#### 1985: Unbroken Quantum Realism, from Microscopic to Macroscopic Levels

D. Bohm and B. J. Hiley: Physics Department, Birkbeck College, University of London, London WC1E 7HX, England.

Received 6 August 1985; published in the issue dated 2 December 1985.

By means of the quantum-potential interpretation we show that there is no need for a break or “cut” in the way we regard reality between quantum and classical levels.

URL: http://link.aps.org/doi/10.1103/PhysRevLett.55.2511 DOI: 10.1103/PhysRevLett.55.2511 PACS: 03.65.Bz

#### 1968: On Hidden Variables—A Reply to Comments by Jauch and Piron and by Gudder

D. BOHM: Birkbeck College, University of London, London, England.

J. BUB: Department of Physical Chemistry, University of Minnesota, Minneapolis, Minnesota.

URL: http://link.aps.org/doi/10.1103/RevModPhys.40.235 DOI: 10.1103/RevModPhys.40.235

#### 1967: Collective Treatment of Liquid Helium

D. BOHM and B. SALT*,†: Birkbeck College, University of London, London, England.

**Abstract:** The problem of liquid helium is treated in terms of the theory of collective coordinates, applying a canonical transformation similar to that used previously for plasmas. This treatment provides further physical and mathematical insights into the origin of phonons, and into the nature of the roton excitations and their interactions. In particular, starting from the many-body Hamiltonian, it directly gives the Feynman-Cohen wave function, and the dipole-dipole interaction of rotons, along with certain further predictions with regard to the roton spectrum and the correlation function. The property of superfluidity is explained by the subsidiary conditions which imply that individual particle excitations are impossible in the lowlying states of the system. The transformed Hamiltonian can thus be shown to have eigenfunctions corresponding to quantized vortices.

URL: http://link.aps.org/doi/10.1103/RevModPhys.39.894 DOI: 10.1103/RevModPhys.39.894 *Present address: Sir John Cass College, University of London, London, England. †This work was done under a grant from the Science Research Council..

#### 1966: A Proposed Solution of the Measurement Problem in Quantum Mechanics by a Hidden Variable Theory

D. BOHM and J. BUB: Birkbeck College, University of London, London, England.

**Abstract:** The measurement problem in quantum mechanics is re-examined and it is shown that it cannot really be solved in a satisfactory way, within the framework of the usual interpretation of the theory. We then discuss von Neumann’s attempts to prove that quantum mechanics is incompatible with the introduction of hidden variables, and develop a more detailed form of Bell’s argument, showing that von Neumann’s analysis is invalid. Using certain ideas that are implicit in the “differential-space” theory of Wiener and Siegel, we go on to propose a new deterministic equation of motion, describing a kind of coupling of the measuring instrument to the observed system that explains in detail how the wave packet is “reduced” during a measurement in a continuous and causally determined way. By averaging over the hidden parameters, we then recover the usual statistical results of quantum mechanics as a special case. However, a more detailed analysis of the theory shows that new experimental and theoretical questions can now be raised, which go outside the framework of the quantum theory as it is now formulated. These questions are examined briefly.

URL: http://link.aps.org/doi/10.1103/RevModPhys.38.453 DOI: 10.1103/RevModPhys.38.453

#### 1966: A Refutation of the Proof by Jauch and Piron that Hidden Variables Can be Excluded in Quantum Mechanics

D. BOHM and J. BUB: Birkbeck College, London University, London, England.

**Abstract:** In this paper, we examine an argument of Jauch and Piron, which aims to prove the impossibility of hidden variables underlying the quantum theory, on the basis of certain assumptions that are weaker than those used by von Neumann for the same purpose. We show that, while the assumptions of Jauch and Piron are in fact weaker than those of von Neumann, the net result is that they actually prove nothing new at all. The conclusions of Jauch and Piron concerning the nonexistence of hidden variables are indeed seen to follow from a false assumption; i.e., that the impossibility of propositions that describe simultaneously the results of measurements of two noncommuting observables is an “empirical fact.” Actually, it is shown that this assumption follows if and only if one first assumes what the authors set out to prove; i.e., that the current linguistic structure of quantum mechanics is the only one that can be used correctly to describe the empirical facts underlying the theory.

URL: http://link.aps.org/doi/10.1103/RevModPhys.38.470 DOI: 10.1103/RevModPhys.38.470

#### 1964: Answer to Fock Concerning the Time Energy Indeterminacy Relation

Y. Aharonov and D. Bohm: Belfer Graduate School of Science, Yeshiva University, New York, New York and Birkbeck College, London, England.

Received 14 February 1964; published in the issue dated June 1964.

**Abstract:** An answer is given to a recent criticizm by Fock, concerning our paper “Time in the Quantum Theory and the Uncertainty Relation for Time and Energy.” It is proved that Fock’s criticizm is wrong, and that our previous conclusion that energy can be measured in an arbitrarily short period of time is valid.

URL: http://link.aps.org/doi/10.1103/PhysRev.134.B1417 DOI: 10.1103/PhysRev.134.B1417

#### 1964: Separation of Motions of Many-Body Systems into Dynamically Independent Parts by Projection onto Equilibrium Varieties in Phase Space. II

G. Carmi*: Belfer Graduate School of Science, Yeshiva University, New York, New York.

D. Bohm: Birkbeck College, London, England.

Received 13 August 1962; revised 10 July 1963; published in the issue dated January 1964.

