Schrödingers Katze und Wigners Freund (zur Beweglichkeit des Heisenbergschen Schnittes)

Schrödinger's Cat and Wigner's Friend (about the Mobility of the Heisenberg Cut) The history of Schrödinger's cat and of Wigner's friend is based on the mobility of “Heisenbergs Schnitt”.     

An Informational Characterization of Schrödinger's Uncertainty Relations

Heisenberg's uncertainty relations employ commutators of observables to set fundamental limits on quantum measurement. The information concerning incompatibility (non-commutativity) of observables is well included but that concerning correlation is missing. Schrödinger's uncertainty relations remedy this defect by supplementing the correlation in terms of anti-commutators. However, both Heisenberg's uncertainty relations and Schrödinger's uncertainty relations are expressed in terms of variances, which are not good measures of uncertainty in general situations (e.g., when mixed states are involved). By virtue of the Wigner–Yanase skew information, we will establish an uncertainty relation along the spirit of Schrödinger from a statistical inference perspective and propose a conjecture. The result may be interpreted as a quantification of certain aspect of the celebrated Wigner–Araki–Yanase theorem for quantum measurement, which states that observables not commuting with a conserved quantity cannot be measured exactly.     

The fate of Schrödinger's cat

Some solutions to the Schrödinger's Cat paradox are proposed, using the possibility of wavefunction collapse at the microscopic level.     

Exact traveling wave solutions of perturbed nonlinear Schrödinger's equation (NLSE) with Kerr law nonlinearity

By using new solutions of nonlinear partial differential equation, a direct algebraic method is described to construct the exact traveling wave solutions for perturbed nonlinear Schrödinger's equation (NLSE). Exact traveling wave solutions are explicitly obtained by this method.     

On inferring quantum energies and expectation values

A maximum entropy-like inference approach is proposed that yields ground and excited states energies and expectation values without solving Schrödinger's equation. The method is tested with reference to the quartic anharmonic oscillator and other model potentials and yields good results without undue numerical effort.     

Reformulating classical and quantum mechanics in terms of a unified set of consistency conditions

This paper imposes consistency conditions on the path of a particle and shows that they imply Hamilton's principle in classical contexts and Schrödinger's equation in quantum mechanical contexts. Thus this paper provides a common, intuitive foundation for classical and quantum mechanics. It also provides a very new perspective on quantum mechanics.     

The Frieden-Soffer principle and wave functions that maximize Shannon's measure

Starting with a wave function ψ of a form that automatically extremizes Shannon's information measure according to Jaynes' maximum entropy principle (MEP), we show how, with the help of Frieden and Soffer's principle of extremal information (EPI), one can devise a potential V(x) that, if taken as a constraint in extremizing the Frieden-Soffer action, transforms ψ into an exact solution of Schrödinger's equation for V(x). The concomitant procedure illustrates the connection between two apparently disparate physical principles (MEP and EPI).     

CONSERVATION LAWS FOR THE SCHRÖDINGER–NEWTON EQUATIONS

In this Letter a first-order Lagrangian for the Schrödinger–Newton equations is derived by modifying a second-order Lagrangian proposed by Christian [Exactly soluble sector of quantum gravity, Phys. Rev. D 56(8) (1997) 4844–4877]. Then Noether's theorem is applied to the Lie point symmetries determined by Robertshaw and Tod [Lie point symmetries and an approximate solution for the Schrödinger–Newton equations, Nonlinearity 19(7) (2006) 1507–1514] in order to find conservation laws of the Schrödinger–Newton equations.     

The Schrödinger-HJW Theorem

A concise presentation of Schrödinger's ancilla theorem (Proc. Camb. Phil. Soc. 32, 446 (1936)) and its several recent rediscoveries.     

Quantization of Brownian Motion

Following our work on the quantization of nonconservative systems using fractional calculus, the canonical quantization of a system with Brownian motion is carried out according to the Dirac method. A suitable Lagrangian corresponding to the Langevin equation is set up. Further, a Hamiltonian is constructed and is transformed to Schrödinger's equation which is solved.     

Reply to “Comments on Bouda and Djama's ‘Quantum Newton's Law’”

In this reply, we hope to bring clarifications about the reservations expressed by Floyd in his comments, give further explanations about the choice of the approach and show that our fundamental result can be reproduced by other ways. We also establish that Floyd's trajectories manifest some ambiguities related to the mathematical choice of the couple of solutions of Schrödinger's equation.     

