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Formulation of quantum mechanics
The phase-space formulation is a formulation of quantum mechanics that places the position and momentum variables on equal footing in phase space. The
Phase-space_formulation
Mathematical structures that allow quantum mechanics to be explained
functions on phase space, but as eigenvalues; more precisely as spectral values of linear operators in Hilbert space. These formulations of quantum mechanics
Mathematical formulation of quantum mechanics
Mathematical_formulation_of_quantum_mechanics
Space of all possible states that a system can take
The phase space of a physical system is the set of all possible physical states of the system when described by a given parameterization. Each possible
Phase_space
Mapping between functions in the quantum phase space
the quantum phase space formulation and Hilbert space operators in the Schrödinger picture. Often the mapping from functions on phase space to operators
Wigner–Weyl_transform
Topics referred to by the same term
referred to as a "microscopic state" Phase space formulation, a formulation of quantum mechanics in phase space Phase (waves), the position of a point in
Phase
Phase-space representation of quantum state vectors is a formulation of quantum mechanics elaborating the phase-space formulation with a Hilbert space
Phase-space_wavefunctions
Key result in Hamiltonian mechanics and statistical mechanics
time-dependent. In the phase-space formulation of quantum mechanics, substituting the Moyal brackets for Poisson brackets in the phase-space analog of the von
Liouville's theorem (Hamiltonian)
Liouville's_theorem_(Hamiltonian)
Computational physics simulation tool
quantum mechanics to represent the phase space distribution of a quantum state such as light in the phase space formulation. It is used in the field of quantum
Husimi_Q_representation
this phase-space formulation. There results a complete phase space formulation of quantum mechanics, completely equivalent to the Hilbert-space operator
Deformation_quantization
Mathematical approach to quantum optics
a suggested way of writing down the phase space distribution of a quantum system in the phase space formulation of quantum mechanics. The P representation
Glauber–Sudarshan P representation
Glauber–Sudarshan_P_representation
Quantum mechanical phenomenon
system, where bounded classical trajectories are confined onto tori in phase space, tunnelling can be understood as the quantum transport between semi-classical
Quantum_tunnelling
Mathematical description of quantum state
Fermion Phase-space formulation Schrödinger equation Wave function collapse Wave packet The functions are here assumed to be elements of L2, the space of square
Wave_function
Description of physical properties at the atomic and subatomic scale
mechanics and quantum mechanics Macroscopic quantum phenomena Phase-space formulation Regularization (physics) Two-state quantum system A momentum eigenstate
Quantum_mechanics
Property of some binary operations
satisfied by operator commutators on a Hilbert space and equivalently in the phase space formulation of quantum mechanics by the Moyal bracket. Let +
Jacobi_identity
Wigner distribution function in physics as opposed to in signal processing
quantum expectation values, and hence quantum mechanics, in phase space (see Phase-space formulation). In most of his correspondence with Moyal in the mid-1940s
Wigner quasiprobability distribution
Wigner_quasiprobability_distribution
Specific quantum state of a quantum harmonic oscillator
location in the complex plane (phase space) is centered at the position and momentum of a classical oscillator of the phase θ and amplitude |α| given by
Coherent_state
Formulation of quantum mechanics
Interaction picture Schrödinger picture Heisenberg–Langevin equations Phase space formulation "Heisenberg representation". Encyclopedia of Mathematics. Retrieved
Heisenberg_picture
Systematic procedure of turning a classical theory into a quantum one
useful and important, as it underlies the alternate equivalent phase space formulation of conventional quantum mechanics. In mathematical physics, geometric
Quantization_(physics)
Suitably normalized antisymmetrization of the phase-space star product
observables in the phase space formulation of quantum mechanics when these observables are described as functions on phase space. It relies on schemes
Moyal_bracket
Dutch theoretical physicist (1910–1996)
who pioneered the largely operator-free formulation of quantum mechanics in phase space known as phase-space quantization. Groenewold was born on 29 June
Hilbrand_J._Groenewold
Process in quantum mechanical theories
symplectic space such that the leading term in the Taylor expansion over ħ of the commutator [A, B] expressed in the phase space formulation is iħ{A, B}
Canonical_quantization
Formulation of quantum mechanics
equation Interaction picture Heisenberg picture Phase space formulation POVM Mathematical formulation of quantum mechanics Schrödinger functional At t
Schrödinger_picture
Formulation of classical mechanics in terms of Hilbert spaces
Koopman observed that the phase space of the classical system can be converted into a Hilbert space. According to this formulation, functions representing
