We show that the minimum Fisher information (MFI) approach to estimating the probability law p(x) on particle position x, over the class of all two-component laws p(x), yields the complex Schrödinger wave equation. Complexity, in particular, traces from an “efficiency scenario” (demanded by MFI) where the two components of p(x) are so separated that their informations add.

For an arbitrary potential V with classical trajectoriesx=g(t), we construct localized oscillating three-dimensional wave lumps ψ(x, t,g) representing a single quantum particle. The crest of the envelope of the ripple follows the classical orbitg(t), slightly modified due to the potential V, and ψ(x, t,g) satisfies the Schrödinger equation. The field energy, momentum, and angular momentum calculated as integrals over all space are equal to the particle energy, momentum, and angular momentum. The relation to coherent states and to Schrödinger waves (...) is also discussed. (shrink)

Quantum mechanics teaches that before detection, knowledge of particle position is, at best, probabilistic, and classical trajectories are seen as a feature of the macroscopic world. These comments refer to detected particles, but we are still free to consider the motions generated by the wave equation. Within hydrogen, the Schrodinger equation allows calculation of kinetic energy at any location, and if this is identified as the energy of the wave, then radial momentum, allowing for spherical harmonics, becomes (...) available. The distance across the real zone of radial momentum is found to match semi-integer wavelengths of the adjusted matter wave, consistent with what is expected from a standing wave condition. The approach is extended to include orbital motions, where it is established that the underlying wave, which has direction and wavelength at each location, forms a series of connected trajectories, which are shown to be ellipses orientated at various angles to the equatorial plane. This suggests that wave trajectories, rather than particle trajectories, are still a feature of the hydrogen atom. The finding allows the reason for the coincidence between energy results derived by Sommerfeld’s classical trajectories and the Schrodinger wave equation to be appreciated. The result has implications when the relativistic situation is considered, as Sommerfeld’s correct deduction of the relativistic energy levels of hydrogen well before Dirac derived his wave equation has long been somewhat puzzling. (shrink)

A Schrödinger-like relativistic wave equation of motion for the Lorentz-scalar potential is formulated based on a Lagrangian formalism of relativistic mechanics with a scaled time as the evolution parameter. Applications of this Schrödinger-like formalism for the Lorentz-scalar potential are given: For the square-step potential, the predictions of this formalism are free from the Klein paradox, and for the Coulomb potential, this formalism yields the exact bound-state eigenenergies and eigenfunctions.

This article is in three parts. Part I gives an account of Erwin Schrödinger's growing up and studies in Vienna, his scientific work—first in Vienna from 1911 to 1920, then in Zurich from 1920 to 1925—on the dielectric properties of matter, atmospheric electricity and radioactivity, general relativity, color theory and physiological optics, and on kinetic theory and statistical mechanics. Part II deals with the creation of the theory of wave mechanics by Schrödinger in Zurich during the early months of (...) 1926; he laid the foundations of this theory in his first two communications toAnnalen der Physik. Part III deals with the early applications of wave mechanics to atomic problems—including the demonstration of equivalence of wave mechanics with the quantum mechanics of Born, Heisenberg, and Jordan, and that of Dirac—by Schrödinger himself and others. The new theory was immediately accepted by the scientific community. (shrink)

This article (Part II) deals with the creation of the theory of wave mechanics by Erwin Schrödinger in Zurich during the early months of 1926; he laid the foundations of this theory in his first two communications toAnnalen der Physik. The background of Schrödinger's work on, and his actual creation of, wave mechanics are analyzed.

This article (Part III) deals with the early applications of wave mechanics to atomic problems—including the demonstration of the formal mathematical equivalence of wave mechanics with the quantum mechanics of Born, Heisenberg, and Jordan, and that of Dirac—by Schrödinger himself and others. The new theory was immediately accepted by the scientific community.

The stochastic approach worked out in earlier papers is applied to the Dirac fluid. It gives a model of the Schrödinger zitterbewegung, from which, by the spinor-vector correspondence, a model of the plane monochromatic wave in the rest frame is derived. The relation of the scheme with quantization is found to have the same character as in the previous papers. The link of spin with relativity is explained.

A Schrödinger-like formalism of relativistic quantum theory is presented based on an alternative Lagrangian formalism of relativistic mechanics with the proper time as the evolution parameter. The Schrödinger-like formalism resolves the great difficulties of negative probability density, Klein paradox, and Zitterbewegung. Ehrenfest's theorem is preserved in the Schrödinger-like formalism.

