Physics Today
Bohr’s molecular model and the melding of classical and quantum mechanics
Anatoly A. Svidzinsky, Marlan O. Scully, Dudley R. Herschbach
Published Aug 1, 2014• 2 citations
Abstract
Svidzinsky, Scully, and Herschbach reply: We agree with M. Y. Amusia that a statement we made about the infinite dimension limit is “lacking in rigor.” It was intended simply to provide heuristic insight into why, for a wide variety of problems, the large-D limit has proven to be a useful starting approximation to obtain results for D = 3.Amusia’s other concerns are answered in a 1975 paper on a D-scaling treatment of helium by David Herrick and Frank Stillinger.11. D. R. Herrick, F. H. Stillinger, Phys. Rev. A 11, 42 (1975). https://doi.org/10.1103/PhysRevA.11.42 They gave a rigorous derivation of the D-dimensional hydrogen atom Hamiltonian, shown as equation 1 in Amusia’s letter. That solution applies for D ≥ 2, as implied in the figure included in box 1 of our article. Herrick and Stillinger also showed that for the correct D → 1 limit, the Z/r term becomes a δ-function. In D-scaling, contrary to Amusia’s assumption, the D = 3 form of Coulomb’s law can be used for dimensional continuation to the large-D limit. The D-dependent similarity transformation affects the Laplacian, not the potential energy. Both theory and application are amply presented in references given in our article (particularly references 8–11). D-scaling, using just elementary algebra, has attained correlation energies for multielectron atoms with accuracy comparable to or better than conventional electronic calculations.22. See, for example, S. Kais, R. Bleil, J. Chem. Phys. 102, 7472 (1995); https://doi.org/10.1063/1.469059D. K. Watson, D. Z. Goodson, Phys. Rev. A 51, R5 (1995), and references therein. https://doi.org/10.1103/PhysRevA.51.R5The papers of Leonard Mlodinow33. See D. R. Herschbach, J. Chem. Phys. 84, 838 (1986), for analysis of the papers cited in reference 1 of Mlodinow’s letter. https://doi.org/10.1063/1.450584 and several other authors, especially the earlier paper by Herrick and Stillinger,11. D. R. Herrick, F. H. Stillinger, Phys. Rev. A 11, 42 (1975). https://doi.org/10.1103/PhysRevA.11.42 fostered the development of D-scaling for electronic structure. Since the kinship of Bohr’s model to dimensional scaling was not recognized until 2005, we did not dwell on that history, other than citing the tutorial article by Edward Witten. The treatment of H2 that Mlodinow cites in the second part of his reference 3 is a deliberately drastic approximation and gives less than 40% of the bond dissociation energy.As emphasized by Petar Grujic, the bold, perplexing enterprise by Niels Bohr motivated much further work melding classical and quantum mechanics. We note that D-scaling has a distinctive character. It might aptly be termed “semiquantum” rather than semiclassical. Although in the large-D limit the equations become classical, quantum mechanics is hidden in the D-dependent units adopted for distance and energy. At that limit, electrons take fixed positions in the D-scaled space and the first-order correction in (1/D) has them execute harmonic vibrations about those positions. In the prequantum era, such behavior was postulated by Gilbert Lewis and Irving Langmuir,44. P. Coffey, Cathedrals of Science: The Personalities and Rivalries That Made Modern Chemistry, Oxford U. Press, New York (2008), chap. 5, esp. p. 139. motivated by chemical arguments but disdained by physicists and considered incompatible with the Bohr model.REFERENCESSection:ChooseTop of pageREFERENCES <<CITING ARTICLES1. D. R. Herrick, F. H. Stillinger, Phys. Rev. A 11, 42 (1975). https://doi.org/10.1103/PhysRevA.11.42, Google ScholarCrossref, ISI2. See, for example, S. Kais, R. Bleil, J. Chem. Phys. 102, 7472 (1995); https://doi.org/10.1063/1.469059, Google ScholarCrossref, ISID. K. Watson, D. Z. Goodson, Phys. Rev. A 51, R5 (1995), and references therein. https://doi.org/10.1103/PhysRevA.51.R5, , Google ScholarCrossref, ISI3. See D. R. Herschbach, J. Chem. Phys. 84, 838 (1986), for analysis of the papers cited in reference 1 of Mlodinow’s letter. https://doi.org/10.1063/1.450584, Google ScholarCrossref, ISI4. P. Coffey, Cathedrals of Science: The Personalities and Rivalries That Made Modern Chemistry, Oxford U. Press, New York (2008), chap. 5, esp. p. 139. Google Scholar© 2014 American Institute of Physics.
