3,090 research outputs found
About maximally localized states in quantum mechanics
We analyze the emergence of a minimal length for a large class of generalized
commutation relations, preserving commutation of the position operators and
translation invariance as well as rotation invariance (in dimension higher than
one). We show that the construction of the maximally localized states based on
squeezed states generally fails. Rather, one must resort to a constrained
variational principle.Comment: accepted for publication in PR
Spacetime could be simultaneously continuous and discrete in the same way that information can
There are competing schools of thought about the question of whether
spacetime is fundamentally either continuous or discrete. Here, we consider the
possibility that spacetime could be simultaneously continuous and discrete, in
the same mathematical way that information can be simultaneously continuous and
discrete. The equivalence of continuous and discrete information, which is of
key importance in information theory, is established by Shannon sampling
theory: of any bandlimited signal it suffices to record discrete samples to be
able to perfectly reconstruct it everywhere, if the samples are taken at a rate
of at least twice the bandlimit. It is known that physical fields on generic
curved spaces obey a sampling theorem if they possess an ultraviolet cutoff.
Most recently, methods of spectral geometry have been employed to show that
also the very shape of a curved space (i.e., of a Riemannian manifold) can be
discretely sampled and then reconstructed up to the cutoff scale. Here, we
develop these results further, and we here also consider the generalization to
curved spacetimes, i.e., to Lorentzian manifolds
On the modification of Hamiltonians' spectrum in gravitational quantum mechanics
Different candidates of Quantum Gravity such as String Theory, Doubly Special
Relativity, Loop Quantum Gravity and black hole physics all predict the
existence of a minimum observable length or a maximum observable momentum which
modifies the Heisenberg uncertainty principle. This modified version is usually
called the Generalized (Gravitational) Uncertainty Principle (GUP) and changes
all Hamiltonians in quantum mechanics. In this Letter, we use a recently
proposed GUP which is consistent with String Theory, Doubly Special Relativity
and black hole physics and predicts both a minimum measurable length and a
maximum measurable momentum. This form of GUP results in two additional terms
in any quantum mechanical Hamiltonian, proportional to and
, respectively, where is the GUP
parameter. By considering both terms as perturbations, we study two quantum
mechanical systems in the framework of the proposed GUP: a particle in a box
and a simple harmonic oscillator. We demonstrate that, for the general
polynomial potentials, the corrections to the highly excited eigenenergies are
proportional to their square values. We show that this result is exact for the
case of a particle in a box.Comment: 11 pages, to appear in Europhysics Letter
Hydrogen atom as an eigenvalue problem in 3D spaces of constant curvature and minimal length
An old result of A.F. Stevenson [Phys. Rev.} 59, 842 (1941)] concerning the
Kepler-Coulomb quantum problem on the three-dimensional (3D) hypersphere is
considered from the perspective of the radial Schr\"odinger equations on 3D
spaces of any (either positive, zero or negative) constant curvature. Further
to Stevenson, we show in detail how to get the hypergeometric wavefunction for
the hydrogen atom case. Finally, we make a comparison between the ``space
curvature" effects and minimal length effects for the hydrogen spectrumComment: 6 pages, v
Perturbation spectrum in inflation with cutoff
It has been pointed out that the perturbation spectrum predicted by inflation
may be sensitive to a natural ultraviolet cutoff, thus potentially providing an
experimentally accessible window to aspects of Planck scale physics. A priori,
a natural ultraviolet cutoff could take any form, but a fairly general
classification of possible Planck scale cutoffs has been given. One of those
categorized cutoffs, also appearing in various studies of quantum gravity and
string theory, has recently been implemented into the standard inflationary
scenario. Here, we continue this approach by investigating its effects on the
predicted perturbation spectrum. We find that the size of the effect depends
sensitively on the scale separation between cutoff and horizon during
inflation.Comment: 6 pages; matches version accepted by PR
Hole spin dynamics and hole factor anisotropy in coupled quantum well systems
Due to its p-like character, the valence band in GaAs-based heterostructures
offers rich and complex spin-dependent phenomena. One manifestation is the
large anisotropy of Zeeman spin splitting. Using undoped, coupled quantum wells
(QWs), we examine this anisotropy by comparing the hole spin dynamics for high-
and low-symmetry crystallographic orientations of the QWs. We directly measure
the hole factor via time-resolved Kerr rotation, and for the low-symmetry
crystallographic orientations (110) and (113a), we observe a large in-plane
anisotropy of the hole factor, in good agreement with our theoretical
calculations. Using resonant spin amplification, we also observe an anisotropy
of the hole spin dephasing in the (110)-grown structure, indicating that
crystal symmetry may be used to control hole spin dynamics
Lorentz-covariant deformed algebra with minimal length
The -dimensional two-parameter deformed algebra with minimal length
introduced by Kempf is generalized to a Lorentz-covariant algebra describing a
()-dimensional quantized space-time. For D=3, it includes Snyder algebra
as a special case. The deformed Poincar\'e transformations leaving the algebra
invariant are identified. Uncertainty relations are studied. In the case of D=1
and one nonvanishing parameter, the bound-state energy spectrum and
wavefunctions of the Dirac oscillator are exactly obtained.Comment: 8 pages, no figure, presented at XV International Colloquium on
Integrable Systems and Quantum Symmetries (ISQS-15), Prague, June 15-17, 200
Discreteness of Space from the Generalized Uncertainty Principle
Various approaches to Quantum Gravity (such as String Theory and Doubly
Special Relativity), as well as black hole physics predict a minimum measurable
length, or a maximum observable momentum, and related modifications of the
Heisenberg Uncertainty Principle to a so-called Generalized Uncertainty
Principle (GUP). We propose a GUP consistent with String Theory, Doubly Special
Relativity and black hole physics, and show that this modifies all quantum
mechanical Hamiltonians. When applied to an elementary particle, it implies
that the space which confines it must be quantized. This suggests that space
itself is discrete, and that all measurable lengths are quantized in units of a
fundamental length (which can be the Planck length). On the one hand, this
signals the breakdown of the spacetime continuum picture near that scale, and
on the other hand, it can predict an upper bound on the quantum gravity
parameter in the GUP, from current observations. Furthermore, such fundamental
discreteness of space may have observable consequences at length scales much
larger than the Planck scale.Comment: 3 pages, revtex4, no figures, to appear in Phys. Lett.
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