Determinism in Classical and Quantal Physics

John Earman

Máximo García-Sucre, Mario Bunge

Abstract Classically every physical thing has some geometry or other, but in the quantum theory the notions of spatial structure, shape, and size seem to become hazy if not outright inapplicable. In particular, elementary entities are by definition structureless and seem to have no size or shape other than those imposed by their environmént—as represented, e.g., by the boundary conditions. However, this does not entail that systems of interacting components, such as molecules, lack a geometry as well, as has been argued in recent times. The purpose of this paper is to investigate this claim. Since the solution to the problem whether a quantum‐mechanical system possesses a geometry depends critically upon the definition of the latter, we elucidate the notions of extension, shape, and spatial configuration of a system. We do so first classically in terms of the energy density, then quantum mechanically in terms of a suitably constructed position distribution function that depends upon the spatial coordinates (and is thus a field quantity) rather than on the particle coordinates. In this way we construct both classical and quantal concepts of the geometry of a system. We also define the notions of spatial structure and chemical structure of a quantal system. Our quantum‐mechanical concepts are quite general and constitute smooth extensions of the corresponding classical concepts. The upshot of this investigation is that a quantal system does have a geometry, albeit not as clearcut a one as a classical system, that is primarily determined by the interactions among its components. (Thus, physics determines geometry of physical systems, not the other way around.) This conclusion is obtained within nonrelativistic quantum mechanics without adding any extra assumptions. Our result vindicates the ball‐and‐spoke models of molecules and thus shows once again that quantum chemistry, although different from classical chemistry, stays close to the chemist's work bench.

John Earman

Viraht Sahni

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.

Mohsen Razavy

The phenomenological optical potential model of nuclear physics is studied to determine how it corresponds to the classical damped motion. Using the Ehrenfest theorem, it is shown that (1) the usual correspondence between the classical and quantal descriptions does not hold in the presence of optical potentials, (2) the number of degrees of freedom for the motion of a particle is twice the dimension of the space in which it is constrained to move, and (3) the canonical and the mechanical momenta are not the same. An example of an optical potential which has a classical analog is studied in some detail, and it is shown that a consistent solution can be found if a set of coordinates equal in number to the degrees of freedom is chosen for the description of the motion. A new Hamiltonian formulation for the linearly damped motion is obtained which is similar to the Hamiltonian proposed by Feshbach and Tikochinsky.

Grete Hermann