**Abstract:** We continue the development started in the preceding paper, in which we treated the many-body problem by separating the motion into an oscillatory part δxi, δpi, and a nonoscillatory part Xi, Pi, the latter being obtained by a noncanonical transformation from xi, pi which is just so tailored as to project out the oscillatory features from xi, pi, and thereby “projecting” xi, pi onto the equilibrium variety Jk(xi,pi)=0 (where Jk is the oscillatory action variable) in phase space. In this paper, we first develop a condensed notation in phase space which facilitates calculations. With the aid of this notation, we then give our results a simple geometrical interpretation in phase space by introducing a certain canonically invariant metrical tensor Oij. This tensor (which is antisymmetric) does not yield the usual orthogonal or pseudo-orthogonal metric, but rather, what is called a “symplectic metric” (i.e., invariant to the symplectic group of transformations). One then sees that the projections that we make are “orthogonal,” in the symplectic sense, to the equilibrium varieties. Likewise, one can see quite generally, that the entire canonical formalism, including the Poisson brackets and the Hamiltonian equations of motion, reduces to simple geometrical relations in phase space, the form of which is suggestive for possible further developments, especially with regard to the treatment in higher approximations. We apply our ideas to the electron gas, and illustrate the dynamics of the plasma with the aid of a comparison with a simple two-dimensional model, possessing all the essential features described above. In this way, we are able to understand many of the basic features of the plasma motions, in terms of concepts such as the generalization of the notion of centrifugal force and Coriolis force to phase space. By going over to a local geodetic frame in the equilibrium variety, we are led in a natural way to the concept of a set of “quasiparticles” for the plasma. If the number of collective oscillatory coordinates is s, t hen there will be 3N-s of these “quasiparticle” coordinates. The latter do not represent any of the actual original particles out of which the system is constituted, but rather, they represent effective pulse-like distributions of charge which move together in a correlated way so as to resemble an actual particle in many respects.

URL: http://link.aps.org/doi/10.1103/PhysRev.133.A332 DOI: 10.1103/PhysRev.133.A332 *This research was supported by the Department of Scientific and Industrial Research of the British Government while the author was at Bristol University, England, and by the U. S. Air Force Office of Scientific Research and the National Science Foundation.

#### 1963: Further Discussion of the Role of Electromagnetic Potentials in the Quantum Theory

Y. Aharonov*: Yeshiva University, New York, New York.

D. Bohm: Birkbeck College, University of London, London, England.

Received 31 October 1962; published in the issue dated May 1963.

**Abstract:** In this paper, we present a further discussion of the role of electromagnetic potentials in the quantum theory, aimed at clarifying some of the points made in earlier papers, and indicating some extensions of these earlier ideas. In addition, we go into the problem of subsidiary conditions and questions concerning the full observability of the potentials. In the course of our discussion, we answer certain recent objections of Belinfante and De Witt.

URL: http://link.aps.org/doi/10.1103/PhysRev.130.1625 DOI: 10.1103/PhysRev.130.1625 *Work supported by the National Science Foundation.

#### 1963: Rotator Model of Elementary Particles Considered as Relativistic Extended Structures in Minkowski Space

Louis De Broglie: Institut Henri Poincaré, Paris, France.

David Bohm: Birkbeck College, London, England.

Pierre Hillion and Francis Halbwachs: Institut Henri Poincaré, Paris, France.

Takehiko Takabayasi: Nagoya University, Nagoya, Japan.

Jean-Pierre Vigier: Institut Henri Poincaré, Paris, France.

Received 7 November 1961; revised 12 April 1962; published in the issue dated January 1963.

**Abstract:** The purpose of this paper is to investigate some consequences of the assumption that elementary particles are not pointlike, but are rather, extended structures in Minkowski space.

In terms of the hypothesis that the internal quantum states of such structures correspond to internal “rotator” levels belonging to the Hilbert space containing all irreducible finite-dimensional representations of the group SO3* of three-dimensional complex rotations (isomorphic to the Lorentz group), we obtain a particle classification which recovers (including leptons) the Nishijima-Gell-Mann classification of elementary particles. In this way, we justify the empirical Nishijima-Gell-Mann relation between isobaric spin, strangeness, baryon number, and charge. Moreover, as will be shown in a second paper, the new internal (“hidden”) degrees of freedom which correspond to isobaric spin, strangeness, and baryon number open up new possibilities for understanding qualitatively and quantitatively the elementary particle interactions and decays; while a simple extension of “fusion” theory yields possible external state vectors and equations associated with any given internal quantized states corresponding to known elementary particles.

URL: http://link.aps.org/doi/10.1103/PhysRev.129.438 DOI: 10.1103/PhysRev.129.438

#### 1962: Remarks on the Possibility of Quantum Electrodynamics without Potentials

Y. Aharonov: Yeshiva University, New York, New York.

D. Bohm: Birkbeck College, London, England.

Received 10 November 1961; published in the issue dated March 1962.

**Abstract:** In this article, we reply to a suggestion of DeWitt for formulating a nonlocal quantum electrodynamics without potentials, showing that his proposal is not a real elimination of potentials, but only a substitution with the aid of which the essential role of potentials is somewhat obscured. Thus, we are not led to change our conclusion that potentials are significant in the formulation of quantum electrodynamics. We do, however, discuss some of the conditions that would have to be satisfied by an essentially nonlocal electrodynamics, and show that such a theory, if it could be developed, would very probably bring about profound modifications in the forms of the laws of quantum mechanics.

