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ResearchEnergy and Momentum Conservation in Bohm's Model for Quantum Mechanics (Summary)Extracted from: Bryan Hall's Ph.D. Thesis
Supervisor & ExaminersI would most sincerely like to thank Dr Rod Sutherland for his continual guidance and suggestions, as well as his careful checking of the text of this thesis. I would also like to thank the examiners for their work in preparing reports and suggesting various minor amendments to the thesis. The examiners made the following favourable comments: Dr Sheldon Goldstein: "This thesis is a very nice piece of work. It is written with exceptional clarity and facility of expression, and marks a significant contribution to the field." Dr David Miller: "The problem with Bohmian mechanics addressed by the thesis concerns energy and momentum conservation and is therefore fundamental. It is timely and important that the problem be addressed. The work in the thesis provides a solution to the problem and therefore makes a significant and original contribution to the discipline." Dr Craig Callender: "I recommend that the degree be awarded. Any errors I found were unimportant ones. My overall impression is that this dissertation is a very fine work in the foundations of physics." AbstractBohm's model for quantum mechanics is examined and a well-known drawback of the model is considered, namely the fact that the model does not conserve energy and momentum. It is shown that the Lagrangian formalism and the use of energy-momentum tensors provide a way of addressing this non-conservation aspect once the model is considered from the point of view of an interacting particle-field system. The full mathematical formulation that is then presented demonstrates that conservation can be reintroduced without disrupting the present agreement of Bohm's model with experiment. Chapter 1: IntroductionThis thesis looks at a particular interpretation of the formalism of quantum mechanics, viz., the model proposed by David Bohm. The aim is not to argue for or against this model, since the whole interpretation question for quantum mechanics is an area of much controversy. Rather, the aim is to resolve a precisely defined physical and mathematical problem that has been highlighted by several authors as being a possible deficiency of the model. It is demonstrated here that this feature of Bohm's model, namely that it does not conserve energy and momentum, can be successfully eliminated if desired. Advocates of Bohm's model can, of course, claim that it is already both empirically adequate and logically consistent without introducing such conservation. Nevertheless, there seems to be a general view, shared by supporters of the model, that the possibility of restoring energy and momentum conservation remains an interesting and aesthetically appealing idea. The structure of the thesis is as follows: Chapter 2 provides a general discussion of the development of quantum mechanics and the problem of its interpretation. It considers the Copenhagen interpretation, the Measurement Problem and the possibility of hidden variables. Chapter 3 summarizes the basic structure of Bohm's model for quantum mechanics. It describes the model's derivation from the equation of continuity and compares the modern minimalist version of the model with Bohm's original version. Expressions for Bohm's "quantum potential" are derived in preparation for later use in the thesis. The fact that Bohm's model does not conserve energy and momentum is then highlighted, this aspect of the model being the main focus of subsequent chapters. Finally, possible extensions to Bohm's model that have been suggested by other authors are discussed. In chapter 4, the Lagrangian formalism is outlined in preparation for applying it to Bohm's model. The eventual aim is to introduce energy and momentum conservation via Noether's theorem. Examples of a Lagrangian for particle motion and Lagrangian densities for free field evolution are first discussed, followed by sample Lagrangian densities for a particle and field in interaction. These expressions serve as possible analogies and guides towards a Lagrangian density for Bohm's model. Finally, an earlier attempt at a Lagrangian formalism for Bohm's model, proposed by Squires, is summarized and discussed. In chapter 5, a Lagrangian density suitable for Bohm's model is introduced. It is then demonstrated that this expression yields the usual equation of motion for the Bohmian particle. Such a Lagrangian formulation characterizes Bohm's model as an interacting particle-field system and pursuing this approach necessarily causes some modification to the Schrodinger equation. It is shown, however, that the particular modification introduced by the Lagrangian density proposed here does not compromise the Schrodinger