Showing papers in "American Journal of Physics in 1965"
3,323 citations
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1,098 citations
TL;DR: The status of the Gibbs and Boltzmann expressions for entropy has been a matter of some confusion in the literature as discussed by the authors, and it has been shown that the Gibbs H function yields the correct entropy as defined in phenomenological thermodynamics.
Abstract: The status of the Gibbs and Boltzmann expressions for entropy has been a matter of some confusion in the literature. We show that: (1) the Gibbs H function yields the correct entropy as defined in phenomenological thermodynamics; (2) the Boltzmann H yields an “entropy” that is in error by a nonnegligible amount whenever interparticle forces affect thermodynamic properties; (3) Boltzmann's other interpretation of entropy, S = k log W, is consistent with the Gibbs H, and derivable from it; (4) the Boltzmann H theorem does not constitute a demonstration of the second law for dilute gases; (5) the dynamical invariance of the Gibbs H gives a simple proof of the second law for arbitrary interparticle forces; (6) the second law is a special case of a general requirement for any macroscopic process to be experimentally reproducible. Finally, the “anthropomorphic” nature of entropy, on both the statistical and phenomenological levels, is stressed.
519 citations
451 citations
TL;DR: In this article, a symmetric tensor operator and the angular momentum vector operator were derived for the SU3 group, in a way analogous to the Runge-Lenz vector for the Kepler problem.
Abstract: A consideration of the eigenvalue problem for the quantum-mechanical three-dimensional isotropic harmonic oscillator leads to a derivation of a conserved symmetric tensor operator, in addition to the angular momentum vector operator. Upon examining the classical limit, it is found that the symmetric tensor completely specifies the orientation of the elliptical orbit, in a way analogous to the Runge-Lenz vector for the Kepler problem. The tensor operator and the angular momentum vector operator are then related to the infinitesimal operators for the SU3 group.
180 citations
TL;DR: In this article, it is shown that the forced quantum oscillator subject to a transient classical force is easily described in terms of the coherent states used in the description of light from optical masers.
Abstract: It is shown that the forced quantum oscillator subject to a transient classical force is easily described in terms of the “coherent states”recently found useful in the description of light from optical masers. The natural role these states play in understanding the notions of the phase of a quantum oscillator and the transition to the classical limit is also explained. Very simple derivations of the state vectors, energy transfer, and various transition probabilities are given.
169 citations
TL;DR: Einstein's First Paper on Quanta as mentioned in this paper was published in English translation in the American Journal of Physics (AJPN) in recognition of the sixtieth anniversary of the appearance of the original work.
Abstract: Of the trio of famous papers that Albert Einstein sent to the Annalen der Physik in 1905 only the paper proposing the photon concept has been unavailable in English translation. The American Journal of Physics is publishing the following translation in recognition of the sixtieth anniversary of the appearance of the original work. Physics teachers may take particular interest in the following aspects: (1) Einstein's keen awareness of the heuristic character of his new conception. (2) His demonstration from thermodynamic and statistical considerations that electromagnetic radiation might be conceived as consisting of finite numbers of discrete corpuscles of energy hv. (3) His prediction of the linear relation between the stopping potential of photoelectrons and the frequency of the incident light. This latter aspect of the photoelectric effect was not included among Lenard's early investigations. It remained for Millikan and others to develop the elegant experimental techniques that confirmed Einstein's bold prediction. Readers interested in pursuing the background in greater depth will find it rewarding to refer to the critical analyses by Martin J. Klein in “Einstein's First Paper on Quanta,” in The Natural Philosopher (Blaisdell Publishing Company, New York, 1963), Vol. II, and “Einstein and the Wave-Particle Duality,” in The Natural Philosopher, Vol. III, 1964. We are grateful to Professor Klein for his criticism and advice regarding this translation and for his generosity in making available to us an unpublished translation of his own.
136 citations
TL;DR: In this paper, it was shown that the corresponding conserved quantity is proportional to the difference in the number of right and left circularly polarized photons in the field, which is the same as Noether's theorem.