URL: http://link.aps.org/doi/10.1103/PhysRev.125.2192 DOI: 10.1103/PhysRev.125.2192

#### 1961: Further Considerations on Electromagnetic Potentials in the Quantum Theory

Y. Aharonov: Department of Physics, Brandeis University, Waltham, Massachusetts.

D. Bohm: H. H. Wills Physics Laboratory, University of Bristol, Bristol, England.

Received 6 April 1961; published in the issue dated August 1961.

**Abstract:** In this article, we discuss in further detail the significance of potentials in the quantum theory, and in so doing, we answer a number of arguments that have been raised against the conclusions of our first paper on the same subject. We then proceed to extend our treatment to include the sources of potentials quantum-mechanically, and we show that when this is done, the same results are obtained as those of our first paper, in which the potential was taken to be a specified function of space and time. In this way, we not only answer certain additional criticisms that have been made of the original treatment, but we also bring out more clearly the importance of the potential in the expression of the local character of the interaction of charged particles and the electromagnetic field.

URL: http://link.aps.org/doi/10.1103/PhysRev.123.1511 DOI: 10.1103/PhysRev.123.1511

#### 1961: Time in the Quantum Theory and the Uncertainty Relation for Time and Energy

Y. Aharonov* and D. Bohm: H. H. Wills Physics Laboratory, Bristol, England.

Received 7 September 1960; published in the issue dated June 1961.

**Abstract:** Because time does not appear in Schrödinger’s equation as an operator but only as a parameter, the time-energy uncertainty relation must be formulated in a special way. This problem has in fact been studied by many authors and we give a summary of their treatments. We then criticize the main conclusion of these treatments; viz., that in a measurement of energy carried out in a time interval, Δt, there must be a minimum uncertainty in the transfer of energy to the observed system, given by Δ(E′-E)>~h/Δt. We show that this conclusion is erroneous in two respects. First, it is not consistent with the general principles of the quantum theory, which require that all uncertainty relations be expressible in terms of the mathematical formalism, i.e., by means of operators, wave functions, etc. Secondly, the examples of measurement processes that were used to derive the above uncertainty relation are not general enough. We then develop a systematic presentation of our own point of view, with regard to the role of time in the quantum theory, and give a concrete example of a measurement process not satisfying the above uncertainty relation.

URL: http://link.aps.org/doi/10.1103/PhysRev.122.1649 DOI: 10.1103/PhysRev.122.1649 *Now at Brandeis University, Waltham, Massachusetts.

#### 1959: Significance of Electromagnetic Potentials in the Quantum Theory

Y. Aharonov and D. Bohm: H. H. Wills Physics Laboratory, University of Bristol, Bristol, England.

[Selected for a Focus in Physics] [Free to Read] Received 28 May 1959; revised 16 June 1959; published in the issue dated August 1959

**Abstract:** In this paper, we discuss some interesting properties of the electromagnetic potentials in the quantum domain. We shall show that, contrary to the conclusions of classical mechanics, there exist effects of potentials on charged particles, even in the region where all the fields (and therefore the forces on the particles) vanish. We shall then discuss possible experiments to test these conclusions; and, finally, we shall suggest further possible developments in the interpretation of the potentials.

URL: http://link.aps.org/doi/10.1103/PhysRev.115.485 DOI: 10.1103/PhysRev.115.485

#### 1958: Relativistic Hydrodynamics of Rotating Fluid Masses

David Bohm: Technion, Haifa, Israel.

Jean-Pierre Vigier: Institut Henri Poincaré, Paris, France.

Received 23 April 1957; published in the issue dated March 1958.

**Abstract:** With the aid of the new notion of center of matter density, we give a relativistic treatment of the behavior of finite-size masses of rotating fluid. This treatment is based on an analysis of the relative motion of this center of matter density and the more familiar center of mass. In this way, we obtain a clear physical interpretation of the equations studied by Mathisson, Weysenhoff, and Möller. We also show that more general types of motions are possible, related to additional degrees of freedom of the relativistic fluid droplet. These degrees of freedom provide a framework for a theory of the quantum numbers of the elementary particles (isotopic spin, strangeness, etc.) which will be developed in a subsequent paper.

URL: http://link.aps.org/doi/10.1103/PhysRev.109.1882 DOI: 10.1103/PhysRev.109.1882

#### 1957: Discussion of Experimental Proof for the Paradox of Einstein, Rosen, and Podolsky

D. Bohm and Y. Aharonov: Technion, Haifa, Israel.

Received 10 May 1957; published in the issue dated November 1957.

**Abstract:** A brief review of the physical significance of the paradox of Einstein, Rosen, and Podolsky is given, and it is shown that it involves a kind of correlation of the properties of distant noninteracting systems, which is quite different from previously known kinds of correlation. An illustrative hypothesis is considered, which would avoid the paradox, and which would still be consistent with all experimental results that have been analyzed to date. It is shown, however, that there already is an experiment whose significance with regard to this problem has not yet been explicitly brought out, but which is able to prove that this suggested resolution of the paradox (as well as a very wide class of such resolutions) is not tenable. Thus, this experiment may be regarded as the first clear empirical proof that the aspects of the quantum theory discussed by Einstein, Rosen, and Podolsky represent real properties of matter.