A brief review of the conceptual difficulties met by the quantum formalism is presented. The main attempts to overcome these difficulties are considered and their limitations are pointed out. A recent proposal based on the assumption of the occurrence of a specific type of wave function collapse is discussed and its consequences for the above-mentioned problems are analyzed.

Reminiscing on the fact that E. Schrödinger was rooted in the same physical tradition as M. Planck and A. Einstein, some aspects of his attitude to quantum mechanics are discussed. In particular, it is demonstrated that the quantum-mechanical paradoxes assumed by Einstein and Schrödinger should not exist, but that otherwise the epistemological problem of physical reality raised in this context by Einstein and Schrödinger is fundamental for our understanding of quantum theory. The nonexistence of such paradoxes just shows that quantum-mechanical (...) effects are due to interference and not to interaction. This line of argument leads consequently to quantum field theories with second quantization, and accordingly quantum theory based both on Planck's constant h and on Democritus's atomism. (shrink)

Starting from a nonlinear relativistic Klein-Gordon equation derived from the stochastic interpretation of quantum mechanics (proposed by Bohm-Vigier, (1) Nelson, (2) de Broglie, (3) Guerra et al. (4) ), one can construct joint wave and particle, soliton-like solutions, which follow the average de Broglie-Bohm (5) real trajectories associated with linear solutions of the usual Schrödinger and Klein-Gordon equations.

Early in 1926 Erwin Schrodinger presented his famous theory of wave mechanics to account for atomic phenomena. It is often assumed that Schrodinger’s work reflected a realist philosophy. In this article, I will argue that this assumption is incorrect.

E. Schrödinger's ideas on interpreting quantum mechanics have been recently re-examined by historians and revived by philosophers of quantum mechanics. Such recent re-evaluations have focused on Schrödinger's retention of space–time continuity and his relinquishment of the corpuscularian understanding of microphysical systems. Several of these historical re-examinations claim that Schrödinger refrained from pursuing his 1926 wave-mechanical interpretation of quantum mechanics under pressure from the Copenhagen and Göttingen physicists, who misinterpreted his ideas in their dogmatic pursuit of the complementarity doctrine and (...) the principle of uncertainty. My analysis points to very different reasons for Schrödinger's decision and, accordingly, to a rather different understanding of the dialogue between Schrödinger and N. Bohr, who refuted Schrödinger's arguments. Bohr's critique of Schrödinger's arguments predominantly focused on the results of experiments on the scattering of electrons performed by Bothe and Geiger, and by Compton and Simon. Although he shared Schrödinger's rejection of full-blown classical entities, Bohr argued that these results demonstrated the corpuscular nature of atomic interactions. I argue that it was Schrödinger's agreement with Bohr's critique, not the dogmatic pressure, which led him to give up pursuing his interpretation for 7 yr. Bohr's critique reflected his deep understanding of Schrödinger's ideas and motivated, at least in part, his own pursuit of his complementarity principle. However, in 1935 Schrödinger revived and reformulated the wave-mechanical interpretation. The revival reflected N. F. Mott's novel wave-mechanical treatment of particle-like properties. R. Shankland's experiment, which demonstrated an apparent conflict with the results of Bothe–Geiger and Compton–Simon, may have been additional motivation for the revival. Subsequent measurements have proven the original experimental results accurate, and I argue that Schrödinger may have perceived even the reformulated wave-mechanical approach as too tenuous in light of Bohr's critique. (shrink)

This article discusses the historical similarities and differences between Schroedinger's and de Broglie's ideas on wave mechanics and gives a biographical account of their scientific relationship. Their arguments over questions such as quantum jumps, the viability of particles within wave mechanics theory, and the inclusion of space, time, and relativity in quantum mechanics are analyzed. The final section of the paper considers the overall role of Schroedinger's ideas in modern quantum mechanics.

Schrödinger’s first proposal for the interpretation of quantum mechanics was based on a postulate relating the wave function on configuration space to charge density in physical space. Schrödinger apparently later thought that his proposal was empirically wrong. We argue here that this is not the case, at least for a very similar proposal with charge density replaced by mass density. We argue that when analyzed carefully, this theory is seen to be an empirically adequate many-worlds theory and not an (...) empirically inadequate theory describing a single world. Moreover, this formulation—Schrödinger’s first quantum theory—can be regarded as a formulation of the many-worlds view of quantum mechanics that is ontologically clearer than Everett’s. (shrink)

There is no paradox with Schrödinger’s cat in a realist interpretation. In particular, a closer look at the temporal aspect shows that the two macroscopic wave functions (alive and dead) of Schrödinger’s cat are not to be compared with two superposed parts of a microscopic quantum wave function.

