A state of a single particle can be represented by a quantum blob in the corresponding phase space, or by a cell in its 2-D subspace. Its area is frequently stated to be no less than one half of the Plank constant, implying that such a cell is an indivisible quantum of the 2-D phase space. But this is generally not true, as is evident, for instance, from representation of some states in the basis of innately discrete observables like angular momentum. Here we consider some dispersed states involving the evanescent waves (EW) different from that in the total internal reflection. Such states are represented by a set of separated point-like cells, but with a large total indeterminacy. An idealized model has a discrete Wigner function forming an infinite periodic array of dots on the phase plane. The question about the total momentum indeterminacy in such state is discussed. We argue that the transverse momentum eigenstates corresponding to the considered EW-s cannot be singled out by any known measurement procedure, and the whole infinite set of the corresponding eigenvalues can contribute only a certain fraction to the observed momentum indeterminacy which remains finite.
A state of a single particle can be represented by a quantum blob in the corresponding phase space, or by a cell in its 2-D subspace. Its area is frequently stated to be no less than one half of the Plank constant, implying that such a cell is an indivisible quantum of the 2-D phase space. But this is generally not true, as is evident, for instance, from representation of some states in the basis of innately discrete observables like angular momentum. Here we consider some dispersed states involving the evanescent waves (EW) different from that in the total internal reflection. Such states are represented by a set of separated point-like cells, but with a large total indeterminacy. An idealized model has a discrete Wigner function forming an infinite periodic array of dots on the phase plane. The question about the total momentum indeterminacy in such state is discussed. We argue that the transverse momentum eigenstates corresponding to the considered EW-s cannot be singled out by any known measurement procedure, and the whole infinite set of the corresponding eigenvalues can contribute only a certain fraction to the observed momentum indeterminacy which remains finite.
Real-space interference of wave packets is governed by their central momentum separation and momentum width, which set the oscillation wavelength and spatial envelope, respectively. Here, we show that in crystalline systems, quantum metric distribution in an isolated band can similarly give rise to real-space interference through the evanescent tails of a localized bound state. The location and width of a quantum metric hot spot in the momentum space set a complex wavevector as one evanescent plane wave component of a bound state. Consequently, with multiple quantum metric hot spots, the density distribution of a bound state or the total density of two spatially overlapped bound states exhibits an interference pattern, where the oscillation period and long-range decay length are determined by the momentum-space separation of the hot spots and the width of the narrowest hot spots, respectively. This quantum-geometric interference is particularly pronounced in (nearly) flat bands, where competing scales from band dispersion are suppressed. Furthermore, we demonstrate that using a Josephson STM the real-space interference can be experimentally observed through the tip-position dependence of the Josephson critical current. Our results reveal a direct correspondence between momentum-space quantum geometry and real-space interference, showing how bound-state profiles encode the momentum-resolved quantum metric.
This paper deals with certain aspects of the phase-space formulation of quantum mechanics. Noting the ambiguities involved in quantizing the classical phase-space dynamics, a generalized quantum-mechanical phase space is introduced in terms of eigenvalues of two mutually incompatible complete sets of operators. General definitions are given for the phase-space distribution function corresponding to a given abstract state of the system and the phase-space function for an arbitrary operator. Our formulation goes over to the Wigner formulation when the complete sets correspond to position and momentum operators in Cartesian coordinates. Explicit examples are discussed with use of action-angle and polar coordinates for simple one-, two-, and three-dimensional systems. Symmetry changes in phase space are discussed by applying Gel'fand-Levitan transformations and it is shown that, in general, the degeneracies of motion in phase space are not reflected in the energy eigenvalues.