- DOI
- 10.1063/pt.3.2462
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Bohr’s molecular model and the melding of classical and quantum mechanics
Leonard D. Mlodinow
Permit me to add some information to the article entitled, “Bohr’s molecular model, a century later.” The authors discuss the idea of “dimensional scaling,” approximating the spectra of atoms and molecules by employing perturbation theory to expand around the infinite-dimension/classical limit. That technique of approximating the Schrödinger equation, originally called the 1/N expansion, was developed and applied to numerous problems in the 1980s, including simple atoms, quarkonium, and the hydrogen molecule.11. L. D. Mlodinow, N. Papanicolaou, Ann. Phys. 128, 314 (1980); https://doi.org/10.1016/0003-4916(80)90323-1 L. D. Mlodinow, N. Papanicolaou, Ann. Phys. 131, 1 (1981). https://doi.org/10.1016/0003-4916(81)90181-0,22. C. M. Bender, L. D. Mlodinow, N. Papanicolaou, Phys. Rev. A 25, 1305 (1982). https://doi.org/10.1103/PhysRevA.25.1305,33. T. Imbo, U. Sukhatme, Phys. Rev. D 31, 2655 (1985); https://doi.org/10.1103/PhysRevD.31.2655M. Kumar, A. Srivastava, J. K. Bhattacharjee, K. Banerjee, Phys. Lett. A 117, 226 (1986). https://doi.org/10.1016/0375-9601(86)90081-2 It grew out of attempts in the 1970s to formulate quantum chromodynamics in the limit of a large number of colors. The use of the 1/N expansion in atomic physics was discussed by Edward Witten (Physics Today, July 1980, page 38), who pointed out that his discussion was based on work that applied the method to hydrogen and helium atoms.11. L. D. Mlodinow, N. Papanicolaou, Ann. Phys. 128, 314 (1980); https://doi.org/10.1016/0003-4916(80)90323-1 L. D. Mlodinow, N. Papanicolaou, Ann. Phys. 131, 1 (1981). https://doi.org/10.1016/0003-4916(81)90181-0The work in reference 11. L. D. Mlodinow, N. Papanicolaou, Ann. Phys. 128, 314 (1980); https://doi.org/10.1016/0003-4916(80)90323-1 L. D. Mlodinow, N. Papanicolaou, Ann. Phys. 131, 1 (1981). https://doi.org/10.1016/0003-4916(81)90181-0 made use of algebraic methods for the analysis, but the coordinate space method was also developed and was initially used to treat the strong-field Zeeman effect.22. C. M. Bender, L. D. Mlodinow, N. Papanicolaou, Phys. Rev. A 25, 1305 (1982). https://doi.org/10.1103/PhysRevA.25.1305 A nice description of the status of the field at the time was given by Laurence Yaffe (Physics Today, August 1983, page 50). Many additional applications of that method in the physics literature may be found in reference 44. A. Chatterjee, Phys. Rep. 186, 249 (1990). https://doi.org/10.1016/0370-1573(90)90048-7.REFERENCESSection:ChooseTop of pageREFERENCES <<1. L. D. Mlodinow, N. Papanicolaou, Ann. Phys. 128, 314 (1980); https://doi.org/10.1016/0003-4916(80)90323-1 , Google ScholarCrossref, ISIL. D. Mlodinow, N. Papanicolaou, Ann. Phys. 131, 1 (1981). https://doi.org/10.1016/0003-4916(81)90181-0, , Google ScholarCrossref, ISI2. C. M. Bender, L. D. Mlodinow, N. Papanicolaou, Phys. Rev. A 25, 1305 (1982). https://doi.org/10.1103/PhysRevA.25.1305, Google ScholarCrossref, ISI3. T. Imbo, U. Sukhatme, Phys. Rev. D 31, 2655 (1985); https://doi.org/10.1103/PhysRevD.31.2655, Google ScholarCrossref, ISIM. Kumar, A. Srivastava, J. K. Bhattacharjee, K. Banerjee, Phys. Lett. A 117, 226 (1986). https://doi.org/10.1016/0375-9601(86)90081-2, , Google ScholarCrossref, ISI4. A. Chatterjee, Phys. Rep. 186, 249 (1990). https://doi.org/10.1016/0370-1573(90)90048-7, Google ScholarCrossref, ISI© 2014 American Institute of Physics.
Physics Today•Aug 1, 2014•1 citation
Bohr’s molecular model and the melding of classical and quantum mechanics
Petar Grujić
Anatoly Svidzinsky, Marlan Scully, and Dudley Herschbach have illustrated well how the molecular theory might have looked had quantum mechanics not been invented. It’s worth noting that Niels Bohr was not alone in his attempts to extend semiclassical mechanics to systems more complex than the hydrogen atom. From 1913 to 1925, many semiclassical models of two-electron systems were proposed.As we see from the authors’ figure 3a, Bohr assumed that electrons in hydrogen molecules move around the molecular axis out of phase. It is a remarkable fact that Werner Heisenberg put forward in 1922 a two-electron atomic model that had electrons moving similarly out of phase and that ascribed a fractional quantum number (½) to the bending mode of the configuration.11. G. Tanner, K. Richter, J.-M. Rost, Rev. Mod. Phys. 72, 497 (2000). https://doi.org/10.1103/RevModPhys.72.497 Although Heisenberg’s model predicted the ground-state helium level, which was in excellent agreement with the spectroscopic data, Bohr opposed the model, since a noninteger quantum number was considered sacrilege. For the same reason, Bohr never published his own results. However, the asynchronous two-electron atomic model turned out to be the fertile one, though the corresponding classical configurations can have very complex structure.22. P. Grujic, N. Simonovic, J. Phys. B 28, 1159 (1995). https://doi.org/10.1088/0953-4075/28/7/011REFERENCESSection:ChooseTop of pageREFERENCES <<1. G. Tanner, K. Richter, J.-M. Rost, Rev. Mod. Phys. 72, 497 (2000). https://doi.org/10.1103/RevModPhys.72.497, Google ScholarCrossref, ISI2. P. Grujic, N. Simonovic, J. Phys. B 28, 1159 (1995). https://doi.org/10.1088/0953-4075/28/7/011, Google ScholarCrossref, ISI© 2014 American Institute of Physics.
