7,861 research outputs found
Wigner quasi-probability distribution for the infinite square well: energy eigenstates and time-dependent wave packets
We calculate the Wigner quasi-probability distribution for position and
momentum, P_W^(n)(x,p), for the energy eigenstates of the standard infinite
well potential, using both x- and p-space stationary-state solutions, as well
as visualizing the results. We then evaluate the time-dependent Wigner
distribution, P_W(x,p;t), for Gaussian wave packet solutions of this system,
illustrating both the short-term semi-classical time dependence, as well as
longer-term revival and fractional revival behavior and the structure during
the collapsed state. This tool provides an excellent way of demonstrating the
patterns of highly correlated Schrodinger-cat-like `mini-packets' which appear
at fractional multiples of the exact revival time.Comment: 45 pages, 16 embedded, low-resolution .eps figures (higher
resolution, publication quality figures are available from the authors);
submitted to American Journal of Physic
On the Method of Interconnection and Damping Assignment Passivity-Based Control for the Stabilization of Mechanical Systems
Interconnection and damping assignment passivity-based control (IDA-PBC) is
an excellent method to stabilize mechanical systems in the Hamiltonian
formalism. In this paper, several improvements are made on the IDA-PBC method.
The skew-symmetric interconnection submatrix in the conventional form of
IDA-PBC is shown to have some redundancy for systems with the number of degrees
of freedom greater than two, containing unnecessary components that do not
contribute to the dynamics. To completely remove this redundancy, the use of
quadratic gyroscopic forces is proposed in place of the skew-symmetric
interconnection submatrix. Reduction of the number of matching partial
differential equations in IDA-PBC and simplification of the structure of the
matching partial differential equations are achieved by eliminating the
gyroscopic force from the matching partial differential equations. In addition,
easily verifiable criteria are provided for Lyapunov/exponential
stabilizability by IDA-PBC for all linear controlled Hamiltonian systems with
arbitrary degrees of underactuation and for all nonlinear controlled
Hamiltonian systems with one degree of underactuation. A general design
procedure for IDA-PBC is given and illustrated with examples. The duality of
the new IDA-PBC method to the method of controlled Lagrangians is discussed.
This paper renders the IDA-PBC method as powerful as the controlled Lagrangian
method
The Quantum Mellin transform
We uncover a new type of unitary operation for quantum mechanics on the
half-line which yields a transformation to ``Hyperbolic phase space''. We show
that this new unitary change of basis from the position x on the half line to
the Hyperbolic momentum , transforms the wavefunction via a Mellin
transform on to the critial line . We utilise this new transform
to find quantum wavefunctions whose Hyperbolic momentum representation
approximate a class of higher transcendental functions, and in particular,
approximate the Riemann Zeta function. We finally give possible physical
realisations to perform an indirect measurement of the Hyperbolic momentum of a
quantum system on the half-line.Comment: 23 pages, 6 Figure
Coherent States for Black Holes
We determine coherent states peaked at classical space-time of the
Schwarzschild black hole in the frame-work of canonical quantisation of general
relativity. The information about the horizon is naturally encoded in the phase
space variables, and the perturbative quantum fluctuations around the classical
geometry depend on the distance from the horizon. For small black holes, space
near the vicinity of the singularity appears discrete with the singular point
excluded from the spectrum.Comment: 48 pages, 18+1 figures, some modifications, references adde
Semiclassical theory for small displacements
Characteristic functions contain complete information about all the moments
of a classical distribution and the same holds for the Fourier transform of the
Wigner function: a quantum characteristic function, or the chord function.
However, knowledge of a finite number of moments does not allow for accurate
determination of the chord function. For pure states this provides the overlap
of the state with all its possible rigid translations (or displacements). We
here present a semiclassical approximation of the chord function for large
Bohr-quantized states, which is accurate right up to a caustic, beyond which
the chord function becomes evanescent. It is verified to pick out blind spots,
which are displacements for zero overlaps. These occur even for translations
within a Planck area of the origin. We derive a simple approximation for the
closest blind spots, depending on the Schroedinger covariance matrix, which is
verified for Bohr-quantized states.Comment: 16 pages, 4 figures
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