Erwin Schrödinger and Werner Heisenberg were the originators of two approaches, known respectively as “wave mechanics” and “matrix mechanics”, to what is now called “quantum mechanics” or “quantum theory”. The two approaches appear to be extremely different, both in their technical forms, and in their philosophical underpinnings. Heisenberg arrived at his theory by effectively renouncing the idea of trying to represent a physical system, such as a hydrogen Bohr's atom model for example, as a structure in space—time, but instead, (...) following the lead of Einstein's 1905 theory of relativity, representing only empirically observable properties, such as the transition amplitudes between the stationary states of the atom. These amplitudes can be arranged in square arrays of numbers. In Heisenberg's scheme these arrays, and other like them, are combined according to certain rules that were later recognized by Max Born to be the rules of matrix multiplication. The whole scheme is abstract and mathematical, and avoids using any space—time picture of what is going on at the atomic level. Schrödinger, on the other hand, represented the electron in an atom by a cloudlike wave surrounding the nucleus. This is a space—time structure that, superficially at least, is more in line with the classical physical theories of the eighteenth and nineteenth centuries. -/- Niels Bohr invited Schrödinger to come to Copenhagen to present his ideas, and to discuss this subject with himself, Heisenberg, and others. Schrödinger arrived in Copenhagen on October 1st, 1926, and was immediately intensively engaged by Bohr and the others in a “debate” that lasted for days. Eventually, Schrödinger become ill, and was confined in Bohr's home to a bed, upon which Bohr sat, continuing the discussion. Schrödinger finally exclaimed “If all the quantum jumping is here to stay, then I am sorry that I ever became involved in quantum mechanics”. Bohr replied, “But we are glad that you did!”. (shrink)

The purpose of this paper is to review and clarify the quantum “measurement problem.” The latter originates in the ambivalent nature of the “observer”: Although the observer is not described by the Schrödinger equation, it should nevertheless be possible to “quantize” him and include him in the wave function if quantum theory is universally valid. The problem is to prove that no contradiction may arise in these two conflicting descriptions. The proof invokes the notion of irreversibility. The validity of (...) the latter is questionable, because the standard rationale for classical irreversibility, namely mixing and coarse graining, does not apply to quantum theory. There is no chaos in a closed, finite quantum system. However, when a system is large enough, it cannot be perfectly isolated from its “environment,” namely from external (or even internal) degrees of freedom which are not fully accounted for in the Hamiltonian of that system. As a consequence, the long-range evolution of such a quantum system is essentially unpredictable. It follows that the notion of irreversibility is a valid one in quantum theory and the “measurement problem” can be brought to a satisfactory solution. (shrink)

Within the framework of Relativistic Schrödinger Theory (RST), the scalar two-particle systems with electromagnetic interactions are treated on the basis of a non-Abelian gauge group U(2) which is broken down to the Abelian subgroup U(1)×U(1). In order that the RST dynamics be consistent with the (non-Abelian) Maxwell equations, there arises a compatibility condition which yields cross relationships for the links between the field strengths and currents of both particles such that self-interactions are eliminated. In the non-relativistic limit, the RST dynamics (...) becomes identical to the well-known Hartree–Fock equations (for spinless particles). Consequently the original RST field equations may be considered as the relativistic generalization of the Hartree–Fock equations, and the “exchange interactions” of the conventional theory (induced by the anti-symmetrization postulate) do reappear here as ordinary gauge interactions due to a broken symmetry. (shrink)

The meaning of the wave function and its evolution are investigated. First, we argue that the wave function in quantum mechanics is a description of random discontinuous motion of particles, and the modulus square of the wave function gives the probability density of the particles being in certain locations in space. Next, we show that the linear non-relativistic evolution of the wave function of an isolated system obeys the free Schrödinger equation due to the requirements of (...) spacetime translation invariance and relativistic invariance. Thirdly, we argue that the random discontinuous motion of particles may lead to a stochastic, nonlinear collapse evolution of the wave function. A discrete model of energy-conserved wavefunction collapse is proposed and shown consistent with existing experiments and our macroscopic experience. Besides, we also give a critical analysis of the de Broglie-Bohm theory, the many-worlds interpretation and other dynamical collapse theories, and briefly discuss the issues of unifying quantum mechanics and relativity. (shrink)