URL: http://link.aps.org/doi/10.1103/PhysRev.108.1070 DOI: 10.1103/PhysRev.108.1070

#### 1957: Role of Subsidiary Conditions in the Collective Description of Electron Interactions

David Bohm: Technion, Haifa, Israel.

Kerson Huang*: Bell Telephone Laboratories, Murray Hill, New Jersey and Institute for Advanced Study, Princeton, New Jersey.

David Pines: Palmer Physical Laboratory, Princeton University, Princeton, New Jersey.

Received 8 March 1957; published in the issue dated July 1957.

**Abstract:** The role of the subsidiary conditions in the Bohm-Pines collective description of electron interactions is discussed in detail. The subsidiary conditions are shown to be compatible with the approximations made in obtaining the Hamiltonian and energy of the many-electron system. Their effect on the ground-state energy and the specific heat is found to be small.

URL: http://link.aps.org/doi/10.1103/PhysRev.107.71 DOI: 10.1103/PhysRev.107.71 *Present address: Institute for Advanced Study, Princeton, New Jersey.

#### 1954: Model of the Causal Interpretation of Quantum Theory in Terms of a Fluid with Irregular Fluctuations

D. Bohm: Faculdade de Filosofia, Ciências e Letras, Universidade de São Paulo, São Paulo, Brazil.

J. P. Vigier: Institut Henri Poincaré, Paris, France.

Received 14 June 1954; published in the issue dated October 1954.

**Abstract:** In this paper, we propose a physical model leading to the causal interpretation of the quantum theory. In this model, a set of fields which are equivalent in many ways to a conserved fluid, with density |ψ|2, and local stream velocity, dξ/dt=∇S/m, act on a particle-like inhomogeneity which moves with the local stream velocity of the equivalent fluid. By introducing the hypothesis of a very irregular and effectively random fluctuation in the motions of the fluid, we are able to prove that an arbitrary probability density ultimately decays into |ψ|2. Thus, we answer an important objection to the causal interpretation, made by Pauli and others. This result is extended to the Dirac equation and to the many-particle problem.

URL: http://link.aps.org/doi/10.1103/PhysRev.96.208 DOI: 10.1103/PhysRev.96.208

#### 1953: A Collective Description of Electron Interactions: III. Coulomb Interactions in a Degenerate Electron Gas

David Bohm: Faculdade de Filosofia, Ciencias e Letras, Universidade de Sao Paulo, Sao Paulo, Brazil.

David Pines: Department of Physics, University of Illinois, Urbana, Illinois.

Received 21 May 1953; published in the issue dated November 1953.

**Abstract:** The behavior of the electrons in a dense electron gas is analyzed quantum-mechanically by a series of canonical transformations. The usual Hamiltonian corresponding to a system of individual electrons with Coulomb interactions is first re-expressed in such a way that the long-range part of the Coulomb interactions between the electrons is described in terms of collective fields, representing organized “plasma” oscillation of the system as a whole. The Hamiltonian then describes these collective fields plus a set of individual electrons which interact with the collective fields and with one another via short-range screened Coulomb interactions. There is, in addition, a set of subsidiary conditions on the system wave function which relate the field and particle variables. The field-particle interaction is eliminated to a high degree of approximation by a further canonical transformation to a new representation in which the Hamiltonian describes independent collective fields, with n′ degrees of freedom, plus the system of electrons interacting via screened Coulomb forces with a range of the order of the inter electronic distance. The new subsidiary conditions act only on the electronic wave functions; they strongly inhibit long wavelength electronic density fluctuations and act to reduce the number of individual electronic degrees of freedom by n′. The general properties of this system are discussed, and the methods and results obtained are related to the classical density fluctuation approach and Tomonaga’s one-dimensional treatment of the degenerate Fermi gas.

URL: http://link.aps.org/doi/10.1103/PhysRev.92.609 DOI: 10.1103/PhysRev.92.609

#### 1953: Proof That Probability Density Approaches |ψ|2 in Causal Interpretation of the Quantum Theory

David Bohm.

Faculdade de Filosofia, Ciencias e Letras, Universidade de São Paulo, São Paulo, Brazil.

Received 25 June 1952; published in the issue dated January 1953.

**Abstract:** In two previous papers a causal interpretation of the quantum theory was developed which involved the hypothesis that a quantum-mechanical system contains a precisely defined particle variable x but that, at present, we are restricted to calculating the probability density P(x, t) that the particle is at the position x. It was shown that the assumption that P(x, t)=|ψ(x, t)|2 is consistent, in the sense that if it holds initially, the equations of motion of the particles will cause this relation to be maintained for all time. In this paper, we extend the theory by showing that as a result of random collisions, an arbitrary probability density will ultimately decay into one with a density of |ψ(x, t)|2. Since all quantum-mechanical experiments to date have been concerned with statistical ensembles of systems that have been colliding with other systems for a very long time, it is therefore inevitable that as we draw samples from such ensembles, the probability density of systems with particles at the point x will be equal to |ψ(x, t)|2.