equation's standard, experimentally-verified predictions. Chapter 6 summarizes the general theory of energy and momentum conservation for particle-field systems in terms of the divergence of energy-momentum tensors. It then tentatively considers the application of this formalism to Bohm's model and highlights some difficulties that arise. Chapter 7 proceeds to resolve these difficulties encountered in the non-relativistic theory by instead formulating a relativistic treatment, using a Klein-Gordon version of Bohm's model published by de Broglie. The mathematical proof of Noether's theorem is then re-derived from first principles for this particular situation. The previous problems are thereby eliminated, with the intention then being to proceed by taking the non-relativistic limit. In preparation for this step, separate expressions are obtained for the energy-momentum tensors of the field, particle and interaction, with the overall divergence being shown to be zero as required. Chapter 8 takes the non-relativistic limit of the formulation in the previous chapter. Particular attention is paid to the appropriate expression for the energy-momentum tensor of the particle, so that certain subtleties can be addressed concerning rest energy and the symmetry of the tensors. Three rules are thereby identified which allow the non-relativistic limits for the field and interaction expressions to be obtained easily. The overall divergence is then confirmed to be zero for the non-relativistic case, showing that energy and momentum conservation have been successfully introduced into Bohm's model. Finally, chapter 9 summarizes all the steps that have been taken in developing the argument and the problems encountered, including some comments on the strengths and weaknesses of the formulation. Chapter 2: Interpretations of Quantum Mechanics & the Measurement Problem
2.1 Historical ContextQuantum physics grew from attempts to understand the behaviour of infinitesimally small sub-atomic entities. As outlined by Heisenberg, in his 1932 Noble Prize address^{R1}, the basic postulates of the quantum theory arose from the fact that atomic systems are capable of assuming only discrete stationary states, and therefore of undergoing only discrete energy changes. Initially the program of quantum mechanics involved attempting to model observable phenomena such as the electromagnetic emission and absorption spectrum of atoms. Classical physics had dealt with "objective" processes occurring in space and time by specifying some initial conditions and modelling the time evolution of such processes. In addressing the quantum problem, Heisenberg observed that, according to the program of classical physics, it ought to be possible to calculate the exact path of electrons "orbiting" atomic nuclei from the measured properties of the emitted and absorbed radiation. However, the program of producing a causal model in which the frequency spectrum is directly related to the path of an electron "orbiting" around an atom met with very considerable difficulties. Heisenbergs' ultimate solution to the problem was to develop the theory of Matrix Mechanics^{R2}, in which any concept which could not be experimentally verified was excluded. Heisenberg observed that by abandoning notions which were not experimentally testable, contradictions between experiment and theory could be avoided. Consequently, Heisenberg argued that classical concepts such as the electron trajectory (position & momentum), which remains unobservable, should be abandoned at the quantum level. Heisenberg emphasised that the existence of entities, which are in-principle unobservable, cannot be objectively established and belief in their existence is therefore a matter of personal choice. Soon afterwards, Schrodinger produced his "Wave Mechanics", in which a quantum mechanical description of a system is presented in terms of a characteristic function known as the wave function. Following the publication of his original paper^{R3}, Schrodinger initially advanced the view that entities such as electrons and photons were, in fact, waves. A wave model, which interpreted the Schrodinger wave function as describing the spatial extent of real physical waves, seems well suited to explaining quantum interference. However, there are a number of difficulties with erecting a wave-based quantum theory to describe individual electrons which can be counted by geiger counters and observed as spots on photographic plates. 