It is usually claimed that the Laguerre polynomials were popularized by Schrödinger when creating wave mechanics; however, we show that he did not immediately identify them in studying the hydrogen atom. In the case of relativistic Dirac equations for an electron in a Coulomb field, Dirac gave only approximations, Gordon and Darwin gave exact solutions, and Pidduck first explicitly and elegantly introduced the Laguerre polynomials, an approach neglected by most modern treatises and articles. That Laguerre polynomials were not very (...) popular before their use in quantum mechanics, probably because they had been little used in classical mathematical physics, is confirmed by the fact that, as we show, they had been rediscovered independently several times during the nineteenth century, in published or unpublished studies of Abel, Murphy, Chebyshev, and Laguerre. (shrink)

The origin of the wave properties of matter is discussed from the point of view of stochastic electrodynamics. A nonrelativistic model of a charged particle with an effective structure embedded in the random zeropoint radiation field reveals that the field induces a high-frequency vibration on the particle; internal consistency of the theory fixes the frequency of this jittering at mc2/ħ. The particle is therefore assumed to interact intensely with stationary zeropoint waves of this frequency as seen from its proper (...) frame of reference; such waves, identified here as de Broglie's phase waves, give rise to a modulated wave in the laboratory frame, with de Broglie's wavelength and phase velocity equal to the particle velocity. The time-independent equation that describes this modulated wave is shown to be the stationary Schrödinger equation (or the Klein-Gordon equation in the relativistic version). In a heuristic analysis appled to simple periodic cases, the quantization rules are recovered from the assumption that for a particle in a stationary state there must correspond a stationary modulation.Along an independent and complementary line of reasoning, an equation for the probability amplitude in configuration space for a particle under a general potential V(x) is constructed, and it is shown that under conditions derived from stochastic electrodynamics it reduces to Schrödinger's equation. This equation reflects therefore the dual nature of the quantum particles, by describing simultaneously the corresponding modulated waveand the ensemble of particles. (shrink)

The connection between quantum mechanics and classical statistical mechanics has motivated in the past the representation of the Schrödinger quantum-wave equation in terms of “projections” onto the quantum configuration space of suitable phase-space asymptotic kinetic models. This feature has suggested the search of a possible exact super-dimensional classical dynamical system, denoted as Schrödinger CDS, which uniquely determines the time-evolution of the underlying quantum state describing a set of N like and mutually interacting quantum particles. In this paper the realization (...) of the same CDS in terms of a coupled set of Hamiltonian systems is established. These are respectively associated with a quantum-hydrodynamic CDS advancing in time the quantum fluid velocity and a further one the RD-CDS, describing the relative dynamics with respect to the quantum fluid. (shrink)

We propose a technical reformulation of the measurement problem of quantum mechanics, which is based on the postulate that the final state of a measurement is classical; this accords with experimental practice as well as with Bohr’s views. Unlike the usual formulation (in which the post-measurement state is a unit vector in Hilbert space), our version actually opens the possibility of admitting a purely technical solution within the confines of conventional quantum theory (as opposed to solutions that either modify this (...) theory, or introduce unusual and controversial interpretative rules and/or ontologies). To that effect, we recall a remarkable phenomenon in the theory of Schrödinger operators (discovered in 1981 by Jona-Lasinio, Martinelli, and Scoppola), according to which the ground state of a symmetric double-well Hamiltonian (which is paradigmatically of Schrödinger’s Cat type) becomes exponentially sensitive to tiny perturbations of the potential as ħ→0. We show that this instability emerges also from the textbook wkb approximation, extend it to time-dependent perturbations, and study the dynamical transition from the ground state of the double well to the perturbed ground state (in which the cat is typically either dead or alive, depending on the details of the perturbation). Numerical simulations show that adiabatically arising perturbations may (quite literally) cause the collapse of the wave-function in the classical limit. Thus, at least in the context of a simple mathematical model, we combine the technical and conceptual virtues of decoherence (which fails to solve the measurement problem but launches the key idea that perturbations may come from the environment) with those of dynamical collapse models à la grw (which do solve the measurement problem but are ad hoc), without sharing their drawbacks: single measurement outcomes are obtained (instead of merely diagonal reduced density matrices), and no modification of quantum mechanics is needed. (shrink)