In the previous papers we also pointed out that, within the conceptual framework of the causal interpretation, it was possible to suggest mathematical theories more general than are permitted by the usual interpretation and that these more general theories might be needed in the domain of 10-13 cm, where present theories seem to fail. However, if these more general theories should apply at the level of 10-13 cm, then there would be a tendency to create discrepancies between P and |ψ|2, a tendency whose cumulative effects should be felt even at the atomic level, where the more general theory ought to approach the usual theory. However, because those discrepancies have been shown to die out as a result of collisions, we can expect that under normal conditions the difference between P and |ψ|2 would be negligible. Conditions are suggested, however, in which this difference might be appreciable, and experiments are indicated which might be able to test for the existence of such discrepancies.

URL: http://link.aps.org/doi/10.1103/PhysRev.89.458 DOI: 10.1103/PhysRev.89.458

#### 1953: Comments on a Letter Concerning the Causal Interpretation of the Quantum Theory

David Bohm.

Faculdade de Filosofia, Ciências e Letras, Universidade de São Paulo, São Paulo, Brazil.

Received 17 November 1952; published in the issue dated January 1953.

URL: http://link.aps.org/doi/10.1103/PhysRev.89.319.2 DOI: 10.1103/PhysRev.89.319.2

#### 1952: Reply to a Criticism of a Causal Re-Interpretation of the Quantum Theory

David Bohm.

Faculdade de Filosofia, Ciências e Letras, Universidade de São Paulo, São Paulo, Brazil.

Received 22 May 1952; published in the issue dated July 1952.

URL: http://link.aps.org/doi/10.1103/PhysRev.87.389.2 DOI: 10.1103/PhysRev.87.389.2

#### 1952: A Suggested Interpretation of the Quantum Theory in Terms of “Hidden” Variables. II

David Bohm*.

Palmer Physical Laboratory, Princeton University.

Received 5 July 1951; published in the issue dated January 1952.

**Abstract:** In this paper, we shall show how the theory of measurements is to be understood from the point of view of a physical interpretation of the quantum theory in terms of “hidden” variables, developed in a previous paper. We find that in principle, these “hidden” variables determine the precise results of each individual measurement process. In practice, however, in measurements that we now know how to carry out, the observing apparatus disturbs the observed system in an unpredictable and uncontrollable way, so that the uncertainty principle is obtained as a practical limitation on the possible precision of measurements. This limitation is not, however, inherent in the conceptual structure of our interpretation. We shall see, for example, that simultaneous measurements of position and momentum having unlimited precision would in principle be possible if, as suggested in the previous paper, the mathematical formulation of the quantum theory needs to be modified at very short distances in certain ways that are consistent with our interpretation but not with the usual interpretation.

We give a simple explanation of the origin of quantum-mechanical correlations of distant objects in the hypothetical experiment of Einstein, Podolsky, and Rosen, which was suggested by these authors as a criticism of the usual interpretation.

Finally, we show that von Neumann’s proof that quantum theory is not consistent with hidden variables does not apply to our interpretation, because the hidden variables contemplated here depend both on the state of the measuring apparatus and the observed system and therefore go beyond certain of von Neumann’s assumptions.

In two appendixes, we treat the problem of the electromagnetic field in our interpretation and answer certain additional objections which have arisen in the attempt to give a precise description for an individual system at the quantum level.

URL: http://link.aps.org/doi/10.1103/PhysRev.85.180 DOI: 10.1103/PhysRev.85.180 *Now at Universidade de São Paulo, Faculdade de Filosofia, Ciencias e Letras, São Paulo, Brasil.

#### 1952: A Suggested Interpretation of the Quantum Theory in Terms of “Hidden” Variables. I

David Bohm*.

Palmer Physical Laboratory, Princeton University, Princeton, New Jersey.

Received 5 July 1951; published in the issue dated January 1952.

**Abstract:** The usual interpretation of the quantum theory is self-consistent, but it involves an assumption that cannot be tested experimentally, viz., that the most complete possible specification of an individual system is in terms of a wave function that determines only probable results of actual measurement processes. The only way of investigating the truth of this assumption is by trying to find some other interpretation of the quantum theory in terms of at present “hidden” variables, which in principle determine the precise behavior of an individual system, but which are in practice averaged over in measurements of the types that can now be carried out. In this paper and in a subsequent paper, an interpretation of the quantum theory in terms of just such “hidden” variables is suggested. It is shown that as long as the mathematical theory retains its present general form, this suggested interpretation leads to precisely the same results for all physical processes as does the usual interpretation. Nevertheless, the suggested interpretation provides a broader conceptual framework than the usual interpretation, because it makes possible a precise and continuous description of all processes, even at the quantum level. This broader conceptual framework allows more general mathematical formulations of the theory than those allowed by the usual interpretation. Now, the usual mathematical formulation seems to lead to insoluble difficulties when it is extrapolated into the domain of distances of the order of 10-13 cm or less. It is therefore entirely possible that the interpretation suggested here may be needed for the resolution of these difficulties. In any case, the mere possibility of such an interpretation proves that it is not necessary for us to give up a precise, rational, and objective description of individual systems at a quantum level of accuracy.

URL: http://link.aps.org/doi/10.1103/PhysRev.85.166 DOI: 10.1103/PhysRev.85.166 *Now at Universidade de São Paulo, Faculdade de Filosofia, Ciencias, e Letras, São Paulo, Brasil.

