Expectation Values of Observables in Time-Dependent Quantum Mechanics

Let U(t) be the evolution operator of the Schrödinger equation generated by a Hamiltonian of the form H ₀(t) + W(t), where H ₀(t) commutes for all twith a complete set of time-independent projectors . Consider the observable A=_j P _jjwhere _j j , >0, for jlarge. Assuming that the matrix elements of W(t) behave as for p>0 large enough, we prove estimates on the expectation value for large times of the type where >0 depends on pand . Typical applications concern the energy expectation H₀(t) in case H ₀(t) H ₀or the expectation of the position operator x²(t) on the lattice where W(t) is the discrete Laplacian or a variant of it and H ₀(t) is a time-dependent multiplicative potential.     

Simple Realism and Canonically Conjugate Observables in Non-Relativistic Quantum Mechanics

In this paper we list some minimal requirements for a physically natural, straightforwardly realist interpretation of non-relativistic quantum mechanics. The goal is to characterize what one might call a simple realism of quantum systems, and of the observables associated with them.Simple realism as developed here is a generalized interpretation-scheme, one that abstracts important shared features of Einsteinian naive realism, the so-called modal interpretations, and the orthodox interpretation itself. Some such schemes run afoul of the classic no-go theorems, while others do not. The role of non-commuting observables plays a major role in this success or failure. In particular, we show that if a simple-realist interpretation attributes simultaneously definite values to canonically conjugate observables, then it necessarily falls prey to Kochen-Specker contradictions.This exercise provides some insight into why modal interpretations work, while more generally placing limits on the scope of simple realism itself. In particular, we find that within the framework of simple realism, the only consistent interpretation of the uncertainty relations is the orthodox one. What's more, we point out that similar conclusions are bound to hold for many other non-commuting observables as well.     

Stochastic Isometries in Quantum Mechanics

The class of stochastic maps, that is, linear, trace-preserving, positive maps between the self-adjoint trace class operators of complex separable Hilbert spaces plays an important role in the representation of reversible dynamics and symmetry transformations. Here a characterization of the isometric stochastic maps is given and possible physical applications are indicated.     

Quantum Observables and Effect Algebras

We study observables on monotone σ-complete effect algebras. We find conditions when a spectral resolution implies existence of the corresponding observable. We characterize sharp observables of a monotone σ-complete homogeneous effect algebra using its orthoalgebraic skeleton. In addition, we study compatibility in orthoalgebras and we show that every orthoalgebra satisfying RIP is an orthomodular poset.     

Probability Representation of Quantum Observables and Quantum States

We introduce the probability distributions describing quantum observables in conventional quantum mechanics and clarify their relations to the tomographic probability distributions describing quantum states. We derive the evolution equation for quantum observables (Heisenberg equation) in the probability representation and give examples of the spin-1/2 (qubit) states and the spin observables. We present quantum channels for qubits in the probability representation.     

A Stochastic Mechanics Based on Bohm‧s Theory and its Connection with Quantum Mechanics

We construct a stochastic mechanics by replacing Bohm‧s first-order ordinary differential equation of motion with a stochastic differential equation where the stochastic process is defined by the set of Bohmian momentum time histories from an ensemble of particles. We show that, if the stochastic process is a purely random process with n-th order joint probability density in the form of products of delta functions, then the stochastic mechanics is equivalent to quantum mechanics in the sense that the former yields the same position probability density as the latter. However, for a particular non-purely random process, we show that the stochastic mechanics is not equivalent to quantum mechanics. Whether the equivalence between the stochastic mechanics and quantum mechanics holds for all purely random processes but breaks down for all non-purely random processes remains an open question.     

Observables,Evolution Equation,and Stationary States Equation in the Joint Probability Representation of Quantum Mechanics

Symplectic and optical joint probability representations of quantum mechanics are considered, in which the functions describing the states are the probability distributions with all random arguments (except the argument of time). The general formalism of quantizers and dequantizers determining the star product quantization scheme in these representations is given. Taking the Gaussian functions as the distributions of the tomographic parameters the correspondence rules for most interesting physical operators are found and the expressions of the dual symbols of operators in the form of singular and regular generalized functions are derived. Evolution equations and stationary states equations for symplectic and optical joint probability distributions are obtained.     

On Schrödinger's Search for a Stochastic Interpretation of Quantum Mechanics

Following Schrödinger a stochastic interpretation of quantum mechanics is given based on the introduction of an intermediate probability in diffusion processes. The Schrödinger equation is derived following Nelson's approach and following a variational approach. Some problems of the quantum theory of measurement are discussed.     

Classical Mechanics and Quantum Mechanics

The Newton equation of motion is derived from quantum mechanics.     

Nonlinear Quantum Mechanics at the Planck Scale

I argue that the linearity of quantum mechanics is an emergent feature at the Planck scale, along with the manifold structure of space-time. In this regime the usual causality violation objections to nonlinearity do not apply, and nonlinear effects can be of comparable magnitude to the linear ones and still be highly suppressed at low energies. This can offer alternative approaches to quantum gravity and to the evolution of the early universe. PACS: 04.60.-m, 03.65.Ta     

Quantum Observables,Lie Algebra Homology and TQFT

Let us consider a Lie (super)algebra G spanned by T where T are quantum observables in BV formalism. It is proved that for every tensor c^... that determines a homology class of the Lie algebra G the expression c^...T...T is again a quantum observable. This theorem is used to construct quantum observables in the BV sigma model. We apply this construction to explain Kontsevich's results about the relation between homology of the Lie algebra of Hamiltonian vector fields and topological invariants of manifolds.     

Stochastic Quantum Scattering and State Reduction at Measurement

A stochastic modification of quantum scattering theory is proposed. It states that the final scattering state reduces alternatively to substates corresponding to transitions with or without energy-momentum transfer. The state reduction at the measuring process comes out as to require.     

Smearing of Observables and Spectral Measures on Quantum Structures

An observable on a quantum structure is any σ-homomorphism of quantum structures from the Borel σ-algebra of the real line into the quantum structure which is in our case a monotone σ-complete effect algebra with the Riesz Decomposition Property. We show that every observable is a smearing of a sharp observable which takes values from a Boolean σ-subalgebra of the effect algebra, and we prove that for every element of the effect algebra there corresponds a spectral measure.     

Geometric Structure for Quantum Mechanics

A geometric connection between quantum mechanics and classical mechanics is described and an operator version of the Poisson bracket is developed.     

Consciousness and Quantum Mechanics

For a quantum mechanical measurement to be complete, John von Neumann and others assumed that a conscious observer must be present to affect a reduction or collapse of the state function. Also, William James believed that the influence of consciousness on physical bodies is required by the demands of biological evolution. The author shows how both of these ideas might be correct if there exists a neurological mechanism that responds to the presence of an inside observer of a kind defined in a previous paper. An experiment is proposed to test the hypothetical mechanism.