Physics Today•Aug 1, 2014•1 citation
Bohr’s molecular model and the melding of classical and quantum mechanics
M. Ya. Amusia
With considerable interest I read the article “Bohr’s molecular model, a century later” by Anatoly Svidzinsky, Marlan Scully, and Dudley Herschbach (Physics Today, January 2014, page 33). It is always interesting to see how our current understanding sheds new light on the revolutionary scientific ideas of the past. I agree with the authors that although quantum mechanics is the real basis for atomic- and molecular-physics computations, the old Bohr model that treats electrons in atoms like tiny planets moving around the sun-nucleus is intuitively clear and very attractive.As I see it, the aim of the article is to show how to use Bohr’s approach to treat not only simple atoms but molecules. An important point in that approach is reconciliation of quantum mechanics with Bohr’s ideas. The authors claim that in infinite dimensions, quantum mechanics “morphs into classical mechanics.”I cannot say, however, that the infinite-dimension system is a clarifying model to describe physical or chemical objects. I do not see that the reference to chromodynamics (and to Edward Witten’s article in Physics Today, July 1980, page 38) is a clarifying one. Far from convincing are statements like “Hence the large-D limit, where 1/D → 0, is closer to the real world (1/D = 1/3) than is the oft-used D = 1 regime. Indeed, results obtained at large D usually resemble those for D = 3.” I confess it sounds too lacking in rigor to be convincing.But the large-D limit per se is not what bothers me. Of great concern is the assumption that the radial part of the D-dimensional Schrödinger equation in Hartree units looks like this:(1)where Z is the nuclear charge and l is the angular momentum.Equation (1) has no sense for D = 1. Indeed, in one dimension, a finite angular momentum requires infinite speed of a rotating particle. Therefore, for D = 1, the correct equation is(2)which does not follow from equation (1) at D = 1.But most important is that equation (1) implies that the Coulomb potential does not depend on D. That could have been considered correct before it became clear that the Coulomb law follows from Maxwell’s equations when one considers a field generated by a point-like electric charge. So to obtain the Coulomb law for a two-dimensional space, one has to consider Maxwell equations in a two-dimensional world. For a point-like charge, those equations lead to a 1/(ln r) instead of 1/r dependence. If one takes into account the D-dependence of the Coulomb potential, derivations performed in the Physics Today article become meaningless. Indeed, when using an unrealistic equation (1), how can one believe that it illuminates Bohr’s quite realistic and physical postulates?© 2014 American Institute of Physics.
Physics Today•Aug 1, 2014•1 citation
Beyond the classical view of atoms
David L. Rosen
The article “Insights from the classical atom,” by Petar Grujić and Nenad Simonović (Physics Today, May 2012, page 41), was interesting but incomplete. It left out stochastic electrodynamics (SED), an approach to classical mechanics that includes the effects of a proposed zero-point radiation background.The zero-point radiation is a Lorentz-invariant random distribution of energy into different modes of electromagnetic radiation. Quantum electrodynamics (QED) uses the hypothesis that the amplitudes of the Lorentz-invariant radiation are quantized, whereas SED uses the hypothesis that they are continuous.The motions of electric charges in SED are modeled using the equations of motion for a particle, often associated with a concrete visualization. SED is considered part of classical physics because of the continuous distribution of amplitudes. However, discrete values for energy and momentum emerge from SED without the ad hoc quantization conditions of QED.Stochastic electrodynamics has been investigated as a classical physics explanation for quantum phenomena. For instance, the blackbody radiation spectrum has been derived without quantum assumptions.11. T. H. Boyer, Phys. Rev. 182, 1374 (1969). https://doi.org/10.1103/PhysRev.182.1374 Lorentz-invariant radiation provides perturbations that mimic quantum phenomena such as the Casimir force, the van der Waals force, the Lamb shift, and the radius of the Bohr atom.22. M. Ibison, B. Haisch, Phys. Rev. A 54, 2737 (1996). https://doi.org/10.1103/PhysRevA.54.2737Physical chemistry provides many mathematical expressions that are often thought to be fundamentally quantum in nature. However, some of those expressions have been shown to emerge from both QED and SED. For example, the Schrödinger equation has been derived from SED,33. G. H. Goedecke, Found. Phys. 13, 1101 (1983); https://doi.org/10.1007/BF00728139 G. H. Goedecke, Found. Phys. 13, 1195 (1983). https://doi.org/10.1007/BF00727993 and the wavefunction in the SED Schrödinger equation is shown to have the a priori significance of position probability amplitude. A physical model based on SED has been developed to increase our understanding of both the diffraction of particle beams and the Pauli exclusion principle.44. A. F. Kracklauer, Found. Phys. Lett. 12, 441 (1999). https://doi.org/10.1023/A:1021629310707Quantum mechanics is often cited to justify chemical bonding on an elementary level. However, the visualizations used for chemical bonding are often based on mathematical expressions that can be derived from either QED or SED. Regardless of which turns out to be more accurate in physics experiments or more