The discrete phase space representation of quantum mechanics involving a characteristic length is investigated. The continuous (1 + 1)-dimensional phase space is first discussed for the sake of simplicity. It is discretized into denumerable infinite number of concentric circles. These circles, endowed with “unit area”, are degenerate phase cells resembling closed strings.Next, Schrödinger wave equation for one particle in the three dimensional space under the influence of a static potential is studied in the discrete phase space representation of wave mechanics. The Schrödinger equation in the arena of discrete phase space is a partial difference equation. The energy eigenvalue problem for a three dimensional oscillator is exactly solved.Next, relativistic wave equations in the scenario of three dimensional discrete phase space and continuous time are explored. Specially, the partial finite difference-differential equation for a scalar field is investigated for the sake of simplicity. The exact relativistic invariance of the partial finite difference-differential version of the Klein-Gordon equation is rigorously proved. Moreover, it is shown that all nine important Green’s functions of the partial finite difference-differential wave equation for the scalar field are non-singular.In the appendix, a possible physical interpretation for the discrete orbits in the phase space as degenerate, string-like phase cells is provided in a mathematically rigorous way.
There has been recent interest in formulating theories of non-representational indeterminacy. The aim of this paper is to clarify the relevance of quantum mechanics to this project. Quantum-mechanical examples of vague objects have been offered by various authors, displaying indeterminate identity, in the face of the famous Evans argument that such an idea is incoherent. It has also been suggested that the quantum-mechanical treatment of state-dependent properties exhibits metaphysical indeterminacy. In both cases it is important to consider the details of the metaphysical account and the way in which the quantum phenomenon is captured within it. Indeed if we adopt a familiar way of thinking about indeterminacy and apply it in a natural way to quantum mechanics, we run into illuminating difficulties and see that the case is far less straightforward than might be hoped.
The evanescent waves named as EW1, EW2, EW3 are described in 3 respective experimental setups: 1) total internal reflection; 2) scattering on an inhomogeneous planar target; and 3) propagation along a waveguide. Some interactions are considered between EW2 and the environment. The latter may include a beam of probing particles and/or the screen on which the EW2 are formed. Some new properties of EW are described, such as complex energy eigenvalues in case of a movable screen, and evanescence exchange between the interacting objects. This reveals the connection between evanescent states and the Gamow states of the studied system. The 4-momentum exchange between EW2 and the probe is highly selective and may collapse the superposition of studied EW2- eigenstates to a single EW-eigenstate of the probing particle. Possible imprints of EW2 in the far field are briefly discussed and a simple experiment is suggested for their observation. .
We study the quantum mechanical harmonic oscillator in two and three dimensions, with particular attention to the solutions as basis states for representing their respective symmetry groups — O(2), O(1,1), O(3), and O(2,1). The goal of this study is to establish a correspondence between Hilbert space descriptions found by solving the Schrodinger equation in polar coordinates, and Fock space descriptions constructed by expressing the symmetry operators in terms of creation/annihilation operators. We obtain wavefunctions characterized by a principal quantum number, the group Casimir eigenvalue, and one group generator whose eigenvalue is m + s , for integer m and real constant parameter s . For the three groups that contain O(2), the solutions split into two inequivalent representations, one associated with s = 0, from which we recover the familiar description of the oscillator as a product of one-dimensional solutions, and the other with s > 0 (in three dimensions, solutions are found for s = 0 and s = 1/2) whose solutions are non-separable in Cartesian coordinates, and are hence overlooked by the standard Fock space approach. The O(1,1) solutions are singlet states, restricted to zero eigenvalue of the symmetry operator, which represents the boost, not angular momentum. For O(2), a single set of creation and annihilation operators forms a ladder representation for the allowed oscillator states for any s, and the degeneracy of energy states is always finite. However, in three dimensions, the integer and half-integer eigenstates are qualitatively different: the former can be expressed as finite dimensional irreducible tensors under O(3) or O(2,1) while the latter exhibit infinite degeneracy. Creation operators that produce the allowed integer states by acting on the non-degenerate ground state are constructed as irreducible tensor products of the fundamental vector representation. However, the half-integer eigenstates are infinite-dimensional, as expected for the non-compact O(2,1), and no Fock space description is found for this case. For all s ≠ 0 solutions, the SU(N) symmetry of the harmonic oscillator Hamiltonian recently discovered by Bars is spontaneously broken by the ground state. The connection of this symmetry breaking to the non-separability into one-dimensional Cartesian solutions is demonstrated. We conclude with a discussion of the O(2,1) solutions, and the problem of timelike ghost excitations.