Standard Quantum Mechanics Research Articles

Multifold paths of neutrons in the three-beam interferometer detected by a tiny energy kick

A neutron optical experiment is presented to investigate the paths taken by neutrons in a three-beam interferometer. In various beam-paths of the interferometer, the energy of the neutrons is partially shifted so that the faint traces are left along the beam-path. By ascertaining an operational meaning to "the particle's path", which-path information is extracted from these faint traces with minimal-perturbations. Theory is derived by simply following the time evolution of the wave function of the neutrons, which clarifies the observation in the framework of standard quantum mechanics. Which-way information is derived from the intensity, sinusoidally oscillating in time at different frequencies, which is considered to result from the interfering cross terms between stationary main component and the energy-shifted which-way signals. Final results give experimental evidence that the (partial) wave functions of the neutrons in each beam path are superimposed and present in multiple locations in the interferometer.

Work drives time evolution

We propose the idea that time evolution of quantum systems is driven by work. The formalism presented here falls within the scope of a recently proposed theory of gravitating quantum matter where extractible work, and not energy, is responsible for gravitation. Our main assumption is that extractible work, and not the Hamiltonian, dictates dynamics. We find that expectation values of meaningful quantities, such as the occupation number, deviate from those predicted by standard quantum mechanics. The scope, applications and validity of this proposal are also discussed.

Quantization and Bifurcation beyond Square-Integrable Wavefunctions.

Probability interpretation is the cornerstone of standard quantum mechanics. To ensure the validity of the probability interpretation, wavefunctions have to satisfy the square-integrable (SI) condition, which gives rise to the well-known phenomenon of energy quantization in confined quantum systems. On the other hand, nonsquare-integrable (NSI) solutions to the Schrödinger equation are usually ruled out and have long been believed to be irrelevant to energy quantization. This paper proposes a quantum-trajectory approach to energy quantization by releasing the SI condition and considering both SI and NSI solutions to the Schrödinger equation. Contrary to our common belief, we find that both SI and NSI wavefunctions contribute to energy quantization. SI wavefunctions help to locate the bifurcation points at which energy has a step jump, while NSI wavefunctions form the flat parts of the stair-like distribution of the quantized energies. The consideration of NSI wavefunctions furthermore reveals a new quantum phenomenon regarding the synchronicity between the energy quantization process and the center-saddle bifurcation process.

Contact geometry and quantum mechanics

We present a generally covariant approach to quantum mechanics in which generalized positions, momenta and time variables are treated as coordinates on a fundamental “phase-spacetime”. We show that this covariant starting point makes quantization into a purely geometric flatness condition. This makes quantum mechanics purely geometric, and possibly even topological. Our approach is especially useful for time-dependent problems and systems subject to ambiguities in choices of clock or observer. As a byproduct, we give a derivation and generalization of the Wigner functions of standard quantum mechanics.

The post-selection operator current

In this paper, we develop the concept of the post-selection operator current and we use it to extend the methodology of quantum mechanics into related fields of science and engineering. We begin by reviewing some results from standard quantum mechanics. We then introduce the post-selection operator current and note that the concept of the operator current provides a common framework for connecting underlying concepts in classical signal theory and quantum mechanics. Next, we explore the geometry of post-selection making use of the Pancharatnam phase. We illustrate the usefulness of the method through a series of simple examples; at each stage we link the results of quantum mechanics to applications in signal processing. We then simplify some results by introducing a density operator. This simplification allows us to present a number of useful observations about weak values. We conclude by using the operator current explore the relationship between post-selection and gauge invariance. The methods developed in this paper enable us to characterize the time evolution of a post-selected operator. The methods can be applied to other evolutionary/transport equations including the Fokker–Planck equation, and the equation of Brownian motion.

Some clarifications about the Bohmian geodesic deviation equation and Raychaudhuri’s equation

One of the important and famous topics in general theory of relativity and gravitation is the problem of geodesic deviation and its related singularity theorems. An interesting subject is the investigation of these concepts when quantum effects are considered. Since the definition of trajectory is not possible in the framework of standard quantum mechanics (SQM), we investigate the problem of geodesic equation and its related topics in the framework of Bohmian quantum mechanics in which the definition of trajectory is possible. We do this in a fixed background and we do not consider the backreaction effects of matter on the space–time metric.