#### 1952: A Collective Description of Electron Interactions: II. Collective vs Individual Particle Aspects of the Interactions

David Pines: Randal Morgan Laboratory of Physics, University of Pennsylvania, Philadelphia, Pennsylvania.

David Bohm*: Palmer Physical Laboratory, Princeton University, Princeton, New Jersey.

Received 28 September 1951; published in the issue dated January 1952.

**Abstract:** The behavior of the electrons in a dense electron gas is analyzed in terms of their density fluctuations. These density fluctuations may be split into two components. One component is associated with the organized oscillation of the system as a whole, the so-called “plasma” oscillation. The other is associated with the random thermal motion of the individual electrons and shows no collective behavior. It represents a collection of individual electrons surrounded by comoving clouds of charge which screen the electron fields within a distance of the order of magnitude of the Debye length. This split up of the density fluctuations corresponds to an effective separation of the Coulomb interaction into long-range and short-range parts; the separation occurs at roughly the Debye length.

The relation between the individual and collective aspects of the electron gas is discussed in detail, and a general physical picture of the behavior of the system is given. It is shown that for phenomena involving distances greater than the Debye length, the system behaves collectively; for distances shorter than this length, it may be treated as a collection of approximately free individual particles, whose interactions may be described in terms of two-body collisions.

This approach is used to study the interaction of a specified electron with the remainder of the electron gas. It is shown that the collective part of the response of this remainder to the field of the specified particle screens this field within a distance of the order of the Debye length; this furnishes a detailed description of the screening process. Moreover, if the specified particle moves with greater than the mean thermal speed, it excites collective oscillations in the form of a wake trailing the particle. The frequency of these collective oscillations and the energy emitted by the particle are calculated. A correspondence theoretical method is used to treat this phenomenon for the electrons in a metal. The results are in good agreement with the experiments of Ruthemann and Lang on the energy loss of kilovolt electrons in this metallic films.

The generalization of these methods to an arbitrary interparticle force is carried out, and a criterion is obtained for the validity of a collective description of the particle interactions. It is shown that strong forces and high particle density favor collective behavior, while high random thermal velocities oppose it.

URL: http://link.aps.org/doi/10.1103/PhysRev.85.338 DOI: 10.1103/PhysRev.85.338 *Now at Physics Department, University of Sao Paulo, Sao Paulo, Brazil.

#### 1951: Application of Collective Treatment of Electron and Ion Vibrations to Theories of Conductivity and Superconductivity

David Bohm.

Princeton, New Jersey.

Palmer Physical Laboratory, Princeton University, Princeton, New Jersey.

Received 21 September 1951; published in the issue dated November 1951.

URL: http://link.aps.org/doi/10.1103/PhysRev.84.836.2 DOI: 10.1103/PhysRev.84.836.2

#### 1951: A Collective Description of Electron Interactions. I. Magnetic Interactions

David Bohm and David Pines*.

Palmer Physical Laboratory, Princeton University, Princeton, New Jersey.

Received 4 December 1950; published in the issue dated June 1951.

**Abstract:** A new approach to the treatment of the interactions in a collection of electrons is developed, which we call the collective description. The collective description is based on the organized behavior produced by the interactions in an electron gas of high density; this organized behavior results in oscillations of the system as a whole, the so-called “plasma oscillations.” The collective description, in contrast to the usual individual particle description, describes in a natural way the long-range correlations in electron positions brought about by their mutual interaction. In this paper we confine our attention to the magnetic interactions between the electrons; the coulomb interactions will be discussed in a subsequent paper.

The transition from the usual single-particle description to the collective description of the electron motion in terms of organized oscillations is obtained by a suitable canonical transformation. The complete hamiltonian for a collection of charges interacting with the transverse electromagnetic field is re-expressed as a sum of three terms. One involves the collective field coordinates and expresses the degree of excitation of organized oscillations. The others represent the kinetic energy of the electrons and the residual particle interaction, which is not describable in terms of the organized oscillations, and corresponds to a screened interparticle force of short range.

Both a classical and a quantum-mechanical treatment are given, and the criteria for the validity of the collective description are discussed.

URL: http://link.aps.org/doi/10.1103/PhysRev.82.625 DOI: 10.1103/PhysRev.82.625 *Now at Randal Morgan Laboratory of Physics, University of Pennsylvania, Philadelphia, Pennsylvania.

#### 1950: Screening of Electronic Interactions in a Metal

David Bohm and David Pines*.

Palmer Physical Laboratory, Princeton University, Princeton, New Jersey.

Received 16 October 1950; published in the issue dated December 1950.

URL: http://link.aps.org/doi/10.1103/PhysRev.80.903.2 DOI: 10.1103/PhysRev.80.903.2 *Now at Randal Morgan Laboratory of Physics, University of Pennsylvania, Philadelphia, Pennsylvania.

#### 1950: Effects of Plasma Boundaries in Plasma Oscillations

D. Bohm and E. P. Gross*.

Department of Physics, Princeton University, Princeton, New Jersey.

Received 28 March 1950; published in the issue dated September 1950.

**Abstract:** The discussion of electron plasma oscillations is extended to include some of the effects of boundaries. It is first shown that an electron taking part in a traveling plasma oscillation will be reflected at a sheath of infinitesimal thickness with velocity appropriate to the oscillation traveling in the reverse direction. This means that standing waves may be built up without loss at the sheaths. This approach is extended to sheaths where a finite time of penetration is necessary before reflection occurs and also to the case of reflection at metallic electrodes. In both cases expressions for the damping are derived and it is concluded that for low pressure discharges damping resulting from imperfect reflection from electrode sheaths may be comparable with collision damping but that damping arising from conducting electrodes is unimportant.

