scholarly journals The uncertainty relation for joint measurement of position and momentum

2004 ◽  
Vol 4 (6&7) ◽  
pp. 546-562
Author(s):  
R.F. Werner
Keyword(s):  
Phase Space ◽  
Uncertainty Relation ◽  
Position Measurement ◽  
Joint Measurement ◽  
Best Constant ◽  
Conjugate Variable ◽  
The Difference

We prove an uncertainty relation, which imposes a bound on any joint measurement of position and momentum. It is of the form (\Delta P)(\Delta Q)\geq C\hbar, where the `uncertainties' quantify the difference between the marginals of the joint measurement and the corresponding ideal observable. Applied to an approximate position measurement followed by a momentum measurement, the uncertainties become the precision \Delta Q of the position measurement, and the perturbation \Delta P of the conjugate variable introduced by such a measurement. We also determine the best constant C, which is attained for a unique phase space covariant measurement.

scholarly journals A phase-space beam position monitor for synchrotron radiation

2015 ◽  
Vol 22 (4) ◽  
pp. 946-955 ◽  
Author(s):  
Nazanin Samadi ◽  
Bassey Bassey ◽  
Mercedes Martinson ◽  
George Belev ◽  
Les Dallin ◽  
...  
Keyword(s):  
Phase Space ◽  
Synchrotron Radiation ◽  
Incident Angle ◽  
Operating Conditions ◽  
Photon Beam ◽  
Profile Measurement ◽  
Beam Position Monitor ◽  
Position Measurement ◽  
Beam Position ◽  
X Ray

The stability of the photon beam position on synchrotron beamlines is critical for most if not all synchrotron radiation experiments. The position of the beam at the experiment or optical element location is set by the position and angle of the electron beam source as it traverses the magnetic field of the bend-magnet or insertion device. Thus an ideal photon beam monitor would be able to simultaneously measure the photon beam's position and angle, and thus infer the electron beam's position in phase space. X-ray diffraction is commonly used to prepare monochromatic beams on X-ray beamlines usually in the form of a double-crystal monochromator. Diffraction couples the photon wavelength or energy to the incident angle on the lattice planes within the crystal. The beam from such a monochromator will contain a spread of energies due to the vertical divergence of the photon beam from the source. This range of energies can easily cover the absorption edge of a filter element such as iodine at 33.17 keV. A vertical profile measurement of the photon beam footprint with and without the filter can be used to determine the vertical centroid position and angle of the photon beam. In the measurements described here an imaging detector is used to measure these vertical profiles with an iodine filter that horizontally covers part of the monochromatic beam. The goal was to investigate the use of a combined monochromator, filter and detector as a phase-space beam position monitor. The system was tested for sensitivity to position and angle under a number of synchrotron operating conditions, such as normal operations and special operating modes where the photon beam is intentionally altered in position and angle at the source point. The results are comparable with other methods of beam position measurement and indicate that such a system is feasible in situations where part of the synchrotron beam can be used for the phase-space measurement.

Superoperator methods of calculating cross sections: III. Unified description of classical and quantal scattering

1980 ◽  
Vol 58 (8) ◽  
pp. 1171-1182 ◽  
Author(s):  
R. E. Turner ◽  
R. F. Snider
Keyword(s):  
Quantum Mechanics ◽  
Phase Space ◽  
Time Dependence ◽  
Stationary State ◽  
Cross Sections ◽  
Differential Cross ◽  
Density Operator ◽  
The Difference ◽  
Unified Description ◽  
Classical And Quantum Mechanics

It is shown how differential cross sections can be obtained from the time dependence of phase space packets. This procedure is valid both for classical and quantum mechanics. Two methods are described. In one the trajectory of the packet is emphasized, while in the second the packet is appropriately spread to infinite size. Both methods are applicable to either mechanics. It is shown how the quantal results agree with those of the stationary state approach as formulated in terms of the density operator. The description is also used to elucidate the difference between the scattered flux and the generalized flux that arises naturally in the superoperator formulation.

