6,913 research outputs found
Unusual quantum states: nonlocality, entropy, Maxwell's daemon, and fractals
This paper analyses the mathematical properties of some unusual quantum
states that are constructed by inserting an impenetrable barrier into a chamber
confining a single particle.Comment: 9 Figure
Geometry of PT-symmetric quantum mechanics
Recently, much research has been carried out on Hamiltonians that are not
Hermitian but are symmetric under space-time reflection, that is, Hamiltonians
that exhibit PT symmetry. Investigations of the Sturm-Liouville eigenvalue
problem associated with such Hamiltonians have shown that in many cases the
entire energy spectrum is real and positive and that the eigenfunctions form an
orthogonal and complete basis. Furthermore, the quantum theories determined by
such Hamiltonians have been shown to be consistent in the sense that the
probabilities are positive and the dynamical trajectories are unitary. However,
the geometrical structures that underlie quantum theories formulated in terms
of such Hamiltonians have hitherto not been fully understood. This paper
studies in detail the geometric properties of a Hilbert space endowed with a
parity structure and analyses the characteristics of a PT-symmetric Hamiltonian
and its eigenstates. A canonical relationship between a PT-symmetric operator
and a Hermitian operator is established. It is shown that the quadratic form
corresponding to the parity operator, in particular, gives rise to a natural
partition of the Hilbert space into two halves corresponding to states having
positive and negative PT norm. The indefiniteness of the norm can be
circumvented by introducing a symmetry operator C that defines a positive
definite inner product by means of a CPT conjugation operation.Comment: 22 Page
Faster than Hermitian Quantum Mechanics
Given an initial quantum state |psi_I> and a final quantum state |psi_F> in a
Hilbert space, there exist Hamiltonians H under which |psi_I> evolves into
|psi_F>. Consider the following quantum brachistochrone problem: Subject to the
constraint that the difference between the largest and smallest eigenvalues of
H is held fixed, which H achieves this transformation in the least time tau?
For Hermitian Hamiltonians tau has a nonzero lower bound. However, among
non-Hermitian PT-symmetric Hamiltonians satisfying the same energy constraint,
tau can be made arbitrarily small without violating the time-energy uncertainty
principle. This is because for such Hamiltonians the path from |psi_I> to
|psi_F> can be made short. The mechanism described here is similar to that in
general relativity in which the distance between two space-time points can be
made small if they are connected by a wormhole. This result may have
applications in quantum computing.Comment: 4 page
Robust non-adiabatic molecular dynamics for metals and insulators
We present a new formulation of the correlated electron-ion dynamics (CEID)
scheme, which systematically improves Ehrenfest dynamics by including quantum
fluctuations around the mean-field atomic trajectories. We show that the method
can simulate models of non-adiabatic electronic transitions, and test it
against exact integration of the time-dependent Schroedinger equation. Unlike
previous formulations of CEID, the accuracy of this scheme depends on a single
tunable parameter which sets the level of atomic fluctuations included. The
convergence to the exact dynamics by increasing the tunable parameter is
demonstrated for a model two level system. This algorithm provides a smooth
description of the non-adiabatic electronic transitions which satisfies the
kinematic constraints (energy and momentum conservation) and preserves quantum
coherence. The applicability of this algorithm to more complex atomic systems
is discussed.Comment: 36 pages, 5 figures. Accepted for publication in Journal of Chemical
Physic
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