Quantum mechanics has a deterministic Schrödinger equation for the wave function. The Göttingen-Copenhagen statistical interpretation is based on the Born Rule that interprets the wave function as a "probability amplitude." A precept of this interpretation is the lack of determinism in quantum mechanics. The Bohm interpretation is that the wave function is a source of a field experienced by the electrons, thereby attributing determinism to quantum theory. In this paper, we present a new perspective on such determinism. The ideas are based on the equations of motion or "Quantal Newtonian" Laws obeyed by each electron. These Laws, derived from the temporal and stationary-state Schrödinger equation, are interpreted in terms of "classical" fields whose sources are quantal expectations of Hermitian operators taken with respect to the wave function. According to the Second Law, each electron experiences an external field-the quantal Coulomb-Lorentz law. It also experiences an internal field representative of properties of the system: correlations due to Coulomb repulsion and Pauli principle; the density; kinetic effects; and an internal magnetic field component. There is a response field. The First Law states that the sum of the external and internal fields experienced by each electron vanishes. These fields are akin to those of classical physics: They pervade all space; their structure is descriptive of the quantum system; the energy of the system is stored in these fields. It is in the classical behavior of these fields, which arise from quantal sources that one may then speak of determinism in quantum mechanics.
Quantum mechanics has a deterministic Schrödinger equation for the wave function. The Göttingen-Copenhagen statistical interpretation is based on the Born Rule that interprets the wave function as a "probability amplitude." A precept of this interpretation is the lack of determinism in quantum mechanics. The Bohm interpretation is that the wave function is a source of a field experienced by the electrons, thereby attributing determinism to quantum theory. In this paper, we present a new perspective on such determinism. The ideas are based on the equations of motion or "Quantal Newtonian" Laws obeyed by each electron. These Laws, derived from the temporal and stationary-state Schrödinger equation, are interpreted in terms of "classical" fields whose sources are quantal expectations of Hermitian operators taken with respect to the wave function. According to the Second Law, each electron experiences an external field-the quantal Coulomb-Lorentz law. It also experiences an internal field representative of properties of the system: correlations due to Coulomb repulsion and Pauli principle; the density; kinetic effects; and an internal magnetic field component. There is a response field. The First Law states that the sum of the external and internal fields experienced by each electron vanishes. These fields are akin to those of classical physics: They pervade all space; their structure is descriptive of the quantum system; the energy of the system is stored in these fields. It is in the classical behavior of these fields, which arise from quantal sources that one may then speak of determinism in quantum mechanics.
The Schrödinger-Pauli (SP) theory of electrons in an electromagnetic field explicitly accounts for the electron spin moment. The many-electron theory is complemented via a new descriptive perspective viz. that of the individual electron via its equation of motion or ‘Quantal Newtonian’ first law. The law is in terms of ‘classical’ fields whose sources are quantum mechanical expectation values of Hermitian operators taken with respect to the system wave function. The law states that each electron experiences an external and an internal field, the sum of which vanish. The external field is the sum of the binding electrostatic and a Lorentz field. The internal field is a sum of fields representative respectively of electron correlations due to the Pauli principle and Coulomb repulsion; the electron density; kinetic effects; and an internal magnetic field. The energy can be expressed in integral virial form in terms of these fields. The law is elucidated by application to the 23S state of a quantum dot in a magnetic field. It is proved that the Hamiltonian is an exactly known and universal functional of the wave function. This generalizes the SP equation, and reveals that its eigenfunctions and eigenvalues may be determined self-consistently. A Quantal density functional theory (QDFT) of the SP system is developed whereby additional properties are determined. A physical interpretation of Spin-DFT based on the QDFT mapping is provided. Further generalizations of the present work to the temporal case, and relativistic Dirac theory, are proposed.