complete in mathematical description, the two theories overlap with regard to physical chemistry. Therefore, I propose that introductory chemistry taught in secondary schools can be rationalized as easily by references to SED as to QED.Either theory produces identical predictions for many experiments involving second-order correlations, squeezing (quadrature noise reduction), and the original Einstein-Podolsky-Rosen proposal.55. S. Chaturvedi, P. D. Drummond, Phys. Rev. A 55, 912 (1997). https://doi.org/10.1103/PhysRevA.55.912 However, SED is not completely equivalent to QED on a mathematical or physical level. The SED Schrödinger equation is incomplete, because there is a companion equation that has no counterpart in ordinary quantum mechanics and restricts initial conditions.33. G. H. Goedecke, Found. Phys. 13, 1101 (1983); https://doi.org/10.1007/BF00728139 G. H. Goedecke, Found. Phys. 13, 1195 (1983). https://doi.org/10.1007/BF00727993 Furthermore, the predictions of QED and SED are different with regard to high-order correlations in nonlinear optics.I agree fully with Grujić and Simonović that the classical atom is alive and well. Models of the atom based on SED are sufficiently accurate for many technological and most pedagogical applications. Even if the classical atom isn’t the most fundamental description of the physical world, it is still an important one.REFERENCESSection:ChooseTop of pageREFERENCES <<CITING ARTICLES1. T. H. Boyer, Phys. Rev. 182, 1374 (1969). https://doi.org/10.1103/PhysRev.182.1374, Google ScholarCrossref, ISI2. M. Ibison, B. Haisch, Phys. Rev. A 54, 2737 (1996). https://doi.org/10.1103/PhysRevA.54.2737, Google ScholarCrossref, ISI3. G. H. Goedecke, Found. Phys. 13, 1101 (1983); https://doi.org/10.1007/BF00728139 , Google ScholarCrossref, ISIG. H. Goedecke, Found. Phys. 13, 1195 (1983). https://doi.org/10.1007/BF00727993, , Google ScholarCrossref, ISI4. A. F. Kracklauer, Found. Phys. Lett. 12, 441 (1999). https://doi.org/10.1023/A:1021629310707, Google ScholarCrossref, ISI5. S. Chaturvedi, P. D. Drummond, Phys. Rev. A 55, 912 (1997). https://doi.org/10.1103/PhysRevA.55.912, Google ScholarCrossref, ISI© 2013 American Institute of Physics.
Physics Today•Apr 30, 2013•2 citations
BCS-to-BEC evolution details
D. M. Eagles
Again an article in Physics Today, (by Carlos Sá de melo October 2008, page 45), has incorrectly implied that Anthony Leggett was the first person to study the crossover from Bardeen-Cooper-Schrieffer to Bose–Einstein condensation. On page 47 of the article, it states that “a clear picture of the BCS-to-BEC evolution at zero temperature didn’t emerge until 1980, when Anthony Leggett realized that the physics could be captured by a simple description in real space of paired fermions with opposite spins.” Although the model I considered in my 1969 paper 1 1. D. M. Eagles, Phys. Rev. 186, 456 (1969). https://doi.org/10.1103/PhysRev.186.456 is slightly different from Leggett’s, figure 4 in my paper clearly shows regions where pairing without superconductivity occurs and where superconductivity is limited by the Bose- condensation temperature of pairs, and on page 458 I discuss a limit at which the diameter of pairs is small compared with the distance between them.I also disagree with a statement in the box on page 47 of Sá de Melo’s article that “the evolution from a Bardeen-Cooper-Schrieffer superfluid to a Bose–Einstein condensation superfluid cannot be studied in … superconductors.” At least in ceramic samples of SrTiO3 with 3% of the titanium replaced by zirconium, the transition has been studied by varying the carrier concentration via differing heat treatments to produce different concentrations of oxygen vacancies. 2 2. R. J. Tainsh, C. Andrikidis, Solid State Commun. 60, 517 (1986) https://doi.org/10.1016/0038-1098(86)90729-5 D. M. Eagles, Solid State Commun. 60, 521 (1986) https://doi.org/10.1016/0038-1098(86)90730-1 D. M. Eagles, R. J. Tainsh, C. Andrikidis, Physica C 157, 48 (1989). https://doi.org/10.1016/0921-4534(89)90466-8 It is possible that such a transition may be found in other superconducting semiconductors when people start to search for suitable materials. However, in three dimensions the pairing strength has to be above some threshold value to obtain the possibility of reaching the Bose-gas regime. Also, many authors think that the BEC regime occurs in underdoped cuprates, 3,4 3. Q. Chen, J. Stajic, S. Tan, K. Levin, Phys. Rep. 412, 1 (2005). https://doi.org/10.1016/j.physrep.2005.02.005 4. A. S. Alexandrov, J. Supercond. Nov. Magn. 20, 481 (2007). https://doi.org/10.1007/s10948-007-0258-z while the consensus is that overdoped samples are BCS-like.REFERENCESSection:ChooseTop of pageREFERENCES <<CITING ARTICLES1. D. M. Eagles, Phys. Rev. 186, 456 (1969). https://doi.org/10.1103/PhysRev.186.456 , Google ScholarCrossref, ISI2. R. J. Tainsh, C. Andrikidis, Solid State Commun. 60, 517 (1986) https://doi.org/10.1016/0038-1098(86)90729-5 , Google ScholarCrossref, ISI D. M. Eagles, Solid State Commun. 60, 521 (1986) https://doi.org/10.1016/0038-1098(86)90730-1 , , Google ScholarCrossref, ISI D. M. Eagles, R. J. Tainsh, C. Andrikidis, Physica C 157, 48 (1989). https://doi.org/10.1016/0921-4534(89)90466-8 , , Google ScholarCrossref, ISI3. Q. Chen, J. Stajic, S. Tan, K. Levin, Phys. Rep. 412, 1 (2005). https://doi.org/10.1016/j.physrep.2005.02.005 , Google ScholarCrossref, ISI4. A. S. Alexandrov, J. Supercond. Nov. Magn. 20, 481 (2007). https://doi.org/10.1007/s10948-007-0258-z , Google ScholarCrossref, ISI© 2009 American Institute of Physics.