The excitation of the plasma by sharp beams is considered briefly and expressions are derived for the energy transfer of a beam to growing and of stationary amplitudes. It is pointed out that beams should excite oscillations only when a regular geometry exists. With irregular geometry bunching pulses are to be expected, of the type observed by Merrill and Webb. A detailed analysis is given of the bunching and of the Merrill and Webb experiments. Good agreement is obtained if one assumes that the pulses are maintained because high harmonic waves in the pulse cannot be shielded out by the plasma. These feed back energy towards the cathode and continuously modulate the beam.

URL: http://link.aps.org/doi/10.1103/PhysRev.79.992 DOI: 10.1103/PhysRev.79.992 *Laboratory for Insulation Research, Massachusetts Institute of Technology, Cambridge, Massachusetts.

#### 1950: Nuclear Size Resonances

K. W. Ford and D. Bohm.

Princeton University, Princeton, New Jersey.

Received 19 June 1950; published in the issue dated August 1950.

URL: http://link.aps.org/doi/10.1103/PhysRev.79.745.3 DOI: 10.1103/PhysRev.79.745.3

#### 1949: Finite Relativistic Charge-Current Distributions

D. Bohm, M. Weinstein, and H. Kouts.

Palmer Physical Laboratory, Princeton University, Princeton, New Jersey.

Received 1 August 1949; published in the issue dated September 1949

URL: http://link.aps.org/doi/10.1103/PhysRev.76.867 DOI: 10.1103/PhysRev.76.867

#### 1949: Theory of Plasma Oscillations. B. Excitation and Damping of Oscillations

D. Bohm and E. P. Gross*.

Department of Physics, Princeton University, Princeton, New Jersey.

Received 27 January 1949; published in the issue dated June 1949.

**Abstract:** The theory of electron oscillations of an unbounded plasma is extended to take into account the effects of collisions and special groups of particles having well-defined ranges of velocities. It is found that as a result of collisions a wave tends to be damped in a time of the order of the mean time between collisions. If beams of sharply defined velocity or groups of particles far above mean thermal speeds are present, however, they introduce a tendency toward instability so that small oscillations grow until limited by effects not taken into account in the linear approximation. An estimate is made of the steady-state amplitude for plasma oscillations in which excitation occurs because of a peak at high velocities in the electron velocity distribution, and in which the main damping arises from collisions. It is also found that in variable density plasmas, waves moving in the direction of decreasing plasma density show even stronger instability.

In absence of plasma oscillations, any beam of well-defined velocity is scattered by the individual plasma electrons acting at random, but, when all particles act in unison in the form of a plasma oscillation, the scattering can become much greater. Because of the instability of the plasma when special beams are present, the beams are scattered by the oscillations which they produce. It is suggested that this type of instability can explain the results of Langmuir, which show that beams of electrons traversing a plasma are scattered much more rapidly than can be accounted for by random collisions alone. It is also suggested that this type of instability may be responsible for radio noises received from the sun’s atmosphere and from interstellar space.

URL: http://link.aps.org/doi/10.1103/PhysRev.75.1864 DOI: 10.1103/PhysRev.75.1864 *Now at Harvard University, Cambridge, Massachusetts.

#### 1949: Theory of Plasma Oscillations. A. Origin of Medium-Like Behavior

D. Bohm and E. P. Gross*.

Department of Physics, Princeton University, Princeton, New Jersey.

Received 13 January 1949; published in the issue dated June 1949.

**Abstract:** A theory of electron oscillations of an unbounded plasma of uniform ion density is developed, taking into account the effects of random thermal motions, but neglecting collisions.

The first problem considered is that of finding the frequencies at which a plasma can undergo organized steady-state oscillations of small enough amplitude so that a linear approximation applies. It is found that long wave-length oscillations of plasmas with a Maxwell distribution of electron velocities are characterized by the steady-state dispersion relation ω2=ωP2+(3κT/m)(2π/λ)2. Here ωP is the plasma frequency, T the absolute temperature of the electron gas, λ the wave-length, and ω the angular frequency of oscillation. It is also shown that organized oscillations of wave-lengths smaller than the Debye length for the electron gas are not possible.

The theory is then extended to describe the processes by which oscillations are set up. It is found that, for a given wave-length, a plasma can oscillate with arbitrary frequency, but that those frequencies not given by the steady-state dispersion relation describe motions in which, after some time, there is no contribution to macroscopic averages. These additional frequencies lead asymptotically only to microscopic fluctuations of the charge density about the organized oscillation of the plasma. In this way, one can describe the manner in which the system develops organized behaviour.

The treatment is then applied to large steady-state oscillations for which the equations are non-linear. One obtains solutions in which particles close to the wave velocity are trapped in the trough of the potential, oscillating back and forth about a mean velocity equal to that of the wave. One can also obtain non-linear traveling pulse solutions in which a group of particles, moving as a pulse, creates a reaction on the surrounding charge, which traps the particles and holds them together.

URL: http://link.aps.org/doi/10.1103/PhysRev.75.1851 DOI: 10.1103/PhysRev.75.1851 *Now at Harvard University, Cambridge, Massachusetts.