This is the first comprehensive study of Schrödinger's scientific and philosophical writings. The task requires a person trained thoroughly in physical science and yet capable of appreciating the sometimes puzzling philosophical ideas Schrödinger put forward. Professor Scott, a physicist, is remarkably successful at communicating both the physical and the philosophical ideas. After a brief summary of Schrödinger's diverse writings, he divides the writings into four groups which are treated in separate chapters. The first group, including very early papers, deals with (...) Schrödinger's work on Statistical Mechanics, and statistical theories in general. The second covers the crucial development of Wave Mechanics. The third concerns Schrödinger's interpretation of quantum mechanics and the important departures from the prevailing views of the Copenhagen school. Finally the fourth group contains Schrödinger's views on life and the self. The author shows that Schrödinger was led to his doctrine of identity by reflecting on the paradox of freedom and determinism. His study "What is Life?" convinced him that living systems are governed by the law of causality but he also believed that men were free. The doctrine of identity was his solution to the paradox. Scott is critical of this solution and a number of other doctrines of Schrödinger. The book is clearly written throughout and is a good introduction to Schrödinger's thought.--R. H. K. (shrink)

We propose a technical reformulation of the measurement problem of quantum mechanics, which is based on the postulate that the final state of a measurement is classical; this accords with experimental practice as well as with Bohr’s views. Unlike the usual formulation (in which the post-measurement state is a unit vector in Hilbert space), our version actually opens the possibility of admitting a purely technical solution within the confines of conventional quantum theory (as opposed to solutions that either modify this (...) theory, or introduce unusual and controversial interpretative rules and/or ontologies).To that effect, we recall a remarkable phenomenon in the theory of Schrödinger operators (discovered in 1981 by Jona-Lasinio, Martinelli, and Scoppola), according to which the ground state of a symmetric double-well Hamiltonian (which is paradigmatically of Schrödinger’s Cat type) becomes exponentially sensitive to tiny perturbations of the potential as ħ→0. We show that this instability emerges also from the textbook wkb approximation, extend it to time-dependent perturbations, and study the dynamical transition from the ground state of the double well to the perturbed ground state (in which the cat is typically either dead or alive, depending on the details of the perturbation).Numerical simulations show that adiabatically arising perturbations may (quite literally) cause the collapse of the wave-function in the classical limit. Thus, at least in the context of a simple mathematical model, we combine the technical and conceptual virtues of decoherence (which fails to solve the measurement problem but launches the key idea that perturbations may come from the environment) with those of dynamical collapse models à la grw (which do solve the measurement problem but are ad hoc), without sharing their drawbacks: single measurement outcomes are obtained (instead of merely diagonal reduced density matrices), and no modification of quantum mechanics is needed. (shrink)

It is shown that the fact that an electron diffraction or interference pattern is not dependent on observer movement may be explained by de Broglie's phase waves, but cannot be explained by Schrödinger waves.

The author has recently proposed a “quasi-classical” theory of particles and interactions in which particles are pictured as extended periodic disturbances in a universal field χ(x, t), interacting with each other via nonlinearity in the equation of motion for χ. The present paper explores the relationship of this theory to nonrelativistic quantum mechanics; as a first step, it is shown how it is possible to construct from χ a configuration-space wave function Ψ(x 1,x 2,t), and that the theory requires (...) that Ψ satisfy the two-particle Schrödinger equation in the case where the two particles are well separated from each other. This suggests that the multiparticle Schrödinger equation can be obtained as a direct consequence of the quasi-classical theory without any use of the usual formalism (Hilbert space, quantization rules, etc.) of conventional quantum theory and in particular without using the classical canonical treatment of a system as a “crutch” theory which has subsequently to be “quantized.” The quasi-classical theory also suggests the existence of a preferred “absolute” gauge for the electromagnetic potentials. (shrink)

In this paper a nonspreading, unnormalizable wave packet satisfying the Schrödinger equation is constructed. A modification of the Schrödinger equation is considered which allows the normalization of the wave packet. The case is generalized for relativistic mechanics.