Physics Today•Jul 31, 2009•4 citations
Svidzinsky, Scully, and Herschbach reply: We agree with M. Y. Amusia that a statement we made about the infinite dimension limit is “lacking in rigor.” It was intended simply to provide heuristic insight into why, for a wide variety of problems, the large-D limit has proven to be a useful starting approximation to obtain results for D = 3.Amusia’s other concerns are answered in a 1975 paper on a D-scaling treatment of helium by David Herrick and Frank Stillinger.11. D. R. Herrick, F. H. Stillinger, Phys. Rev. A 11, 42 (1975).
https://doi.org/10.1103/PhysRevA.11.42 They gave a rigorous derivation of the D-dimensional hydrogen atom Hamiltonian, shown as equation 1 in Amusia’s letter. That solution applies for D ≥ 2, as implied in the figure included in box 1 of our article. Herrick and Stillinger also showed that for the correct D → 1 limit, the Z/r term becomes a δ-function. In D-scaling, contrary to Amusia’s assumption, the D = 3 form of Coulomb’s law can be used for dimensional continuation to the large-D limit. The D-dependent similarity transformation affects the Laplacian, not the potential energy. Both theory and application are amply presented in references given in our article (particularly references 8–11). D-scaling, using just elementary algebra, has attained correlation energies for multielectron atoms with accuracy comparable to or better than conventional electronic calculations.22. See, for example, S. Kais, R. Bleil, J. Chem. Phys. 102, 7472 (1995);
https://doi.org/10.1063/1.469059D. K. Watson, D. Z. Goodson, Phys. Rev. A 51, R5 (1995), and references therein.
https://doi.org/10.1103/PhysRevA.51.R5The papers of Leonard Mlodinow33. See D. R. Herschbach, J. Chem. Phys. 84, 838 (1986), for analysis of the papers cited in reference 1 of Mlodinow’s letter.
https://doi.org/10.1063/1.450584 and several other authors, especially the earlier paper by Herrick and Stillinger,11. D. R. Herrick, F. H. Stillinger, Phys. Rev. A 11, 42 (1975).
https://doi.org/10.1103/PhysRevA.11.42 fostered the development of D-scaling for electronic structure. Since the kinship of Bohr’s model to dimensional scaling was not recognized until 2005, we did not dwell on that history, other than citing the tutorial article by Edward Witten. The treatment of H2 that Mlodinow cites in the second part of his reference 3 is a deliberately drastic approximation and gives less than 40% of the bond dissociation energy.As emphasized by Petar Grujic, the bold, perplexing enterprise by Niels Bohr motivated much further work melding classical and quantum mechanics. We note that D-scaling has a distinctive character. It might aptly be termed “semiquantum” rather than semiclassical. Although in the large-D limit the equations become classical, quantum mechanics is hidden in the D-dependent units adopted for distance and energy. At that limit, electrons take fixed positions in the D-scaled space and the first-order correction in (1/D) has them execute harmonic vibrations about those positions. In the prequantum era, such behavior was postulated by Gilbert Lewis and Irving Langmuir,44. P. Coffey, Cathedrals of Science: The Personalities and Rivalries That Made Modern Chemistry, Oxford U. Press, New York (2008), chap. 5, esp. p. 139. motivated by chemical arguments but disdained by physicists and considered incompatible with the Bohr model.REFERENCESSection:ChooseTop of pageREFERENCES <<CITING ARTICLES1. D. R. Herrick, F. H. Stillinger, Phys. Rev. A 11, 42 (1975).
https://doi.org/10.1103/PhysRevA.11.42, Google ScholarCrossref, ISI2. See, for example, S. Kais, R. Bleil, J. Chem. Phys. 102, 7472 (1995);
https://doi.org/10.1063/1.469059, Google ScholarCrossref, ISID. K. Watson, D. Z. Goodson, Phys. Rev. A 51, R5 (1995), and references therein.
https://doi.org/10.1103/PhysRevA.51.R5, , Google ScholarCrossref, ISI3. See D. R. Herschbach, J. Chem. Phys. 84, 838 (1986), for analysis of the papers cited in reference 1 of Mlodinow’s letter.
https://doi.org/10.1063/1.450584, Google ScholarCrossref, ISI4. P. Coffey, Cathedrals of Science: The Personalities and Rivalries That Made Modern Chemistry, Oxford U. Press, New York (2008), chap. 5, esp. p. 139. Google Scholar© 2014 American Institute of Physics.