Abstract The Schrödinger–Pauli (SP) theory of electrons in an electromagnetic field explicitly accounts for the electron spin moment. The many‐electron theory is complemented via a new descriptive perspective namely, that of the individual electron via its equation of motion or “Quantal Newtonian” first law. The law is in terms of ‘classical’ fields whose sources are quantum mechanical expectation values of Hermitian operators taken with respect to the system wave function. The law states that each electron experiences an external and an internal field, the sum of which vanish. The external field is the sum of the binding electrostatic and a Lorentz field. The internal field is a sum of fields representative respectively of electron correlations due to the Pauli principle and Coulomb repulsion; the electron density; kinetic effects; and an internal magnetic field. The energy can be expressed in integral virial form in terms of these fields. The law is elucidated by application to the 23S state of a quantum dot in a magnetic field. It is proved that the Hamiltonian is an exactly known and universal functional of the wave function. This generalizes the SP equation, and reveals that its eigenfunctions and eigenvalues may be determined self‐consistently. A quantal density functional theory (QDFT) of the SP system is developed whereby additional properties are determined. A physical interpretation of Spin‐DFT based on the QDFT mapping is provided. Further generalizations of the present work to the temporal case, and relativistic Dirac theory, are proposed.
Since the early twentieth century, quantum mechanics has sought an interpretation that offers a consistent worldview. In the course of that, many proposals were advanced, but all of them introduce, at some point, interpretation elements (semantics) that find no correlate in the formalism (syntactics). This distance from semantics and syntactics is one of the major reasons for finding so abstruse and diverse interpretations of the formalism. To overcome this issue, we propose an alternative stochastic interpretation, based exclusively on the formal structure of the Schrödinger equation, without resorting to external assumptions such as the collapse of the wave function or the role of the observer. We present four (mathematically equivalent) mathematical derivations of the Schrödinger equation based on four constructs: characteristic function, Boltzmann entropy, Central Limit Theorem (CLT), and Langevin equation. All of them resort to axioms already interpreted and offer complementary perspectives to the quantum formalism. The results show the possibility of deriving the Schrödinger equation from well-defined probabilistic principles and that the wave function represents a probability amplitude in the configuration space, with dispersions linked to the CLT. It is concluded that quantum mechanics has a stochastic support, originating from the separation between particle and field subsystems, allowing an objective description of quantum behavior as a mean-field theory, analogous, but not equal, to Brownian motion, without the need for arbitrary ontological entities.
The present paper is based upon equations obtained in an earlier paper by the author devoted to a new formulation of quantum electrodynamics. The equations describe the structure of the electron as well as its motion in external fields, interaction with a measuring apparatus inclusive, in a deterministic manner without any jumps. Quantum mechanics is an approximate theory because its equations follow from the above equations upon neglecting the self-field of the electron itself. Just this leads to paradoxes, seeming contradictions and jumps in quantum mechanics. The quantum mechanical wavefunction has a dual interpretation. In some problems the square of its modulus represents a real distribution of the electronic density while in others the same square determines the probability distribution of coordinates. It is shown why, given the different interpretations of the wavefunction, it satisfies one and the same Dirac or Schrödinger equation. Description of many-electron systems is also considered in the starting approach as well as in quantum mechanics. Neutrinos are discussed in brief.
Einstein’s 1919 distinction between “principle theories” and ”constructive theories” has been applied by Jeffrey Bub to classify the Copenhagen interpretation of quantum mechanics (QM) as a principle theory agree with this classification. Additionally, I argue that Bohm’s interpretation of QM fits Einstein’s concept of a constructive theory. Principle theories include empirically established laws or principles, such as the first and second laws of thermodynamics or the principles of special relativity, including the Born Rule of QM. According to Einstein, principle theories offer ”security in their foundations and logical perfection”. However, ultimate understanding requires constructive theories, which build complex phenomena from simpler models. Constructive theories provide intelligible models of physical phenomena. Bohm’s QM, with its added microstructure, presents such a model. In this framework, quantum phenomena appear from statistical ensembles of microparticles in motion, with deterministic particle trajectories guided by the wave function. This reveals how Bohm’s account offers a constructive model for understanding quantum phenomena.
The predictions of quantum mechanics are probabilistic. Quantum probabilities are extracted using a postulate of the theory called the Born rule, the status of which is central to the problem of quantum mechanics. Efforts to justify the Born rule from other physical principles, and thus elucidate the measurement process, have involved lengthy statistical or information-theoretic arguments. Here we show that Bohm's deterministic formulation of quantum mechanics allows the Born rule for measurements on a single system to be derived, without any statistical assumptions. We solve a simple example where the creation of an ensemble of identical quantum states, together with position measurements on those states, are described by Bohm's quantum dynamics. The calculated measurement outcomes agree with the Born-rule probabilities, which are thus a consequence of deterministic evolution. Our results demonstrate that quantum probabilities can emerge from simple dynamical laws alone, and they support the view that there is no underlying indeterminism in quantum phenomena.