The Schrödinger equation for a particle of rest mass $m$ and electrical charge $ne$ interacting with a four-vector potential $A_i$ can be derived as the non-relativistic limit of the Klein–Gordon equation $\left( \Box '+m^2\right) \varPsi =0$ for the wave function $\varPsi $ , where $\Box '=\eta ^{jk}\partial '_j\partial '_k$ and $\partial '_j=\partial _j -\mathrm {i}n e A_j$ , or equivalently from the one-dimensional action $S_1=-\int m ds +\int neA_i dx^i$ for the corresponding point particle in the semi-classical approximation $\varPsi (...) \sim \exp {(\mathrm {i}S_1)}$ , both methods yielding the equation $\mathrm {i}\partial _0\varPsi \approx \left( \frac{1}{2m}\eta ^{\alpha \beta }\partial '_{\alpha }\partial '_{\beta } + m + n e\phi \right) \varPsi $ in Minkowski space–time , where $\alpha ,\beta =1,2,3$ and $\phi =-A_0$ . We show that these two methods generally yield equations that differ in a curved background space–time $g_{ij}$ , although they coincide when $g_{0\alpha }=0$ if $m$ is replaced by the effective mass $\mathcal{M}\equiv \sqrt{m^2-\xi R}$ in both the Klein–Gordon action $S$ and $S_1$ , allowing for non-minimal coupling to the gravitational field, where $R$ is the Ricci scalar and $\xi $ is a constant. In this case $\mathrm {i}\partial _0\varPsi \approx \left( \frac{1}{2\mathcal{M}'} g^{\alpha \beta }\partial '_{\alpha }\partial '_{\beta } + \mathcal{M}\phi ^{(\mathrm g)} + n e\phi \right) \varPsi $ , where $\phi ^{(\mathrm g)} =\sqrt{g_{00}}$ and $\mathcal{M}'=\mathcal{M}/\phi ^{(\mathrm g)} $ , the correctness of the gravitational contribution to the potential having been verified to linear order $m\phi ^{(\mathrm g)} $ in the thermal-neutron beam interferometry experiment due to Colella et al. Setting $n=2$ and regarding $\varPsi $ as the quasi-particle wave function, or order parameter, we obtain the generalization of the fundamental macroscopic Ginzburg-Landau equation of superconductivity to curved space–time. Conservation of probability and electrical current requires both electromagnetic gauge and space–time coordinate conditions to be imposed, which exemplifies the gravito-electromagnetic analogy, particularly in the stationary case, when div ${{\varvec{A}}}=\hbox {div}{{\varvec{A}}}^{(\mathrm g)}=0$ , where ${{\varvec{A}}}^{\alpha }=-A^{\alpha }$ and ${{\varvec{A}}}^{(\mathrm g)\alpha }=-\phi ^{(\mathrm g)}g^{0\alpha }$ . The quantum-cosmological Schrödinger (Wheeler–DeWitt) equation is also discussed in the $\mathcal{D}$ -dimensional mini-superspace idealization, with particular regard to the vacuum potential $\mathcal V$ and the characteristics of the ground state, assuming a gravitational Lagrangian $L_\mathcal{D}$ which contains higher-derivative terms up to order $\mathcal{R}^4$ . For the heterotic superstring theory , $L_\mathcal{D}$ consists of an infinite series in $\alpha '\mathcal{R}$ , where $\alpha '$ is the Regge slope parameter, and in the perturbative approximation $\alpha '|\mathcal{R}| \ll 1$ , $\mathcal V$ is positive semi-definite for $\mathcal{D} \ge 4$ . The maximally symmetric ground state satisfying the field equations is Minkowski space for $3\le {\mathcal {D}}\le 7$ and anti-de Sitter space for $8 \le \mathcal {D} \le 10$. (shrink)

This is the first comprehensive study of Schrödinger's scientific and philosophical writings. The task requires a person trained thoroughly in physical science and yet capable of appreciating the sometimes puzzling philosophical ideas Schrödinger put forward. Professor Scott, a physicist, is remarkably successful at communicating both the physical and the philosophical ideas. After a brief summary of Schrödinger's diverse writings, he divides the writings into four groups which are treated in separate chapters. The first group, including very early papers, deals with (...) Schrödinger's work on Statistical Mechanics, and statistical theories in general. The second covers the crucial development of Wave Mechanics. The third concerns Schrödinger's interpretation of quantum mechanics and the important departures from the prevailing views of the Copenhagen school. Finally the fourth group contains Schrödinger's views on life and the self. The author shows that Schrödinger was led to his doctrine of identity by reflecting on the paradox of freedom and determinism. His study "What is Life?" convinced him that living systems are governed by the law of causality but he also believed that men were free. The doctrine of identity was his solution to the paradox. Scott is critical of this solution and a number of other doctrines of Schrödinger. The book is clearly written throughout and is a good introduction to Schrödinger's thought.--R. H. K. (shrink)