#### 1949: Note on a Theorem of Bloch Concerning Possible Causes of Superconductivity

D. Bohm.

Physics Department, Princeton University, Princeton, New Jersey.

Received 13 September 1948; published in the issue dated February 1949.

**Abstract:** Attention is called to a theorem of Bloch, from which it is shown that even when interelectronic interactions are taken into account, the state of lowest electronic free energy corresponds to a zero net current. This result contradicts the hypothesis that superconductivity is caused by spontaneous currents.

URL: http://link.aps.org/doi/10.1103/PhysRev.75.502 DOI: 10.1103/PhysRev.75.502

#### 1948: The Self-Oscillations of a Charged Particle

D. Bohm and M. Weinstein.

Palmer Physical Laboratory, Princeton University, Princeton, New Jersey.

Received 19 July 1948; published in the issue dated December 1948.

**Abstract:** It is found that certain rigid charge distributions can oscillate without radiation even when no forces are present, other than their own retarded fields. The periods are of the order of the time required for light to cross the particle. The energy of these oscillations is always positive, and there are therefore no exponentially increasing unstable motions of the type possessed by the Dirac classical electron.

The frequencies of these oscillations are such, that when quantized, the energy of the first excited state is of the order of the meson self-energy. Hence, it is suggested that some kinds of mesons may be electrons in such an excited state of self-oscillation.

It is indicated that the principle of causality may have to be reformulated in terms of causal connections over finite intervals of time if one wishes to regard the electron plus its associated electromagnetic field as a single system.

URL: http://link.aps.org/doi/10.1103/PhysRev.74.1789 DOI: 10.1103/PhysRev.74.1789

#### 1948: Plasma Oscillations as a Cause of Acceleration of Cosmic-Ray Particles

David Bohm and E. P. Gross.

Princeton University, Princeton, New Jersey.

Received 12 July 1948; published in the issue dated September 1948.

URL: http://link.aps.org/doi/10.1103/PhysRev.74.624 DOI: 10.1103/PhysRev.74.624

#### 1947: Theory of the Synchro-Cyclotron

David Bohm and L. L. Foldy.

Radiation Laboratory, Department of Physics, University of California, Berkeley, California.

Received 31 May 1947; published in the issue dated October 1947.

**Abstract:** In the synchro-cyclotron (or frequency-modulated cyclotron) the higher energies available are obtained at the expense of a decrease in the ion current compared with that available from the conventional cyclotron. This decrease results from the fact that during only a small fraction of the frequency-modulation cycle is it possible for ions to be captured into phase stable orbits that do not return to the center during the first phase oscillation. By solving the phase equation, it is possible to obtain a general expression for this fraction, which is defined as the capture efficiency. At a constant dee voltage and varying rate of frequency modulation, the capture efficiency has a maximum at an equilibrium phase angle of 30° (corresponding to an energy gain per turn equal to half the maximum available). For larger equilibrium phase angles the efficiency decreases as a result of the smaller range of phase stability, while for smaller phase angles it decreases as a result of return of particles to the center. The maximum efficiency is proportional to the square root of the dee voltage or alternatively to the square root of the rate of frequency modulation, and depends on the charge and mass of the ions only through the ratio of charge to mass. Comparisons of the theoretical expectations with available experimental data show satisfactory agreement. Capture efficiencies for present designs of synchro-cyclotrons are of the order of 0.1 to 2 percent.

URL: http://link.aps.org/doi/10.1103/PhysRev.72.649 DOI: 10.1103/PhysRev.72.649

#### 1947: On the Neutron-Proton Scattering Cross Section

David Bohm and C. Richman.

Radiation Laboratory, Department of Physics, University of California, Berkeley, California.

Received 22 January 1947; published in the issue dated May 1947.

**Abstract:** The sensitivity of the theoretical neutron-proton scattering cross section to possible variations in the quantities defining the potential has been investigated in the energy range from 0 to 6 Mev. It is found that in this energy range the variations in exchange or tensor character and range of the potential which are consistent with what is known about the interaction do not result in so large a modification of the predicted cross section as do variations in shape.

A well resembling a Yukawa potential (A exp(-αr)/r) is found to yield a cross section at 6 Mev which is 10 percent lower than that given by a square well fitted to the epithermal neutron cross section and the binding energy of deuteron. The Yukawa well is in better agreement with experiment than is the square well. One of the constants entering into the determination of the potential is the epithermal neutron scattering cross section in hydrogen. We have determined this by extrapolating the data of Frisch with the aid of the theory, and obtained a value of 20.8×10-24 cm2.

URL: http://link.aps.org/doi/10.1103/PhysRev.71.567 DOI: 10.1103/PhysRev.71.567

#### 1946: The Low Energy β-Spectrum of Cu64

Harold Lewis and David Bohm.

Department of Physics, University of California, Berkely, California.

Received 12 January 1946; published in the issue dated February 1946.

URL: http://link.aps.org/doi/10.1103/PhysRev.69.129 DOI:10.1103/PhysRev.69.129

#### 1946: Theory of Synchrotron

1946, David Bohm (With L. Foldy) Theory of synchrotron, Phys. Rev. 70, 249-258.

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An overview of Leslie Foldy’s work can be found at the Soft Condensed Matter Theory Group (PDF File).