Permit me to add some information to the article entitled, “Bohr’s molecular model, a century later.” The authors discuss the idea of “dimensional scaling,” approximating the spectra of atoms and molecules by employing perturbation theory to expand around the infinite-dimension/classical limit. That technique of approximating the Schrödinger equation, originally called the 1/N expansion, was developed and applied to numerous problems in the 1980s, including simple atoms, quarkonium, and the hydrogen molecule.11. L. D. Mlodinow, N. Papanicolaou, Ann. Phys. 128, 314 (1980); https://doi.org/10.1016/0003-4916(80)90323-1 L. D. Mlodinow, N. Papanicolaou, Ann. Phys. 131, 1 (1981). https://doi.org/10.1016/0003-4916(81)90181-0,22. C. M. Bender, L. D. Mlodinow, N. Papanicolaou, Phys. Rev. A 25, 1305 (1982). https://doi.org/10.1103/PhysRevA.25.1305,33. T. Imbo, U. Sukhatme, Phys. Rev. D 31, 2655 (1985); https://doi.org/10.1103/PhysRevD.31.2655M. Kumar, A. Srivastava, J. K. Bhattacharjee, K. Banerjee, Phys. Lett. A 117, 226 (1986). https://doi.org/10.1016/0375-9601(86)90081-2 It grew out of attempts in the 1970s to formulate quantum chromodynamics in the limit of a large number of colors. The use of the 1/N expansion in atomic physics was discussed by Edward Witten (Physics Today, July 1980, page 38), who pointed out that his discussion was based on work that applied the method to hydrogen and helium atoms.11. L. D. Mlodinow, N. Papanicolaou, Ann. Phys. 128, 314 (1980); https://doi.org/10.1016/0003-4916(80)90323-1 L. D. Mlodinow, N. Papanicolaou, Ann. Phys. 131, 1 (1981). https://doi.org/10.1016/0003-4916(81)90181-0The work in reference 11. L. D. Mlodinow, N. Papanicolaou, Ann. Phys. 128, 314 (1980); https://doi.org/10.1016/0003-4916(80)90323-1 L. D. Mlodinow, N. Papanicolaou, Ann. Phys. 131, 1 (1981). https://doi.org/10.1016/0003-4916(81)90181-0 made use of algebraic methods for the analysis, but the coordinate space method was also developed and was initially used to treat the strong-field Zeeman effect.22. C. M. Bender, L. D. Mlodinow, N. Papanicolaou, Phys. Rev. A 25, 1305 (1982). https://doi.org/10.1103/PhysRevA.25.1305 A nice description of the status of the field at the time was given by Laurence Yaffe (Physics Today, August 1983, page 50). Many additional applications of that method in the physics literature may be found in reference 44. A. Chatterjee, Phys. Rep. 186, 249 (1990). https://doi.org/10.1016/0370-1573(90)90048-7.REFERENCESSection:ChooseTop of pageREFERENCES <<1. L. D. Mlodinow, N. Papanicolaou, Ann. Phys. 128, 314 (1980); https://doi.org/10.1016/0003-4916(80)90323-1 , Google ScholarCrossref, ISIL. D. Mlodinow, N. Papanicolaou, Ann. Phys. 131, 1 (1981). https://doi.org/10.1016/0003-4916(81)90181-0, , Google ScholarCrossref, ISI2. C. M. Bender, L. D. Mlodinow, N. Papanicolaou, Phys. Rev. A 25, 1305 (1982). https://doi.org/10.1103/PhysRevA.25.1305, Google ScholarCrossref, ISI3. T. Imbo, U. Sukhatme, Phys. Rev. D 31, 2655 (1985); https://doi.org/10.1103/PhysRevD.31.2655, Google ScholarCrossref, ISIM. Kumar, A. Srivastava, J. K. Bhattacharjee, K. Banerjee, Phys. Lett. A 117, 226 (1986). https://doi.org/10.1016/0375-9601(86)90081-2, , Google ScholarCrossref, ISI4. A. Chatterjee, Phys. Rep. 186, 249 (1990). https://doi.org/10.1016/0370-1573(90)90048-7, Google ScholarCrossref, ISI© 2014 American Institute of Physics.
Anatoly Svidzinsky, Marlan Scully, and Dudley Herschbach have illustrated well how the molecular theory might have looked had quantum mechanics not been invented. It’s worth noting that Niels Bohr was not alone in his attempts to extend semiclassical mechanics to systems more complex than the hydrogen atom. From 1913 to 1925, many semiclassical models of two-electron systems were proposed.As we see from the authors’ figure 3a, Bohr assumed that electrons in hydrogen molecules move around the molecular axis out of phase. It is a remarkable fact that Werner Heisenberg put forward in 1922 a two-electron atomic model that had electrons moving similarly out of phase and that ascribed a fractional quantum number (½) to the bending mode of the configuration.11. G. Tanner, K. Richter, J.-M. Rost, Rev. Mod. Phys. 72, 497 (2000). https://doi.org/10.1103/RevModPhys.72.497 Although Heisenberg’s model predicted the ground-state helium level, which was in excellent agreement with the spectroscopic data, Bohr opposed the model, since a noninteger quantum number was considered sacrilege. For the same reason, Bohr never published his own results. However, the asynchronous two-electron atomic model turned out to be the fertile one, though the corresponding classical configurations can have very complex structure.22. P. Grujic, N. Simonovic, J. Phys. B 28, 1159 (1995). https://doi.org/10.1088/0953-4075/28/7/011REFERENCESSection:ChooseTop of pageREFERENCES <<1. G. Tanner, K. Richter, J.-M. Rost, Rev. Mod. Phys. 72, 497 (2000). https://doi.org/10.1103/RevModPhys.72.497, Google ScholarCrossref, ISI2. P. Grujic, N. Simonovic, J. Phys. B 28, 1159 (1995). https://doi.org/10.1088/0953-4075/28/7/011, Google ScholarCrossref, ISI© 2014 American Institute of Physics.