Koopman–von Neumann classical mechanics
Koopman–von_Neumann_classical_mechanics
Quantum characteristics are phase-space trajectories that arise in the phase space formulation of quantum mechanics through the Wigner transform of Heisenberg
Method of quantum characteristics
Method_of_quantum_characteristics
Equations that describe the behavior of a physical system
operators and the classical Poisson bracket by the commutator, the phase space formulation closely follows classical Hamiltonian mechanics, placing position
Equations_of_motion
Concept in statistics
arise naturally in the study of quantum mechanics when treated in phase space formulation, commonly used in quantum optics, time-frequency analysis, and
Quasiprobability_distribution
Formulation of quantum mechanics
This formulation has proven crucial to the subsequent development of theoretical physics, because manifest Lorentz covariance (time and space components
Path_integral_formulation
Complex number whose squared absolute value is a probability
Matter wave Phase space formulation Uncertainty principle Ward's probability amplitude Wave packet The spanning set of a Hilbert space does not suffice for
Probability_amplitude
Signal processing
The method used to transform a distribution is borrowed from the phase space formulation of quantum mechanics, even though the subject matter of this article
Transformation between distributions in time–frequency analysis
Transformation_between_distributions_in_time–frequency_analysis
Formulation of classical mechanics using momenta
and thereby describe quantum mechanical diffusion in phase space (See Phase space formulation and Wigner–Weyl transform). This more algebraic approach
Hamiltonian_mechanics
Recipe for constructing a quantum analog of a classical physical theory
representation change, however, Weyl's map underlies the alternate phase-space formulation of conventional quantum mechanics. Half-form Lagrangian foliation
Geometric_quantization
Approximation or recovery of classical mechanics in certain theories
two into a common mathematical framework in various ways. In the phase space formulation of quantum mechanics, which is statistical in nature, logical connections
Classical_limit
Branch of mathematics
algebraic geometry Noncommutative torus Noncommutative topology Phase space formulation Quasi-free algebra Spectral action principle Spectral triple Connes
Noncommutative_geometry
Quantum mechanical model
}e^{(2n+1)\left(x-{\tfrac {1}{2}}\sinh(2x)\right)}dx~.} In the phase space formulation of quantum mechanics, eigenstates of the quantum harmonic oscillator
Quantum_harmonic_oscillator
Equations describing classical electromagnetism
mechanics, or for use in quantum mechanics. The covariant formulation (on spacetime rather than space and time separately) makes the compatibility of Maxwell's
Maxwell's_equations
Computational materials scientist and Condensed-matter physicist
phonon Boltzmann equation. They later derived, from the Wigner phase-space formulation of quantum mechanics, a unified transport equation that seamlessly
Nicola_Marzari
Property of a mathematical function
space. Directional derivative Partial derivative Gradient Gateaux derivative Fréchet derivative Derivative (generalizations) Phase space formulation § Star
Semi-differentiability
Foundational principle in quantum physics
discussion of this important but technical distinction.) In the phase space formulation of quantum mechanics, the Robertson–Schrödinger relation follows
Uncertainty_principle
and statistics, among other fields. Moyal helped establish the phase space formulation of quantum mechanics in 1949 by bringing together the ideas of
José_Enrique_Moyal
States of matter for water as a solid
properties. In space, amorphous ice is the most common form as confirmed by observation. Thus, it is theorized to be the most common phase in the universe
Phases_of_ice
Relativistic quantum mechanical wave equation
}{2}}} . While this was still a non-relativistic formulation, he believed that a fully relativistic formulation possibly required a more complicated model for
Dirac_equation
Mathematical model combining space and time
December 2006. Feng, W. X. (3 August 2022). "Gravothermal phase transition, black holes and space dimensionality". Physical Review D. 106 (4) L041501. arXiv:2207
Spacetime
Iranian astrophysicist and theoretical physicist
Astrophysics 470 (3), 1111–1116, 2007 Y. Sobouti, S. Nasiri, A phase space formulation of quantum state functions International journal of modern physics
Yousef_Sobouti
Formulation of quantum mechanics
Matrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925. It was the first conceptually
Matrix_mechanics
Array of antennas creating a steerable beam
Beam Antennas for Space Applications Principle of Phased Array 'Phased Array' microphone system of Tony Faulkner Principles of Phased Array systems - Tutorial
Phased_array
Type of vector space in math
Hilbert space as "the Hilbert space" or just "Hilbert space". In the Wightman formulation of relativistic quantum field theory, the state space is likewise
Hilbert_space
value of an operator in Hilbert space and the expectation value of its associated function in the phase space formulation with respect to a quasiprobability
Optical_equivalence_theorem
NASA infrared space telescope