Abstract: In the absence of charges and currents, Maxwell's equations are invariant under the transformation E′ = E cosθ+B sinθ, B′ = −E sinθ+B cosθ. Using Noether's theorem, we show that the corresponding conserved quantity is proportional to the difference in the number of right and left circularly polarized photons in the field.
TL;DR: In this paper, a phase shift analysis of quantum mechanical scattering problems in one dimension is developed, which leads to the familiar relation between the scattering amplitude in the lth partial wave and the phase shift δl. The optical theorem is evident as an immediate consequence of the conservation of probability.
Abstract: A phase-shift analysis of quantum mechanical scattering problems in one dimension is developed. Many of the techniques and results of three dimensions remain even in one dimension, while the discussion is vastly simplified, and thus accessible to the student much earlier, due to the absence of special functions (in particular, Bessel and Legendre functions) with their complicated asymptotic formulas and orthogonality relations. A “partial wave” analysis leads to the familiar relation fl∼eiδl sinδl between the scattering amplitude in the lth partial wave and the phase shift δl. The optical theorem is evident as an immediate consequence of the conservation of probability. The method is applied to an analysis of the well-known square-well scattering resonances.
TL;DR: In this paper, a rigorous solution of the inverted pendulum requires several mathematical techniques which are not in the repetoire of the typical student of classical mechanics at the senior level, however, the problem is tractable by Lagrange's equations and the physics of the situation is sufficiently simple so that it is an excellent example to use to extend the student's analytical abilities.
Abstract: A rigorous solution of the inverted pendulum requires several mathematical techniques which are not in the repetoire of the typical student of classical mechanics at the senior level. However, the problem is tractable by Lagrange's equations and the physics of the situation is sufficiently simple so that it is an excellent example to use to extend the student's analytical abilities. Two unexpected dividends are derived from this problem: (1) the vertically driven case is an example of a time-dependent potential energy function, a rare situation in mechanics; (2) it is a vivid example of a classical system undergoing strong focusing and therefore can be used to introduce a discussion of the strong-focusing synchrotron.
TL;DR: In this paper, it was shown that Newtonian mechanics is quite sufficient to discuss the dynamics of the expanding universe, and this is not a cleverly contrived classical model, or a crude nonrelativistic approximation, but a completely correct discussion of the dynamics in a region where both general relativity and Newtonian physics are equally valid.
Abstract: The general relativistic solution to the cosmological problem is often considered to be one of its most beautiful results. It is shown that Newtonian mechanics is quite sufficient to discuss the dynamics of the expanding universe. Furthermore, this is not a cleverly contrived classical model, or a crude nonrelativistic approximation, but a completely correct discussion of the dynamics of the expansion in a region where both general relativity and Newtonian mechanics are equally valid.
TL;DR: In this article, the calculated geometrical appearance of objects moving at relativistic speeds and subtending large angles at the observer is illustrated by diagrams of a plane grid and perspective views of a group of boxes.
Abstract: The calculated geometrical appearance of objects moving at relativistic speeds and subtending large angles at the observer is illustrated by diagrams of a plane grid and perspective views of a group of boxes. In addition to the distortion of scales in the direction of motion, planes perpendicular to the motion appear as hyperboloids. Contrary to an impression which might be taken from some papers on the subject, the Lorentz contraction is visible under suitable conditions, in particular for observations approximately at right angles to the motion.
TL;DR: In this article, a further study of the degeneracy of the two-dimensional harmonic oscillator is made, both in the isotropic and anisotropic cases, by considering the Hamiltonian as a linear operator acting through the Poisson bracket on functions of the coordinates and momenta, a method applicable generally to bilinear Hamiltonians.