Primary Assumptions and Guidance Laws in Wave Mechanics

In an article written by Louis de Broglie in 1959 (30 years after the Nobel prize rewarding his foundation of Wave Mechanics), the most challenging problem raised by the Bohr, Heisenberg and Born Standard Quantum Mechanics (SQM) was pointed out in the renunciation to describe “a permanent localization in space, and therefore a well-defined trajectory” for any moving particle. This challenge is taken up in the present paper, showing that de Broglie’s Primary Assumption p=hk, predicting the wave-particle duality, does also allow to obtain from the energy-dependent form of the Schrödinger and/or Klein-Gordon equations the Guidance Laws piloting particles along well-defined trajectories. The energy-independent equations, on the other hand, may only give rise—both in SQM and in the Bohmian approach—to probabilistic descriptions, overshadowing the role of de Broglie’s matter waves in physical space.

Tunneling time in space fractional quantum mechanics

We calculate the time taken by a wave packet to travel through a classically forbidden region of space in space fractional quantum mechanics. We obtain the close form expression of tunneling time from a rectangular barrier by stationary phase method. We show that tunneling time depends upon the width b of the barrier for b→∞ and therefore Hartman effect doesn't exist in space fractional quantum mechanics. Interestingly we found that the tunneling time monotonically reduces with increasing b. The tunneling time is smaller in space fractional quantum mechanics as compared to the case of standard quantum mechanics. We recover the Hartman effect of standard quantum mechanics as a special case of space fractional quantum mechanics.

Quantum origins of space-time

Chunjun Cao and Sean Carroll at the California Institute of Technology in Pasadena have tried to extract the kind of space-time we would find in the vicinity of our solar system from standard quantum mechanics. This type of space-time is one whose curvature is mostly flat, but with small undulations due to weak gravitational fields. To see if space-time can emerge from this quantum description, Cao and Carroll used an abstract mathematical concept called Hilbert space that can be split into different tiny parts, such that each one corresponds to a single point in 3D space. The researchers relate entanglement to geometry by further assuming that the greater the entanglement between two parts, the closer they are. The entire system is also assumed to be in some state of equilibrium, such that increasing the entanglement in one region decreases it elsewhere, and vice versa. Given a handful of such assumptions, Cao and Carroll have shown that the equations governing the dynamics of entanglement are similar to Einstein's equations of general relativity. In other words, space-time and gravity emerge from entanglement.

Spinfoam cosmology with the proper vertex amplitude

The proper vertex amplitude is derived from the Engle–Pereira–Rovelli–Livine vertex by restricting to a single gravitational sector in order to achieve the correct semi-classical behaviour. We apply the proper vertex to calculate a cosmological transition amplitude that can be viewed as the Hartle–Hawking wavefunction. To perform this calculation we deduce the integral form of the proper vertex and use extended stationary phase methods to estimate the large-volume limit. We show that the resulting amplitude satisfies an operator constraint whose classical analogue is the Hamiltonian constraint of the Friedmann–Robertson–Walker cosmology. We find that the constraint dynamically selects the relevant family of coherent states and demonstrate a similar dynamic selection in standard quantum mechanics. We investigate the effects of dynamical selection on long-range correlations.

Two and three particles interacting in a one-dimensional trap

We outline a procedure for using matrix mechanics to compute energy eigenvalues and eigenstates for two and three interacting particles in a confining trap, in one dimension. Such calculations can bridge a gap in the undergraduate physics curriculum between single-particle and many-particle quantum systems, and can also provide a pathway from standard quantum mechanics course material to understanding current research on cold-atom systems. In particular, we illustrate the notion of “fermionization” and how it occurs not only for the ground state in the presence of strong repulsive interactions, but also for excited states, in both the strongly attractive and strongly repulsive regimes.

