In deformation quantization (a.k.a. the Wigner-Weyl-Moyal formulation of quantum mechanics), we consider a single quantum particle moving freely in one dimension, except for the presence of one infinite potential wall. Dias and Prata pointed out that, surprisingly, its stationary-state Wigner function does not obey the naive equation of motion, i.e. the naive stargenvalue (*-genvalue) equation. We review our recent work on this problem, that treats the infinite wall as the limit of a Liouville potential. Also included are some new results: (i) we show explicitly that the Wigner-Weyl transform of the usual density matrix is the physical solution, (ii) we prove that an effective-mass treatment of the problem is equivalent to the Liouville one, and (iii) we point out that self-adjointness of the operator Hamiltonian requires a boundary potential, but one different from that proposed by Dias and Prata.
We discuss a recent method proposed by Kryukov and Walton to address boundary-value problems in the context of deformation quantization. We compare their method with our own approach and establish a connection between the two formalisms.
We examine the deformation quantization of a single particle moving in one dimension (i) in the presence of an infinite potential wall, (ii) confined by an infinite square well, and (iii) bound by a delta function potential energy. In deformation quantization, considered as an autonomous formulation of quantum mechanics, the Wigner function of stationary states must be found by solving the so-called $\*$-genvalue (``stargenvalue'') equation for the Hamiltonian. For the cases considered here, this pseudo-differential equation is difficult to solve directly, without an ad hoc modification of the…
Normally, the half-harmonic oscillator is active when $x>0$ and absent when $x<0$. From a canonical quantization perspective, this leads to odd eigenfunctions being present while even eigenfunctions are absent. In that case, only the usual odd eigenfunctions will appear if the wall slides to negative infinity. However, if an affine quantization is used, sliding the wall away shows that all the odd and even eigenfunctions are encountered, exactly like any full-harmonic oscillator. We provide numerical support for this.
The nonlinear quantization of the domain wall (relativistic membrane of codimension 1) is considered. The membrane dust equation is considered as an analogue of the Hamilton-Jacobi equation, which allows us to construct its quantum analogue. The resulting equation has the form of a nonlinear Klein-Fock-Gordon equation. It can be interpreted as the mean field approximation for a quantum domain wall. Dispersion relations are obtained for small perturbations (in a linear approximation). The group speed of perturbations does not exceed the speed of light. For perturbations propagating along the do…
We give a modified Hamiltonian for a particle in a box with infinite potential walls that takes into account wall effects. The Hamiltonian is expressed in both the position and momentum representation. In the momentum representation the eigenvalue problem for energy is an integral equation.
A serious objection made by Wallstrom against quantum interpretations based flow variables, in particular Nelsonian stochastics, is their empirical inequivalence with quantum theory: They are unable to obtain a quantization condition for flows around zeros which is automatically fulfilled by the wave function. It is found that the quantization condition follows from a simple additional postulate: The Laplace operator of the density has to be finite and positive at zeros of the density. This postulate is a quite natural consequence of subquantum theories, as far as they conform to a simple prin…
In this work the evolution of a wavefunction in an infinite potential well with time dependent boundaries is investigated. Previous methods for wells with walls moving at a constant velocity are summarised. These methods are extended to wells with slowly accelerating walls. The location and time of the revivals in the well of the initial wavefunction are derived using Jacobi's elliptic theta function.
It is shown that adiabatic cycles excite a quantum particle, which is confined in a one-dimensional region and is initially in an eigenstate. During the cycle, an infinitely sharp wall is applied and varied its strength and position. After the completion of the cycle, the state of the particle arrives another eigenstate. It is also shown that we may vary the final adiabatic state by choosing the parameters of the cycle. With a combination of these adiabatic cycles, we can connect an arbitrary pair of eigenstates. Hence, these cycles may be regarded as basis of the adiabatic excitations. A deta…
The Schrödinger equation is solved for the case of a particle confined to a small region of a box with infinite walls. If walls of the well are moved, then, due to an effective quantum nonlocal interaction with the boundary, even though the particle is nowhere near the walls, it will be affected. It is shown that this force apart from a minus sign is equal to the expectation value of the gradient of the quantum potential for vanishing at the walls boundary condition. Variation of this force with time is studied. A selection of Bohmian trajectories of the confined particle is also computed.
