5,150 research outputs found
Thermalisation of Quantum States
An exact stochastic model for the thermalisation of quantum states is
proposed. The model has various physically appealing properties. The dynamics
are characterised by an underlying Schrodinger evolution, together with a
nonlinear term driving the system towards an asymptotic equilibrium state and a
stochastic term reflecting fluctuations. There are two free parameters, one of
which can be identified with the heat bath temperature, while the other
determines the characteristic time scale for thermalisation. Exact expressions
are derived for the evolutionary dynamics of the system energy, the system
entropy, and the associated density operator.Comment: 8 pages, minor corrections. To appear in JM
Information Content for Quantum States
A method of representing probabilistic aspects of quantum systems is
introduced by means of a density function on the space of pure quantum states.
In particular, a maximum entropy argument allows us to obtain a natural density
function that only reflects the information provided by the density matrix.
This result is applied to derive the Shannon entropy of a quantum state. The
information theoretic quantum entropy thereby obtained is shown to have the
desired concavity property, and to differ from the the conventional von Neumann
entropy. This is illustrated explicitly for a two-state system.Comment: RevTex file, 4 pages, 1 fi
Information of Interest
A pricing formula for discount bonds, based on the consideration of the
market perception of future liquidity risk, is established. An
information-based model for liquidity is then introduced, which is used to
obtain an expression for the bond price. Analysis of the bond price dynamics
shows that the bond volatility is determined by prices of certain weighted
perpetual annuities. Pricing formulae for interest rate derivatives are
derived.Comment: 12 pages, 3 figure
The Quantum Canonical Ensemble
The phase space of quantum mechanics can be viewed as the complex projective
space endowed with a Kaehlerian structure given by the Fubini-Study metric and
an associated symplectic form. We can then interpret the Schrodinger equation
as generating a Hamiltonian dynamics. Based upon the geometric structure of the
quantum phase space we introduce the corresponding natural microcanonical and
canonical ensembles. The resulting density matrix for the canonical ensemble
differs from density matrix of the conventional approach. As an illustration,
the results are applied to the case of a spin one-half particle in a heat bath
with an applied magnetic field.Comment: 8 pages, minor corrections. to appear in JMP vol. 3
Note on exponential families of distributions
We show that an arbitrary probability distribution can be represented in
exponential form. In physical contexts, this implies that the equilibrium
distribution of any classical or quantum dynamical system is expressible in
grand canonical form.Comment: 5 page
Efficient Simulation of Quantum State Reduction
The energy-based stochastic extension of the Schrodinger equation is a rather
special nonlinear stochastic differential equation on Hilbert space, involving
a single free parameter, that has been shown to be very useful for modelling
the phenomenon of quantum state reduction. Here we construct a general closed
form solution to this equation, for any given initial condition, in terms of a
random variable representing the terminal value of the energy and an
independent Brownian motion. The solution is essentially algebraic in
character, involving no integration, and is thus suitable as a basis for
efficient simulation studies of state reduction in complex systems.Comment: 4 pages, No Figur
On optimum Hamiltonians for state transformations
For a prescribed pair of quantum states |psi_I> and |psi_F> we establish an
elementary derivation of the optimum Hamiltonian, under constraints on its
eigenvalues, that generates the unitary transformation |psi_I> --> |psi_F> in
the shortest duration. The derivation is geometric in character and does not
rely on variational calculus.Comment: 5 page
Random Hamiltonian in thermal equilibrium
A framework for the investigation of disordered quantum systems in thermal
equilibrium is proposed. The approach is based on a dynamical model--which
consists of a combination of a double-bracket gradient flow and a uniform
Brownian fluctuation--that `equilibrates' the Hamiltonian into a canonical
distribution. The resulting equilibrium state is used to calculate quenched and
annealed averages of quantum observables.Comment: 8 pages, 4 figures. To appear in DICE 2008 conference proceeding
Metric approach to quantum constraints
A new framework for deriving equations of motion for constrained quantum
systems is introduced, and a procedure for its implementation is outlined. In
special cases the framework reduces to a quantum analogue of the Dirac theory
of constrains in classical mechanics. Explicit examples involving spin-1/2
particles are worked out in detail: in one example our approach coincides with
a quantum version of the Dirac formalism, while the other example illustrates
how a situation that cannot be treated by Dirac's approach can nevertheless be
dealt with in the present scheme.Comment: 13 pages, 1 figur
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