Proof of the Spin Statistics Connection 2: Relativistic Theory

The traditional standard theory of quantum mechanics is unable to solve the spin–statistics problem, i.e. to justify the utterly important “Pauli Exclusion Principle” but by the adoption of the complex standard relativistic quantum field theory. In a recent paper (Santamato and De Martini in Found Phys 45(7):858–873, 2015) we presented a proof of the spin–statistics problem in the nonrelativistic approximation on the basis of the “Conformal Quantum Geometrodynamics”. In the present paper, by the same theory the proof of the spin–statistics theorem is extended to the relativistic domain in the general scenario of curved spacetime. The relativistic approach allows to formulate a manifestly step-by-step Weyl gauge invariant theory and to emphasize some fundamental aspects of group theory in the demonstration. No relativistic quantum field operators are used and the particle exchange properties are drawn from the conservation of the intrinsic helicity of elementary particles. It is therefore this property, not considered in the standard quantum mechanics, which determines the correct spin–statistics connection observed in Nature (Santamato and De Martini in Found Phys 45(7):858–873, 2015). The present proof of the spin–statistics theorem is simpler than the one presented in Santamato and De Martini (Found Phys 45(7):858–873, 2015), because it is based on symmetry group considerations only, without having recourse to frames attached to the particles. Second quantization and anticommuting operators are not necessary.

Electromagnetic coupling of strongly non-local quantum mechanics

Although standard quantum mechanics has some non-local features, the probability current of the Schrödinger equation is locally conserved, and this allows minimal electromagnetic coupling. For some important extensions of the Schrödinger equation, however, the probability current is not locally conserved. We show that in these cases the correct electromagnetic coupling requires a relatively simple extension of Maxwell theory which has been known for some time and recently improved by covariant integration of a scalar degree of freedom. We discuss some general properties of the solutions and examine in particular the case of an oscillating dipolar source. Remarkable mathematical and physical differences emerge with respect to Maxwell theory, as a consequence of additional current terms present in the equations for ∇·E and ∇×B. Several possible applications are mentioned.

The orbital: a pivotal concept in the relationship between chemistry and physics? A comment to the work by Fortin and coauthors

The present work is a comment of a recent paper by Fortin and coauthors (Fortin et al. in Found Chem 19:43–59, 2017) in which the authors propose the introduction of Bohmian mechanics (BM) in the philosophy of chemistry and the use of standard quantum mechanics (SQM) as a mere instrument of prediction. This way would allow overcoming the obstacles found in linking molecular chemistry and quantum mechanics. Starting from some remarks on the orbital concept, we highlight and discuss some general issues that need to be taken into account when two scientific theories coexist within the same investigation field, i.e. SQM and BM.

Measurable signatures of quantum mechanics in a classical spacetime

We propose an optomechanics experiment that can search for signatures of a fundamentally classical theory of gravity and in particular of the many-body Schroedinger-Newton (SN) equation, which governs the evolution of a crystal under a self-gravitational field. The SN equation predicts that the dynamics of a macroscopic mechanical oscillator's center of mass wavefunction differ from the predictions of standard quantum mechanics. This difference is largest for low-frequency oscillators, and for materials, such as Tungsten or Osmium, with small quantum fluctuations of the constituent atoms around their lattice equilibrium sites. Light probes the motion of these oscillators and is eventually measured in order to extract valuable information on the pendulum's dynamics. Due to the non-linearity contained in the SN equation, we analyze the fluctuations of measurement results differently than standard quantum mechanics. We revisit how to model a thermal bath, and the wavefunction collapse postulate, resulting in two prescriptions for analyzing the quantum measurement of the light. We demonstrate that both predict features, in the outgoing light's phase fluctuations' spectrum, which are separate from classical thermal fluctuations and quantum shot noise, and which can be clearly resolved with state of the art technology.