Nonlinear Schrödinger equations from prequantum classical statistical field theory

We derive some important features of the standard quantum mechanics from a certain classical-like model—prequantum classical statistical field theory, PCSFT. In this approach correspondence between classical and quantum quantities is established through asymptotic expansions. PCSFT induces not only linear Schrödinger's equation, but also its nonlinear generalizations. This coupling with “nonlinear wave mechanics” is used to evaluate the small parameter of PCSFT.     

Prequantum Classical Statistical Field Theory: Complex Representation, Hamilton-Schrödinger Equation, and Interpretation of Stationary States

We develop a prequantum classical statistical model in that the role of hidden variables is played by classical (vector) fields. We call this model Prequantum Classical Statistical Field Theory (PCSFT). The correspondence between classical and quantum quantities is asymptotic, so we call our approach asymptotic dequantization. We construct the complex representation of PCSFT. In particular, the conventional Schrödinger equation is obtained as the complex representation of the system of Hamilton equations on the infinite-dimensional phase space. In this note we pay the main attention to interpretation of so called pure quantum states (wave functions) in PCSFT, especially stationary states. We show, see Theorem 2, that pure states of QM can be considered as labels for Gaussian measures concentrated on one dimensional complex subspaces of phase space that are invariant with respect to the Schrödinger dynamics. “A quantum system in a stationary state ψ” in PCSFT is nothing else than a Gaussian ensemble of classical fields (fluctuations of the vacuum field of a very small magnitude) which is not changed in the process of Schrödinger's evolution. We interpret in this way the problem of stability of hydrogen atom. One of unexpected consequences of PCSFT is the infinite dimension of physical space on the prequantum scale.     

SU(1, 1) and SU(2) Perelomov number coherent states: algebraic approach for general systems

We study some properties of the SU(1, 1) Perelomov number coherent states. The Schrödinger's uncertainty relationship is evaluated for a position and momentum-like operators (constructed from the Lie algebra generators) in these number coherent states. It is shown that this relationship is minimized for the standard coherent states. We obtain the time evolution of the number coherent states by supposing that the Hamiltonian is proportional to the third generator K₀ of the su(1, 1) Lie algebra. Analogous results for the SU(2) Perelomov number coherent states are found. As examples, we compute the Perelomov coherent states for the pseudoharmonic oscillator and the two-dimensional isotropic harmonic oscillator.     

Bright and dark solitons for the resonant nonlinear Schrödinger's equation with time-dependent coefficients

In this paper, the resonant nonlinear Schrödinger's equation is studied with five forms of nonlinearity. This equation is also considered with time-dependent coefficients and additionally time-dependent linear attenuation is considered. The ansatz method approach is used to carry out the integration. Both bright and dark soliton solutions are obtained in this paper. The constraint conditions for the existence of soliton solutions are also given.     

A critical analysis of Helmholtz's argument against Weber's electrodynamics

We present Helmholtz's argument against Weber's electrodynamics. It is related with a fixed charged nonconducting spherical shell and a charged particle moving inside it. Then we utilize Weber's electrodynamics plus Schrödinger's expression for gravitational interactions in order to obtain the equation of motion and to study this situation. We show that this approach avoids the problems pointed out by Helmholtz. Moreover, it indicates that the effective inertial mass of the charged particle will depend not only on the electrostatic potential of the shell but also on its velocity. This is a relevant aspect of Weber's theory.     

Critical binding of diatomic molecules

The behaviour of two bodies that are just bound or nearly bound is discussed. A class of potentials is given for which Schrödinger's equation has exact solutions at critical binding (zero binding energy). This class includes the known solution for the 6–10 potential. For a general potential characterized by a coupling parameter α, it is shown that the bound state energy tends to zero as -(α - α₀)², where α₀ is the critical value of the coupling parameter. Small energy scattering of atoms which are near critical binding (e.g. helium atoms) is examined. It is shown that determination of the total cross-section up to terms of order k ² is in principle sufficient to distinguish between bound and virtual states of the diatomic molecule.     

Classical spin in a potential field

We consider an ensemble of restricted discrete random walks in 2+1 dimensions. The restriction on the walks is such as to given particles an intrinsic angular momentum. The walks are embedded in a field which affects the mean free path of the walks. We show that the dynamics of the walks is such that second-order effects are described by a discrete form of Schrödinger's equation for particles in a potential field. This provides a classical context of the equation which is independent of its quantum context.