Abstract: A further study of the degeneracy of the two dimensional harmonic oscillator is made, both in the isotropic and anisotropic cases. By regarding the Hamiltonian as a linear operator acting through the Poisson bracket on functions of the coordinates and momenta, a method applicable generally to bilinear Hamiltonians, it is shown how all possible rational constants of the motion may be generated. They are formed from eigenfunctions of the Hamiltonian which are linear combinations of the coordinates and momenta, and which belong to negative pairs of eigenvalues. Canonical coordinates, which may be visualized geometrically for the isotropic oscillator in terms of the Hopf mapping, place the symmetry group responsible for the accidental degeneracy clearly in evidence. Surprisingly, one finds that the unitary unimodular group SU2, is the symmetry group in all cases, even including that of an anisotropic oscillator with incommensurable frequencies. The lack of a quantum-mechanical analogy in the latter case is du...
TL;DR: In this article, the Ritz theory of electromagnetism and optics is criticized in relation to the experimental evidence after the introduction of a simple and natural modification of the hypothesis concerning the velocity of radiation scattered by the electrons of a medium.
Abstract: The Ritz theory of electromagnetism and optics is criticized in relation to the experimental evidence after the introduction of a simple and natural modification of the hypothesis concerning the velocity of radiation scattered by the electrons of a medium. It is argued that the theory is then in harmony with the electron theory of dispersion, accounts satisfactorily for aberration, the first-order Doppler effect from moving sources and interferometer experiments on binary stars. There is no evidence from binary stars which contradicts it. It is compatible with the second-order Doppler effect and possibly the Fizeau experiment since arguments are advanced which indicate that these phenomena depend essentially on the momentum and energy of radiation. Other phenomena are discussed. It is concluded that the best evidence against the theory comes from experiments on the lifetimes of fast mesons and the velocity of γ rays and light from moving sources. The justifications for the discussion are the desirability ...
TL;DR: In this paper, the Hartree-Fock method was applied to the one-particle Schrodinger equation and the results were shown to be exactly the conditions for optimization of an approximate wave function consisting of a product of atomic orbitals.
Abstract: According to Hartree's self-consistent-field (SCF) model of the atom, the motion of each electron in the effective field of the N-1 others is governed by a one-particle Schrodinger equation. Self-consistency of the electronic charge distribution with its own electrostatic field leads to a set of coupled integrodifferential equations (Hartree equations) for N one-particle wavefunctions (atomic orbitals). The Hartree equations were subsequently shown to be precisely the conditions for optimization of an approximate wavefunction consisting of a product of atomic orbitals. An improved formalism, due to Slater and to Fock, represents the atomic wavefunction by a determinant built of atomic spin-orbitals and is thereby consistent with the Pauli principle. Application of the variational principle to a Slater determinant leads to a set of N coupled equations (Hartree-Fock equations), quite similar to Hartree's equations but containing, in addition, exchange interactions—an effect having no classical analog. The error inherent in the Hartree-Fock method, known as electron correlation, arises from smoothing-out of interelectronic repulsive interactions into effective Coulomb and exchange potentials. It accounts for roughly a 1% error in the total energy but is magnified in energy differences, which are more directly related to experimental quantities. A significant improvement in computational facility is achieved if the orbital functions are expanded in terms of a finite set of basis functions. The integrodifferential equations are thereby transformed into algebraic equations (Roothaan's equations) for the expansion coefficients. The analytic approach makes it possible to apply the self-consistent-field method to molecular systems. To date, SCF calculations have been carried out, in some form, for all the atoms in the periodic table and for a growing list of diatomic and polyatomic molecules.
TL;DR: The theory of special relativity is based on a principle of relativity for inertial frames as applied to mechanical phenomena only as discussed by the authors, and the experimental result that the free-space speed of light with respect to the earth is equal to k then implies the same free space speed for every inertial frame, the usual starting point of the theory.
Abstract: A development of the theory of special relativity is based on a principle of relativity for inertial frames as applied to mechanical phenomena only. The Lorentz-transformation equations and the familiar mass-variation formula are deduced from the observation that mass does vary with speed; but in these the speed of light c is everywhere replaced by a universal speed k—namely, the speed at which the mass of any particle must become infinite—which arises as a consequence of the assumptions. The experimental result that the free-space speed of light with respect to the earth is equal to k then implies the same free-space speed of light with respect to every inertial frame—the usual starting point of the theory.