8,130 research outputs found
Entropy and Temperature of a Quantum Carnot Engine
It is possible to extract work from a quantum-mechanical system whose
dynamics is governed by a time-dependent cyclic Hamiltonian. An energy bath is
required to operate such a quantum engine in place of the heat bath used to run
a conventional classical thermodynamic heat engine. The effect of the energy
bath is to maintain the expectation value of the system Hamiltonian during an
isoenergetic expansion. It is shown that the existence of such a bath leads to
equilibrium quantum states that maximise the von Neumann entropy. Quantum
analogues of certain thermodynamic relations are obtained that allow one to
define the temperature of the energy bath.Comment: 4 pages, 1 figur
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
Automatic Generation of Matrix Element Derivatives for Tight Binding Models
Tight binding (TB) models are one approach to the quantum mechanical many
particle problem. An important role in TB models is played by hopping and
overlap matrix elements between the orbitals on two atoms, which of course
depend on the relative positions of the atoms involved. This dependence can be
expressed with the help of Slater-Koster parameters, which are usually taken
from tables. Recently, a way to generate these tables automatically was
published. If TB approaches are applied to simulations of the dynamics of a
system, also derivatives of matrix elements can appear. In this work we give
general expressions for first and second derivatives of such matrix elements.
Implemented in a computer program they obviate the need to type all the
required derivatives of all occuring matrix elements by hand.Comment: 11 pages, 2 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
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