Extensions of Born’s rule to non-linear quantum mechanics, some of which do not imply superluminal communication

Nonlinear modifications of quantum mechanics have a troubled history. They were initially studied for many promising reasons: resolving the measurement problem, formulating a theory of quantum mechanics and gravity, and understanding the limits of standard quantum mechanics. However, certain non-linear theories have been experimentally tested and failed. More significantly, it has been shown that, in general, deterministic non-linear theories can be used for superluminal communication. We highlight another serious issue: the distribution of measurement results predicted by non-linear quantum mechanics depends on the formulation of quantum mechanics. In other words, Born’s rule cannot be uniquely extended to non-linear quantum mechanics. We present these generalizations of Born’s rule, and then examine whether some exclude superluminal communication. We determine that a large class do not allow for superluminal communication, but many lack a consistent definition. Nonetheless, we find a single extension of Born’s rule with a sound operational definition, and that does not exhibit superluminal communication. The non-linear time-evolution leading to a certain measurement event is driven by the state conditioned on measurements that lie within the past light cone of that event.

Contracted Schrödinger equation in quantum phase-space.

The phase space formulation of quantum mechanics is equivalent to standard quantum mechanics where averages are calculated by way of phase space integration as in the case of classical statistical mechanics. We derive the quantum hierarchy equations, often called the contracted Schrödinger equation, in the phase space representation of quantum mechanics which involves quasi-distributions of position and momentum. We use the Wigner distribution for the phase space function and the Moyal phase space eigenvalue formulation to derive the hierarchy. We show that the hierarchy equations in the position, momentum, and position-momentum representations are very similar in structure. © 2017 Wiley Periodicals, Inc.

VIP-2 at LNGS: An experiment on the validity of the Pauli Exclusion Principle for electrons

We are experimentally investigating possible violations of standard quantum mechanics predictions in the Gran Sasso underground laboratory in Italy. We test with high precision the Pauli Exclusion Principle and the collapse of the wave function (collapse models). We present our method of searching for possible small violations of the Pauli Exclusion Principle (PEP) for electrons, through the search for “anomalous” X-ray transitions in copper atoms. These transitions are produced by “new” electrons (brought inside the copper bar by circulating current) which can have the possibility to undergo Pauli-forbidden transition to the 1s level already occupied by two electrons. We describe the VIP2 (VIolation of the Pauli Exclusion Principle) experimental data taking at the Gran Sasso underground laboratories. The goal of VIP2 is to test the PEP for electrons in agreement with the Messiah-Greenberg superselection rule with unprecedented accuracy, down to a limit in the probability that PEP is violated at the level of 10−31. We show preliminary experimental results and discuss implications of a possible violation.

Reply to “Comment on ‘Particle path through a nested Mach-Zehnder interferometer’ ”

The correctness of the consistent histories analysis of weakly interacting probes, related to the path of a particle, is maintained against the criticisms in the Comment, and against the alternative approach described there, which receives no support from standard (textbook) quantum mechanics.

Measurement and potential barrier evolution analysis of cold field emission in fracture fabricated Si nanogap

Cold field emission characteristics of a fracture fabricated Si nanogap (∼100 nm) were investigated with an ascending electric field (voltage) sweep. The nanogap was formed by controlled fracture of a free-standing silicon micro-beam along 〈111〉 direction by a microelectromechanical device, which results in flat, smooth, and conformal electrode pairs. This facilitates simultaneous large area emission, which gives rise to a significant current at low bias voltage, which usually remains indiscernible in nanogaps of this size. The measured emission current–voltage (I–V) characteristics clearly depict two distinct regimes: a linear (I ∝ V) regime at low bias voltage and a nonlinear [ln(I/V2) ∝ V−1] regime at high bias voltage, separated by a transition point. We propose that the linear regime is owed to direct tunneling of electrons, whereas the nonlinear regime is due to Fowler–Nordheim type emission. This proposition essentially implies that the tunneling potential barrier gradually evolved from a rectangular shape to a triangular shape with increasing field (V). This type of evolution is usually observed in molecular size gaps. We have attempted to correlate the I–V curves acquired through the experiments with the electric field induced barrier shape evolution by numerical calculations involving standard quantum mechanics. The observed linear regime at low bias voltage (<5 V) in a relatively large size gap (∼100 nm) is attributed to the fabrication method adopted in this study. The reported study and the fabricated device are significant for developing a futuristic thermotunneling refrigerator that will find a wide range of application in nanoelectronic devices.