2.2 Mathematical Structure and Statistical InterpretationThe mathematical structures of the Heisenberg and Schrodinger formulations of quantum mechanics are well understood and their formal equivalence was established very early on by Schrodinger and Dirac. Consistent with the original formulations, the general Hilbert Space representation was developed. In deducing the correct statistical meaning for the normalised Schrodinger wave function, Max Born provided the central, experimentally verified tenet of non-relativistic quantum mechanics. Born's postulate requires that the volume integral of the square of the Schrodinger wave function's modulus give the probability of finding the particle in that volume. In a similar manner, the statistical distribution of measurement results for any other observable quantity may be determined by switching the wave function to the representation corresponding to that observable. The desired distribution is then given by the squared modulus of the transformed wave function. In this scheme, physical quantities are incorporated as representation-dependent, self-adjoint^{N1} mathematical operators. The point must be made emphatically that, in terms of Born's Interpretation, the Schrodinger wave function, or state function, describes the statistical behaviour of an aggregated collection. This quantum mechanical statistical algorithm need not constrain individual ensemble members. 2.3 The Correspondence PrincipleThe Correspondence Principle requires that, under appropriate limiting circumstances (usually expressed as lim ö ® 0) quantum mechanics reduce to classical mechanics. In this way, quantum mechanics can be viewed as a mathematical generalisation, which for large objects is consistent with classical mechanics. 2.4 The Copenhagen InterpretationWhile the mathematical structure of quantum mechanics is extremely well understood, it is clear that this structure provides very little insight into the nature of any "continuously existing reality." Schrodinger^{R4} eventually conceded that the quantum mechanical formalism developed provides only a statistical algorithm for making predictions about measurement results and does not provide any clear picture of entities existing between measurement events. It therefore gives no insight into the nature of any possible underlying reality and fails to support a "principle of causality" in any form. Adherents to the "Copenhagen interpretation" of quantum mechanics assert that a "complete description of reality" is in fact provided by Born's experimentally verified statistical hypothesis (above) and that models describing the time evolution of individual entities between observations are neither useful nor possible. Niels Bohr summarises the Copenhagen view well claiming that "in quantum mechanics we are not dealing with an arbitrary renunciation of a more detailed analysis of atomic phenomena, but with a recognition that such an analysis is in principle excluded.^{R5}" In an effort to challenge the Copenhagen interpretation, which proposed that quantum theory provided a complete description of individual quantum entities rather than a statistical algorithm for determining the behaviour of quantum ensembles, Einstein and others developed a variety of objections to various peculiarities inherent in the Copenhagen viewpoint. Schrodinger's Cat and the non-locality following from the EPR paradox rank as the most famous of these challenges. In contrast with Bohr, Einstein asserted that the wave function provided a description of only quantum ensembles and not of individual quantum entities.^{R6} Toulmin^{R7} observes that much of the unfocussed and unresolved controversy concerning the interpretation of quantum mechanics has its roots in the fact that Einstein and his supporters have refused to accept the change in standards of "what needs explaining" which has been made with the development of the Copenhagen interpretation of quantum mechanics. In Einstein's view, these changes require one to restrict the horizon of scientific endeavour in an unjustifiable way. Einstein's opponents, on the other hand, claim that his objections show only that he has not properly understood the theory. Toulmin does not deal with the substance of the dispute but draws significant attention to the language in which the dispute is carried on. The dispute is couched in terms of the question, "Is a quantum mechanical description of a physical system complete or not?" Toulmin argues that this way of posing the problem confuses the issue, giving it too sharp an appearance of opposition. A complete or exhaustive description of a physical system is one from which one can, using the currently accepted laws of nature, infer all properties of the system for which it is a physicist's ambition to account. Where two physicists do not share a common standard of what does and does not need to be explained, there is no hope on their agreeing that the corresponding description can be called complete. The use of the word complete, with its implicit reference to particular criteria of completeness, may serve to conceal rather than reveal the point at issue. A similar moral holds more