EFFECT OF THE GENERALIZED UNCERTAINTY RELATION ON THE BLACK HOLE ENTROPY

2002 ◽  
Vol 17 (33) ◽  
pp. 2209-2219
Author(s):  
XIANG LI
Keyword(s):  
Black Hole ◽  
Uncertainty Relation ◽  
Black Hole Entropy ◽  
Brick Wall ◽  
Wall Model ◽  
Dirac Fields ◽  
Brick Wall Model ◽  
Generalized Uncertainty Relation ◽  
Klein Gordon ◽  
The Difference

The quantum entropies of the black hole, due to the massless Klein–Gordon and Dirac fields, are investigated by Rindler approximation. The difference from the brick wall model is that we take into account the effect of the generalized uncertainty relation on the state counting. The divergence appearing in the brick wall model is removed and the entropies proportional to the horizon area come from the contributions of the modes in the vicinity of the horizon. Here we take the units G=c=ℏ=kB=1.

Uncertainty Relation: From Inequality to Equality

1997 ◽  
Vol 52 (1-2) ◽  
pp. 49-52 ◽  
Author(s):  
Georg Süssmann
Keyword(s):  
Phase Space ◽  
Quantum State ◽  
Uncertainty Relation ◽  
The Other ◽  
Mixed States ◽  
Von Neumann ◽  
Other Hand ◽  
Pure Quantum State ◽  
Minimum Uncertainty

Abstract The uncertainty area δ (p, q): - [∫ W(p, q)2 dp dq] - 1 is proposed in place of δ p • δ q, and it is shown that each pure quantum state is a minimum uncertainty state in this sense: δ (p, q) = 2 π ħ. For mixed states, on the other hand, δ(p, q) > 2π ħ. In a phase space of 2F(=6N) dimensions, S: = k B • log[δF (p,q)/(2 π ħ)F] whit δF (p,q):= [∫ W(p, q)2 dF p dF q]-1 is considered as an alternative to von Neumann`s entropy S̃:= kB • trc [ρ̂ log (ρ̂-1)].

scholarly journals Nonclassical Atomic System Dynamics Time-dependently Interacts With Finite Entangled Pair Coherent Parametric Converter Cavity Fields

2021 ◽  
Author(s):  
A.B. mohamed ◽  
E. M. Khalil ◽  
M. Y. Abd-Rabbou
Keyword(s):  
Phase Space ◽  
Time Dependent ◽  
Physical Parameters ◽  
Atomic Coherence ◽  
Wehrl Entropy ◽  
Location Parameters ◽  
Parametric Converter ◽  
Dependent Model ◽  
The Difference ◽  
Quantum Phenomena

Abstract We consider a time-dependent model that describes a qubit time-dependently interacts with a cavity containing finite entangled pair coherent parametric converter fields. The dynamics of some quantum phenomena, as: phase space information, quantum entanglement and squeezing, are explored by atomic Husimi function, atomic Wehrl entropy, variance, and entropy squeezing. The influences of the unitary qubit-cavity interaction, the difference between the two-mode photon numbers, the initial atomic coherence, and the time-dependent qubit location are investigated. It is found that the regularity, the amplitudes and the frequency of the quantum phenomena can be controlled by the physical parameters. For the initial atomic pure state, the qubit-cavity entanglement, the qubit phase space information, and atomic squeezing can be generated strongly compared to those of the initial atomic mixed state. The time-dependent location parameters enhance the generated quantum phenomena, and its effect can be enhanced by the parameters of the two-mode photon numbers and the initial atomic coherence.

scholarly journals Simulation of the long-term behaviour of a fault with two asperities

2010 ◽  
Vol 17 (6) ◽  
pp. 777-784 ◽  
Author(s):  
M. Dragoni ◽  
S. Santini
Keyword(s):  
Phase Space ◽  
Stress Distribution ◽  
Analytical Solution ◽  
Infinite Number ◽  
Fault System ◽  
Seismic Events ◽  
Recurrence Pattern ◽  
Stress Distributions ◽  
The Difference

Abstract. A system made of two sliding blocks coupled by a spring is employed to simulate the long-term behaviour of a fault with two asperities. An analytical solution is given for the motion of the system in the case of blocks having the same friction. An analysis of the phase space shows that orbits can reach a limit cycle only after entering a particular subset of the space. There is an infinite number of different limit cycles, characterized by the difference between the forces applied to the blocks or, as an alternative, by the recurrence pattern of block motions. These results suggest that the recurrence pattern of seismic events produced by the equivalent fault system is associated with a particular stress distribution which repeats periodically. Admissible stress distributions require a certain degree of inhomogeneity, which depends on the geometry of fault system. Aperiodicity may derive from stress transfers from neighboring faults.

Sign in / Sign up

Export Citation Format

Share Document