed to the physical systems themselves. Quantum predictions of experimental results are statements of how a microphysical process shows up in the macrophysical domain. These traces of microphysical processes in the macrophysical domain, the only source of human knowledge about such processes, do not obey deterministic laws. Earlier traces of a microphysical process do not determine later traces uniquely, but only probabilistically. Quantum theory teaches us that there are inherent limitations to human knowledge. [pp. 661-662]
The other major revision is no less fundamental:
The second point, that of the profoundly holistic nature (of the understanding) of quantum physical systems, is not often emphasized, even though it is an obvious consequence of the quantum-mechanical description of physical systems. Although holism has already become rather widely accepted in other disciplines (e.g., psychology), it has been resisted by the physicists, who seem to be influenced by the success of atomism in classical physics. The quantum physical system is a structured whole described by the mathematical structure of an algebra of operators. From the laws of the combination of quantum physical systems, it follows that there are observables – of the combination of the two subsystems that are incompatible with all observables of either subsystem. Thus, in quantum physics there exist holistic properties that cannot be obtained as combinations of the properties of the subsystems. In this sense the whole is not the sum of the parts. In the atomistic approach understanding comes from the reduction of the complex system to simpler subsystems by ever finer separations until one comes to the ultimate constituents. In quantum physics the presence of holistic properties prevents this reduction process, and the notion of ultimate constituents loses it meaning. Atomism belongs to classical physics. A quantum physical system such as a molecule cannot be fully understood by dissecting it into nuclei and electrons, although, in the tradition of our scientific heritage, it is tempting to do this. What one arrives at in this way, however, is only a classical analogue of the quantum physical system, as in the Kepler system of proton and electron for the classical analogue of the hydrogen atom. An electron in an atom ‘is’ something different from an electron in a linear accelerator, and the whole picture of the electron can only be displayed by giving its different aspects as they are mathematically described by the various basis systems in the space of physical states. [p. 662]
Note however one’s decision about what product Hilbert space to use to represent the molecular system is influenced by classical atomistic intuition i.e. reductionistic, only the analysis then must proceed from a global holistic point of view. Nevertheless, Bohm’s main point may be sustained. So we get a curious mixture of reduction and holism – Bohm’s remarks, incidentally, go to show as well just how glib and superficial atheists of the stamp of Lawrence Krauss can be, in that for all their ringing proclamations of reductionism they flagrantly fail to comprehend even the very science upon which they wish to base themselves!
Conclusion: Arno Bohm’s meticulous monograph on the foundations of quantum mechanics is excellent but not suitable as a first text unless one be a hot shot (somewhat akin to trying to teach oneself graduate-level algebra from Serge Lang).
Chapter four is good on the experimental basis of the rotator and the vibrating rotator – Bohm does not just derive the spectrum in a toy model, but gives energy level diagrams to compare with the experimental results on CO, HCl etc. Chapter five on the Wigner-Eckart theorem, a topic not usually covered even in graduate-level textbooks. He explains what a tensor operator is, then states but does not prove the Wigner-Eckart theorem – important because it allows one to express what one wants to know in terms of a much smaller number of reduced matrix elements (he shows how this works for scalar and vector operators, pp. 178-181). Again the following section §5.4 on parity is exhaustive and hardly ever covered by other authors yet vital if one wants any connection to experiment. Chapter 6 on the hydrogen atom follows Pauli’s original derivation in matrix mechanics via the Lenz vector and Wigner-Eckart theorem – note, the usual q and p are not physical observables; thus, in accord with his pedagogical principles, Bohm will not just write out and solve the Hamiltonian in the position coordinates as is usually done. Chapter seven is devoted to alkali atoms, which played an important role in the development of quantum mechanics (appealed to by Sommerfeld in the old quantum mechanics, and by Pauli when working out his exclusion principle). Again, level diagrams are shown for lithium, sodium and cesium.
In all this, one will appreciate Bohm’s constant emphasis on the physical basis in which system can be prepared versus convenient theoretical basis, leading to Clebsch-Gordon coefficients by which to interrelate the two, with many explicit formulae quoted. The important point to which most everyone else is oblivious: the q resp. p observables represent not just a convenient basis but in many cases a physical one that is good for representing states in which we are interested (the completion including high lying states is unphysical). To draw out the content of this last parenthetical comment: if one were to use any arbitrary and very complicated observable to generate the orthonormal basis, it would be uninterpretable. The question as to why the position and momentum bases exist is very profound and would repay reflection; it has to do with the ineluctability of spatial intuition in making the world known to us. Yet, as Niels Bohr never ceases to point out, there are limits to spatial intuition, especially in the quantum world. How is it to be complemented, and by what? This too marks out a very deep field for reflection and inquiry.
A last thing to note is that Bohm wants to prepare the ground for a far more extensive treatment of scattering theory than is the rule in beginning graduate-level textbooks. Hence, in chapter three he includes a very little on rigged Hilbert spaces, a subject to which Bohm has contributed original research memoirs. Chapter eight presents perturbation theory in a format designed to streamline passage from the discrete (Wigner-Brillouin resp. Rayleigh-Schrödinger) to the continuous case (Lippmann-Schwinger equation), which figures importantly in chapters eighteen to twenty-one. The exposition nevertheless is very compressed, e.g. introduces the principal-value integral without explanation (for which, see Gelfand or Reed-Simon). Chapters 18-20 on resonances, the S-matrix, analyticity and causality conditions get a little detailed and one loses sight of the flow of the derivation, especially in chapter 21 where Gamov vectors are explained in terms of rigged Hilbert spaces, Hardy class functions and the Titchmarsch and Paley-Wiener theorems. Here, the rapid-fire quotation without explanation of major results of real analysis leaves one with, at best, an impressionistic feeling for what is going on: to have a hope of comprehending all this one will have to refer to Walter Thirring, in the third volume on quantum mechanics of atoms and molecules in his A Course in Mathematical Physics or to Reed-Simon, in the third volume on scattering theory of Methods of Modern Mathematical Physics.