generally where, in the absence of any explanation, the term "reality" is frequently used. Heisenberg highlighted this when asked directly the question: "Is there a fundamental level of reality?" He responded as follows: "This is just the point; I do not know what the words fundamental reality mean. They are taken from our daily life situation where they have a good meaning, but when we use such terms we are usually extrapolating from our daily lives into an area very remote from it, where we cannot expect the words to have a meaning. This is perhaps one of the fundamental difficulties of philosophy: that our thinking hangs in the language. Anyway, we are forced to use the words so far as we can; we try to extend their use to the utmost, and then we get into situations in which they have no meaning.^{R8}" 2.5 Hidden Variable TheoriesIn spite of the Copenhagen interpretation, there have been extensive efforts to introduce theories providing a deeper description of nature. Principally these theories have taken the form of "hidden variable" theories in which certain properties of individual quantum entities always pre-exist before an act of measurement. One motivation for the hidden variables program is that the Copenhagen interpretation of the Schrodinger equation is unable to account in a satisfying way for the process of measurement wherein a discontinuous transition from a spread-out state to a definite experimental result occurs. Schrodinger wave functions evolve continuously and smoothly through time and after a particle and an apparatus interact they are described by a single, overall wave function from then on. This transition to a correlated state should result in the state of the apparatus becoming less definite, rather than the particle's state becoming more definite. However, at the macroscopic level, definite measurement results are always obtained. The Copenhagen "analysis" of the measurement process simply invokes Von Neumann's Projection Postulate, which asserts that the state vector evolves according to the Schrodinger equation while the system is isolated, but changes discontinuously during measurement to an eigenstate of the observable that is measured^{R9}. Because of the apparent necessity that the postulate apply only for "measurement" interactions, not for "non-measurement" interactions, there has been much controversy concerning this infamous Measurement Problem and the Copenhagen interpretation in general since they were first proposed. Hidden variables programs frequently take their motivation from other areas of physics such as the classical theory of gases, which is understood as a macroscopic approximation arising statistically from the aggregated behaviour of a large number of microscopic gas molecules. On the other hand, advocates of the Copenhagen interpretation have attempted to produce "impossibility proofs" intended to demonstrate the incompatibility of hidden variables theories with quantum mechanics. Von Neumann claimed to present a proof that hidden variables theories were not possible, but the proof failed since it made the incorrect assumption that an algebraic rule which must hold in the mean for non-commuting observables must also hold for the individual hidden values^{R10}. Since the formalism of quantum mechanics does not necessarily imply the Copenhagen interpretation, the possibility of constructing different models that are observationally equivalent to conventional quantum mechanics remains open^{R11}. Although certain types of hidden variables models can be ruled out, it is not possible to invalidate all hidden variables models. Writing in Physics Today (1998), Goldstein^{R12} claims that the Bohr-Einstein debate has actually been resolved in favour of Einstein since a number of observer-free formulations of quantum mechanics, in which the process of measurement can be analysed in terms of more fundamental concepts, have been produced. Examples of observer-free formulations include: Decoherent Histories, Spontaneous Localisation and Pilot Wave theories (including Bohm's Model). Chapter 9: Discussion and ConclusionsIn this thesis, it has been demonstrated that the well-known Bohmian model for quantum mechanics can be made compatible with the laws of conservation of energy and momentum. This has been achieved by constructing a Lagrangian formulation of the model, so that the required conservation is then assured by Noether's theorem. Although this conservation is then known to be present in general terms, extracting a detailed description of it in terms of energy-momentum tensors was found to be not at all straightforward. First, it was necessary to realize that the usual energy-momentum tensors T^{mn}_{field} and T^{mn}_{particle} that appear in such a formulation had, in this case, to be augmented by a third tensor