Quantum mechanics posits that the wave function of a one-particle system evolves with time according to the Schrödinger equation, and furthermore has a square modulus that serves as a probability density function for the position of the particle. It is natural to wonder if this stochastic characterization of the particle's position can be framed as a univariate continuous Markov process, sometimes also called a classical diffusion process, whose temporal evolution is governed by the classically transparent equations of Langevin and (...) Fokker-Planck. It is shown here that this cannot generally be done in a consistent way, despite recent suggestions to the contrary. (shrink)

It has been long known that a year after Schrödinger published his equation, Madelung also published a hydrodynamics version of Schrödinger equation. Quantum diffusion is studied via dissipative Madelung hydrodynamics. Initially the wave packet spreads ballistically, than passes for an instant through normal diffusion and later tends asymptotically to a sub‐diffusive law. In this paper we will review two different approaches, including Madelung hydrodynamics and also Bohm potential. Madelung formulation leads to diffusion interpretation, which after a generalization yields to (...) Ermakov equation. Since Ermakov equation cannot be solved analytically, then we try to find out its solution with Mathematica package. It is our hope that these methods can be verified and compared with experimental data. But we admit that more researches are needed to fill all the missing details. (shrink)

We discuss two recent attempts two solve Schrodinger's cat paradox. One is the modal interpretation developed by Kochen, Healey, Dieks, and van Fraassen. It allows for an observable which pertains to a system to possess a value even when the system is not in an eigenstate of that observable. The other is a recent theory of the collapse of the wave function due to Ghirardi, Rimini, and Weber. It posits a dynamics which has the effect of collapsing the (...) state of macroscopic systems. We argue that the modal interpretation cannot account for non-accurate measurements and that both accounts have the consequence that in ordinary measurement situations the observables that ends up well defined are not quite the ones that we want to be well defined. (shrink)

The gravity-related model of spontaneous wave function collapse, a longtime hypothesis, damps the massive Schrödinger Cat states in quantum theory. We extend the hypothesis and assume that spontaneous wave function collapses are responsible for the emergence of Newton interaction. Superfluid helium would then show significant and testable gravitational anomalies.

In 1927 Louis de Broglie proposed an alternative approach to standard quantum mechanics known as the double solution program (DSP) where particles are represented as bunched fields or solitons guided by a base (weaker) wave. DSP evolved as the famous de Broglie-Bohm pilot wave interpretation (PWI) also known as Bohmian mechanics but the general idea to use solitons guided by a base wave to reproduce the dynamics of the PWI was abandoned. Here we propose a nonlinear scalar (...) field theory able to reproduce the PWI for the Schrödinger and Klein–Gordon guiding waves. Our model relies on a relativistic ‘phase harmony’ condition locking the phases of the solitonic particle and the guiding wave. We also discuss an extension of the theory for the N particles cases in presence of entanglement and external (classical) electromagnectic fields. (shrink)

Multi-time wave functions are wave functions for multi-particle quantum systems that involve several time variables. In this paper we contrast them with solutions of wave equations on a space–time with multiple timelike dimensions, i.e., on a pseudo-Riemannian manifold whose metric has signature such as \ or \, instead of \. Despite the superficial similarity, the two behave very differently: whereas wave equations in multiple timelike dimensions are typically mathematically ill-posed and presumably unphysical, relevant Schrödinger equations for (...) multi-time wave functions possess for every initial datum a unique solution on the spacelike configurations and form a natural covariant representation of quantum states. (shrink)

It is assumed that the motion of a particle in spacetime does not depend on the motion relative to it of any observer or of any frame of reference. Thus if the particle has an internal vibration of the type hypothesized by de Broglie, the phase of that vibration at any point in spacetime must appear to be the same to all observers, i.e., the same in all frames of reference. Each observer or reference frame will have its own de (...) Broglie wave for the particle. The phase of the particle's vibration must, by definition, be the same as that of all possible de Broglie waves at the point where the particle is. By superimposing all these possible de Broglie waves, a wave packet is formed centered in space on the particle. The formation of such a packet is discussed with the help of spacetime diagrams; the packet does not spread with time. The relevance of this packet to the wave mechanics of Schrödinger is discussed; it is also pointed out that any vibration can lead to such a packet. (shrink)