Chapter thirteen contains a good analysis of the Stern-Gerlach experiment, far more thorough than is the rule in textbook treatments. One will find here a lucid though terse explanation of the Bell inequalities, not very philosophically oriented but just the derivation of the relevant formulae. Compare Bohm’s derivation of the cross section from the Lippmann-Schwinger equation [p 362ff] with that given, say, by Cohen-Tannoudji. Chapter 22 covers the Berry phase, first off with a reprise of its general derivation but what’s good in Bohm’s account is §22.3 with an explicit computation in the case of a spinning quantum system in a slowly varying magnetic field (an analogue of Dirac’s magnetic monopole).
As Bohm goes on to show in the succeeding section, the Berry phase is an adiabatic approximation to the non-adiabatic exact Aharanov-Anandan phase: he splits the phase factor into geometrical and dynamical parts. Finally chapter 23 investigates this circle of ideas in the context of the Born-Oppenheimer approximation in molecular systems, in a gauge-theoretical formalism that includes the Berry phase which Born and Oppenheimer miss. Bohm describes conceptually what is at stake: the antireductionist spirit mentioned in the introduction [see p. 642, resp. 571]. Compare Bohm’s treatment with the original paper [Zur Quantentheorie der Molekeln, Annalen der Physik 389, 457-484 (1927)]. In line with his overall pedagogical intent, Bohm rounds out his discussion with a concrete derivation of the Berry connection and its curvature in diatomic molecules [p. 645ff] also yielding thereby a satisfying physical realization of a Dirac monopole (not electromagnetic of course).
The epilogue gets a little philosophical. Bohm highlights the two revisions of the quantum mechanical outlook over that of classical mechanics. First of all,
Probability statements in classical physics are always associated with insufficient knowledge, i.e., they are statements about the observer’s knowledge and not about the physical system, which according to the principles of classical physics can be known to unlimited accuracy. In quantum theory one can only say with what probability certain values can be expected, even if one knows the state as well as possible, i.e., even if the system is in a pure state. Thus in quantum theory statements are inherently probabilistic; the occurrence of probability functions is not just a consequence of the observer’s insufficient knowledge, but a property attributed to the physical systems themselves. Quantum predictions of experimental results are statements of how a microphysical process shows up in the macrophysical domain. These traces of microphysical processes in the macrophysical domain, the only source of human knowledge about such processes, do not obey deterministic laws. Earlier traces of a microphysical process do not determine later traces uniquely, but only probabilistically. Quantum theory teaches us that there are inherent limitations to human knowledge. [pp. 661-662]
The other major revision is no less fundamental:
The second point, that of the profoundly holistic nature (of the understanding) of quantum physical systems, is not often emphasized, even though it is an obvious consequence of the quantum-mechanical description of physical systems. Although holism has already become rather widely accepted in other disciplines (e.g., psychology), it has been resisted by the physicists, who seem to be influenced by the success of atomism in classical physics. The quantum physical system is a structured whole described by the mathematical structure of an algebra of operators. From the laws of the combination of quantum physical systems, it follows that there are observables – of the combination of the two subsystems that are incompatible with all observables of either subsystem. Thus, in quantum physics there exist holistic properties that cannot be obtained as combinations of the properties of the subsystems. In this sense the whole is not the sum of the parts. In the atomistic approach understanding comes from the reduction of the complex system to simpler subsystems by ever finer separations until one comes to the ultimate constituents. In quantum physics the presence of holistic properties prevents this reduction process, and the notion of ultimate constituents loses it meaning. Atomism belongs to classical physics. A quantum physical system such as a molecule cannot be fully understood by dissecting it into nuclei and electrons, although, in the tradition of our scientific heritage, it is tempting to do this. What one arrives at in this way, however, is only a classical analogue of the quantum physical system, as in the Kepler system of proton and electron for the classical analogue of the hydrogen atom. An electron in an atom ‘is’ something different from an electron in a linear accelerator, and the whole picture of the electron can only be displayed by giving its different aspects as they are mathematically described by the various basis systems in the space of physical states. [p. 662]
Note however one’s decision about what product Hilbert space to use to represent the molecular system is influenced by classical atomistic intuition i.e. reductionistic, only the analysis then must proceed from a global holistic point of view. Nevertheless, Bohm’s main point may be sustained. So we get a curious mixture of reduction and holism – Bohm’s remarks, incidentally, go to show as well just how glib and superficial atheists of the stamp of Lawrence Krauss can be, in that for all their ringing proclamations of reductionism they flagrantly fail to comprehend even the very science upon which they wish to base themselves!
Conclusion: Arno Bohm’s meticulous monograph on the foundations of quantum mechanics is excellent but not suitable as a first text unless one be a hot shot (somewhat akin to trying to teach oneself graduate-level algebra from Serge Lang).
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