With considerable interest I read the article “Bohr’s molecular model, a century later” by Anatoly Svidzinsky, Marlan Scully, and Dudley Herschbach (Physics Today, January 2014, page 33). It is always interesting to see how our current understanding sheds new light on the revolutionary scientific ideas of the past. I agree with the authors that although quantum mechanics is the real basis for atomic- and molecular-physics computations, the old Bohr model that treats electrons in atoms like tiny planets moving around the sun-nucleus is intuitively clear and very attractive.As I see it, the aim of the article is to show how to use Bohr’s approach to treat not only simple atoms but molecules. An important point in that approach is reconciliation of quantum mechanics with Bohr’s ideas. The authors claim that in infinite dimensions, quantum mechanics “morphs into classical mechanics.”I cannot say, however, that the infinite-dimension system is a clarifying model to describe physical or chemical objects. I do not see that the reference to chromodynamics (and to Edward Witten’s article in Physics Today, July 1980, page 38) is a clarifying one. Far from convincing are statements like “Hence the large-D limit, where 1/D → 0, is closer to the real world (1/D = 1/3) than is the oft-used D = 1 regime. Indeed, results obtained at large D usually resemble those for D = 3.” I confess it sounds too lacking in rigor to be convincing.But the large-D limit per se is not what bothers me. Of great concern is the assumption that the radial part of the D-dimensional Schrödinger equation in Hartree units looks like this:(1)where Z is the nuclear charge and l is the angular momentum.Equation (1) has no sense for D = 1. Indeed, in one dimension, a finite angular momentum requires infinite speed of a rotating particle. Therefore, for D = 1, the correct equation is(2)which does not follow from equation (1) at D = 1.But most important is that equation (1) implies that the Coulomb potential does not depend on D. That could have been considered correct before it became clear that the Coulomb law follows from Maxwell’s equations when one considers a field generated by a point-like electric charge. So to obtain the Coulomb law for a two-dimensional space, one has to consider Maxwell equations in a two-dimensional world. For a point-like charge, those equations lead to a 1/(ln r) instead of 1/r dependence. If one takes into account the D-dependence of the Coulomb potential, derivations performed in the Physics Today article become meaningless. Indeed, when using an unrealistic equation (1), how can one believe that it illuminates Bohr’s quite realistic and physical postulates?© 2014 American Institute of Physics.
The article “Insights from the classical atom,” by Petar Grujić and Nenad Simonović (Physics Today, May 2012, page 41), was interesting but incomplete. It left out stochastic electrodynamics (SED), an approach to classical mechanics that includes the effects of a proposed zero-point radiation background.The zero-point radiation is a Lorentz-invariant random distribution of energy into different modes of electromagnetic radiation. Quantum electrodynamics (QED) uses the hypothesis that the amplitudes of the Lorentz-invariant radiation are quantized, whereas SED uses the hypothesis that they are continuous.The motions of electric charges in SED are modeled using the equations of motion for a particle, often associated with a concrete visualization. SED is considered part of classical physics because of the continuous distribution of amplitudes. However, discrete values for energy and momentum emerge from SED without the ad hoc quantization conditions of QED.Stochastic electrodynamics has been investigated as a classical physics explanation for quantum phenomena. For instance, the blackbody radiation spectrum has been derived without quantum assumptions.11. T. H. Boyer, Phys. Rev. 182, 1374 (1969). https://doi.org/10.1103/PhysRev.182.1374 Lorentz-invariant radiation provides perturbations that mimic quantum phenomena such as the Casimir force, the van der Waals force, the Lamb shift, and the radius of the Bohr atom.22. M. Ibison, B. Haisch, Phys. Rev. A 54, 2737 (1996). https://doi.org/10.1103/PhysRevA.54.2737Physical chemistry provides many mathematical expressions that are often thought to be fundamentally quantum in nature. However, some of those expressions have been shown to emerge from both QED and SED. For example, the Schrödinger equation has been derived from SED,33. G. H. Goedecke, Found. Phys. 13, 1101 (1983); https://doi.org/10.1007/BF00728139 G. H. Goedecke, Found. Phys. 13, 1195 (1983). https://doi.org/10.1007/BF00727993 and the wavefunction in the SED Schrödinger equation is shown to have the a priori significance of position probability amplitude. A physical model based on SED has been developed to increase our understanding of both the diffraction of particle beams and the Pauli exclusion principle.44. A. F. Kracklauer, Found. Phys. Lett. 12, 441 (1999). https://doi.org/10.1023/A:1021629310707Quantum mechanics is often cited to justify chemical bonding on an elementary level. However, the visualizations used for chemical bonding are often based on mathematical expressions that can be derived from either QED or SED. Regardless of which turns out to be more accurate in physics experiments or more complete in mathematical description, the two theories overlap with