A Gedanken experiment is described to explore a counter-intuitive property of quantum mechanics. A particle is placed in a one-dimensional infinite well. The barrier on one side of the well is suddenly removed and the chamber dramatically enlarged. At specific, periodically recurring, times the particle can be found with probability one at the opposite end of the enlarged chamber in an interval of the same size as the initial well. With the help of symmetry considerations these times are calculated and shown to be dependent on the mass of the particle and the size of the enlarged chamber. Para…
Employing a transformation to hyperbolic space, we derive in a simple way exact solutions for the Klein-Gordon equation in an infinite square-well potential with one boundary moving at constant velocity, for the massless as well as for the massive case.
Deformation quantization conventionally is described in terms of multidifferential operators. Jet manifold technique is well-known provide the adequate formulation of theory of differential operators. We extended this formulation to the multidifferential ones, and consider their infinite order jet prolongation. The infinite order jet manifold is endowed with the canonical flat connection that provides the covariant formula of a deformation star-product.
Phases of a rotating Bose-Einstein condensate trapped in a hard-wall potential
Université Sorbonne Paris Nord（巴黎索邦大学）；CNRS（法国国家科学研究中心）；Hellenic Mediterranean University（希腊地中海大学）；HMU Research Center, Institute of Emerging Technologies（HMU新兴技术研究所）
The one-dimensional scattering of a two body interacting system by an infinite wall is studied in a quantum-mechanical framework. This problem contains some of the dynamical features present in the collision of atomic, molecular and nuclear systems. The scattering problem is solved exactly, for the case of a harmonic interaction between the fragments. The exact result is used to assess the validity of two different approximations to the scattering process. The adiabatic approximation, which considers that the relative co-ordinate is frozen during the scattering process, is found to be inadequa…
We investigate numerically the quantum collision between a stable Helium nanodrop and an infinitely hard wall in one dimension. The scattering outcome is compared to the same event omitting the quantum pressure. Only the quantum process reflects the effect of diffraction of wave packets in space and time.
Exact solution of the Schrödinger equation is given for a particle inside a hard sphere whose wall is moving with a constant velocity. Numerical computations are presented for both contracting and expanding spheres. The propagator is constructed and compared with the propagator of a particle in an infinite square well with one wall in uniform motion.
It has been recently realized that, in the case of polynomial potentials, the exact WKB method can be reformulated in terms of a system of TBA equations. In this paper we study this method in various examples. We develop a graphical procedure due to Toledo, which provides a fast and simple way to study the wall-crossing behavior of the TBA equations. When complemented with exact quantization conditions, the TBA equations can be used to solve spectral problems exactly in Quantum Mechanics. We compute the quantum corrections to the all-order WKB periods in many examples, as well as the exact spe…
The main goal of this work is to study the Dirac oscillator as a quantum field using the canonical formalism of quantum field theory and to develop the canonical quantization procedure for this system in $(1+1)$ and $(3+1)$ dimensions. This is possible because the Dirac oscillator is characterized by the absence of the Klein paradox and by the completeness of its eigenfunctions. We show that the Dirac oscillator field can be seen as constituted by infinite degrees of freedom which are identified as decoupled quantum linear harmonic oscillators. We observe that while for the free Dirac field th…
Within the context of infinite-dimensional representations of the rotation group the Dirac monopole problem is studied in details. Irreducible infinite-dimensional representations, being realized in the indefinite metric Hilbert space, are given by linear unbounded operators in infinite-dimensional topological spaces, supplied with a weak topology and associated weak convergence. We argue that an arbitrary magnetic charge is allowed, and the Dirac quantization condition can be replaced by a generalized quantization rule yielding a new quantum number, the so-called topological spin, which is re…
We describe a quantum system consisting of a one-dimensional linear chain of n identical harmonic oscillators coupled by a nearest neighbor interaction. Two boundary conditions are taken into account: periodic boundary conditions (where the nth oscillator is coupled back to the first oscillator) and fixed wall boundary conditions (where the first oscillator and the $n$th oscillator are coupled to a fixed wall). The two systems are characterized by their Hamiltonian. For their quantization, we treat these systems as Wigner Quantum Systems (WQS), allowing more solutions than just the canonical q…