T^{mn}_{interaction}. Furthermore, attempting to obtain a specific expression for T^{mn}_{interaction} by re-deriving Noether's proof from first principles was found to lead to difficulties and ambiguities in the non-relativistic case. The chosen way forward was to formulate the details of a relativistic Lagrangian model and then take the non-relativistic limit to obtain the appropriate formalism corresponding to Bohm's model. Although the relativistic case proved to be straightforward, the taking of the non-relativistic limit was also found to involve subtleties and to require care. During this procedure, it was necessary to scrutinize the physical interpretation of the various expressions that arose and to clarify the physical meaning of symmetric and non-symmetric energy-momentum tensors. The resulting formalism was then found to have the desired properties for demonstrating conservation. The construction of a Lagrangian formulation of Bohm's model necessarily leads to some modification of the relevant field equation, i.e., of the Schrodinger equation. This has resulted in perhaps the most intriguing aspect of the present approach, namely that the modified Schrodinger equation is found to be automatically of a special form that reduces back to the standard Schrodinger equation^{N1} in the usual, non-relativistic case of no creation or annihilation of particles, thereby maintaining compatibility with all the relevant experimental evidence. Concerning this restriction to conserved particle number in the non-relativistic case, it may be observed that the divergences of the various parts of the energy-momentum tensor become somewhat trivial under this assumption. However, this does not affect the basic result that energy and momentum conservation have been successfully introduced into Bohm's model. Of course, the fact also remains that Bohmian mechanics is not a widely accepted interpretation of quantum mechanics. Nevertheless, the aim here has simply been to answer the question of whether this model can be compatible with the usual conservation laws (this question having been published in several places), not to argue for the superiority of the model in other ways. Finally it should also be noted that, from a metaphysical point of view, the present work establishes that the laws of energy and momentum conservation are quite compatible with attempts to formulate an interpretation of quantum mechanics incorporating realism. Chapter 2 Reference Notes: ^{R1} Heisenberg Werner, Nobel Prize in Physics Address: A General History of the Development of Quantum Mechanics, 1932. Published by Elsevier Publishing Co, with the permission of the Nobel Foundation. Cited from The World of Physics vol. 2 pp. 353-367. JH Weaver. Published by Simon and Schuster, New York (1987). ^{R2} Heisenberg W., Z. Physik vol. 33, p. 879 (1925). ^{R3} Schrodinger, E., Ann. Physik vol. 79, pp. 361 and 489 (1925); vol. 80, p. 437 (1926); vol. 81, p. 109 (1926) ^{R4} Schrodinger E. Science & Humanism; Physics in Our Time, Cambridge 1951. Cited from Causality and Wave Mechanics in The World of Mathematics Vol II. pp 1056-1068. Newnam JR. Published by George Allen & Unwin Ltd 1960. ^{R5} Goldstein S. Quantum Theory Without Observers - Part One. Physics Today (March 1998) p. 42-46. Published by American Institute of Physics. ^{R6} Einstein A, Franklin J. Physics and Reality (1936). Cited from Dewitt BS and Graham NR. Resource Letter IQM-1 on the Interpretation of Quantum Mechanics. American Journal of Physics. vol. 39 pp. 724-738 (see especially pp. 730 & 731) (July 1971). ^{R7} Toulmin S. The Philosophy of Science. pp. 118-9. Published Hutchison and Company London. (Sixth Impression 1962). ^{R8} Buckley P and Peat FD. A Question of Physics. Conversations in Physics and Biology. pp. 3-16 (see especially p. 9). Published by Routledge and Kegan Paul. London and Henley. 1979. ^{R9} Ballentine LE. Resource Letter IQM-2: Foundations of quantum mechanics since the Bell inequalities. American Journal of Physics. vol. 55 no. 9, (September 1987). Published by the American Association of Physics Teachers. ^{R10} Von Neumann, J. Mathematical Foundations of Quantum Mechanics, Princeton U. P. New Jersey (1955). Also, Bell, J.S., Rev. Mod. Phys. vol.38, p. 447 (1966). ^{R11} Cushing JT. Quantum Mechanics: Historical Contingency & the Copenhagen Hegemony. p. 42. Published by University of Chicago Press (1994). ^{R12} Goldstein S. Quantum Theory Without Observers - Part One. Physics Today (March 1998) p. 42-46. Published by American Institute of Physics. Chapter 2 Supplementary Notes: ^{N1} Self-adjoint or Hermitian operators have real eigenvalues and hence are the only class of operators which can represent real physical magnitudes. Chapter 9 Supplementary Notes: ^{N1} once the Bohmian constraint v = (grad S)/m is applied. |