regard to physical chemistry. Therefore, I propose that introductory chemistry taught in secondary schools can be rationalized as easily by references to SED as to QED.Either theory produces identical predictions for many experiments involving second-order correlations, squeezing (quadrature noise reduction), and the original Einstein-Podolsky-Rosen proposal.55. S. Chaturvedi, P. D. Drummond, Phys. Rev. A 55, 912 (1997). https://doi.org/10.1103/PhysRevA.55.912 However, SED is not completely equivalent to QED on a mathematical or physical level. The SED Schrödinger equation is incomplete, because there is a companion equation that has no counterpart in ordinary quantum mechanics and restricts initial conditions.33. G. H. Goedecke, Found. Phys. 13, 1101 (1983); https://doi.org/10.1007/BF00728139 G. H. Goedecke, Found. Phys. 13, 1195 (1983). https://doi.org/10.1007/BF00727993 Furthermore, the predictions of QED and SED are different with regard to high-order correlations in nonlinear optics.I agree fully with Grujić and Simonović that the classical atom is alive and well. Models of the atom based on SED are sufficiently accurate for many technological and most pedagogical applications. Even if the classical atom isn’t the most fundamental description of the physical world, it is still an important one.REFERENCESSection:ChooseTop of pageREFERENCES <<CITING ARTICLES1. T. H. Boyer, Phys. Rev. 182, 1374 (1969). https://doi.org/10.1103/PhysRev.182.1374, Google ScholarCrossref, ISI2. M. Ibison, B. Haisch, Phys. Rev. A 54, 2737 (1996). https://doi.org/10.1103/PhysRevA.54.2737, Google ScholarCrossref, ISI3. G. H. Goedecke, Found. Phys. 13, 1101 (1983); https://doi.org/10.1007/BF00728139 , Google ScholarCrossref, ISIG. H. Goedecke, Found. Phys. 13, 1195 (1983). https://doi.org/10.1007/BF00727993, , Google ScholarCrossref, ISI4. A. F. Kracklauer, Found. Phys. Lett. 12, 441 (1999). https://doi.org/10.1023/A:1021629310707, Google ScholarCrossref, ISI5. S. Chaturvedi, P. D. Drummond, Phys. Rev. A 55, 912 (1997). https://doi.org/10.1103/PhysRevA.55.912, Google ScholarCrossref, ISI© 2013 American Institute of Physics.
Again an article in Physics Today, (by Carlos Sá de melo October 2008, page 45), has incorrectly implied that Anthony Leggett was the first person to study the crossover from Bardeen-Cooper-Schrieffer to Bose–Einstein condensation. On page 47 of the article, it states that “a clear picture of the BCS-to-BEC evolution at zero temperature didn’t emerge until 1980, when Anthony Leggett realized that the physics could be captured by a simple description in real space of paired fermions with opposite spins.” Although the model I considered in my 1969 paper 1 1. D. M. Eagles, Phys. Rev. 186, 456 (1969). https://doi.org/10.1103/PhysRev.186.456 is slightly different from Leggett’s, figure 4 in my paper clearly shows regions where pairing without superconductivity occurs and where superconductivity is limited by the Bose- condensation temperature of pairs, and on page 458 I discuss a limit at which the diameter of pairs is small compared with the distance between them.I also disagree with a statement in the box on page 47 of Sá de Melo’s article that “the evolution from a Bardeen-Cooper-Schrieffer superfluid to a Bose–Einstein condensation superfluid cannot be studied in … superconductors.” At least in ceramic samples of SrTiO3 with 3% of the titanium replaced by zirconium, the transition has been studied by varying the carrier concentration via differing heat treatments to produce different concentrations of oxygen vacancies. 2 2. R. J. Tainsh, C. Andrikidis, Solid State Commun. 60, 517 (1986) https://doi.org/10.1016/0038-1098(86)90729-5 D. M. Eagles, Solid State Commun. 60, 521 (1986) https://doi.org/10.1016/0038-1098(86)90730-1 D. M. Eagles, R. J. Tainsh, C. Andrikidis, Physica C 157, 48 (1989). https://doi.org/10.1016/0921-4534(89)90466-8 It is possible that such a transition may be found in other superconducting semiconductors when people start to search for suitable materials. However, in three dimensions the pairing strength has to be above some threshold value to obtain the possibility of reaching the Bose-gas regime. Also, many authors think that the BEC regime occurs in underdoped cuprates, 3,4 3. Q. Chen, J. Stajic, S. Tan, K. Levin, Phys. Rep. 412, 1 (2005). https://doi.org/10.1016/j.physrep.2005.02.005 4. A. S. Alexandrov, J. Supercond. Nov. Magn. 20, 481 (2007). https://doi.org/10.1007/s10948-007-0258-z while the consensus is that overdoped samples are BCS-like.REFERENCESSection:ChooseTop of pageREFERENCES <<CITING ARTICLES1. D. M. Eagles, Phys. Rev. 186, 456 (1969). https://doi.org/10.1103/PhysRev.186.456 , Google ScholarCrossref, ISI2. R. J. Tainsh, C. Andrikidis, Solid State Commun. 60, 517 (1986) https://doi.org/10.1016/0038-1098(86)90729-5 , Google ScholarCrossref, ISI D. M. Eagles, Solid State Commun. 60, 521 (1986) https://doi.org/10.1016/0038-1098(86)90730-1 , , Google ScholarCrossref, ISI D. M. Eagles, R. J. Tainsh, C. Andrikidis, Physica C 157, 48 (1989). https://doi.org/10.1016/0921-4534(89)90466-8 , , Google ScholarCrossref, ISI3. Q. Chen, J. Stajic, S. Tan, K. Levin, Phys. Rep. 412, 1 (2005). https://doi.org/10.1016/j.physrep.2005.02.005 , Google ScholarCrossref, ISI4. A. S. Alexandrov, J. Supercond. Nov. Magn. 20, 481 (2007). https://doi.org/10.1007/s10948-007-0258-z , Google ScholarCrossref, ISI© 2009 American Institute of Physics.