NASA's request of US$14 million, allowing the mission to enter the "formulation phase" in February 2016. On 18 February 2016, NASA announced that Roman
Nancy Grace Roman Space Telescope
Nancy_Grace_Roman_Space_Telescope
Phase of a cycle
mechanics, the geometric phase (also known as the Pancharatnam–Berry phase, Pancharatnam phase, or Berry phase) is a phase difference acquired over the
Geometric_phase
British quantum physicist (1935–2025)
limit, the shadow phase spaces converge to one unique phase space. In their algebraic formulation of quantum mechanics the equation of motion takes on
Basil_Hiley
Phenomenon resulting from the superposition of two waves
cancel if they have the same amplitude and their phases are spaced equally in angle. Using phasors, each wave can be represented as A e i φ n {\displaystyle
Wave_interference
Branch of differential geometry and differential topology
Symplectic geometry has its origins in the Hamiltonian formulation of classical mechanics where the phase space of certain classical systems takes on the structure
Symplectic_geometry
Thought experiment in special relativity
acceleration can be eliminated in formulations of the twin paradox in curved spacetime, where the twins can fall freely along space-time geodesics between meetings"
Twin_paradox
Theorem in the mathematical formulation of quantum mechanics
on the Hilbert space of states. The physical states in a quantum theory are represented by unit vectors in Hilbert space up to a phase factor, i.e. by
Wigner's_theorem
Description of a quantum-mechanical system
predictions. Other formulations of quantum mechanics include matrix mechanics, introduced by Werner Heisenberg, and the path integral formulation, developed chiefly
Schrödinger_equation
Limit on measurements at quantum scales
mechanics, the Ehrenfest theorem notwithstanding. Nevertheless, in the phase space formulation of quantum mechanics, such limits are more systematic and practical
Quantum_limit
Mathematical tool used in quantum mechanics
distribution Glauber–Sudarshan P representation Husimi Q representation Phase space formulation Negative probability Signed measure Generalized probabilistic theory
Margenau-Hill quasiprobability distribution
Margenau-Hill_quasiprobability_distribution
Space of possible positions for all objects in a physical system
{\displaystyle q} for a point in configuration space; this is the convention in both the Hamiltonian formulation of classical mechanics, and in Lagrangian
Configuration_space_(physics)
Abandoned antidepressant
an extended-release formulation with enhanced pharmacokinetics suitable for once-daily administration. The combination formulation employs tianeptine as
Tianeptine/naloxone
Certain dynamical systems will eventually return to (or approximate) their initial state
volume-preserving if the volume of each set in phase space is invariant under the flow. Poincaré's formulation of the theorem then states: If a flow preserves
Poincaré_recurrence_theorem
Type of manifold in differential geometry
the phase space of the system. Symplectic manifolds arise from classical mechanics; in particular, they are a generalization of the phase space of a
Symplectic_manifold
Sets of coordinates on phase space which can be used to describe a physical system
phase space which can be used to describe a physical system at any given point in time. Canonical coordinates are used in the Hamiltonian formulation
Canonical_coordinates
Data manipulation in radiology
Thus k-space information is somewhat redundant; an image can be reconstructed using only one half of the k-space. Such is in either the PE (Phase Encode)
K-space in magnetic resonance imaging
K-space_in_magnetic_resonance_imaging
Mathematical model of a system in control engineering
too. The state space (also called time-domain approach and equivalent to phase space in certain dynamical systems) is a geometric space where the axes
State-space_representation
microscopy Phase distortion Phase factor Phase noise Phase offset modulation Phase plane Phase portrait Phase problem Phase retrieval Phase space Phase space formulation
Index_of_physics_articles_(P)
Austrian mathematician and mathematical physicist
Laurikainen Honorary Symposium 2013 / 2 April 2014 Dragoman, D. (2005). "Phase Space Formulation of Quantum Mechanics. Insight into the Measurement Problem". Physica
Maurice_A._de_Gosson
Medical food, probiotic formulation
The De Simone Formulation is a probiotic formula and manufacturing method developed by Claudio De Simone. The De Simone Formulation has been clinically
De_Simone_Formulation
Formulation of classical mechanics
In physics, Lagrangian mechanics is an alternate formulation of classical mechanics founded on the d'Alembert principle of virtual work. It was introduced
Lagrangian_mechanics
Formulation of the principle of stationary action
In physics, Hamilton's principle is William Rowan Hamilton's formulation of the principle of stationary action. It states that the dynamics of a physical
Hamilton's_principle
Lowest possible energy of a quantum system or field
energy quanta only when they reached the boundaries of finite cells in phase space, where their energies became integer multiples of hν. This theory led
Zero-point_energy
Mathematical model of the time dependence of a point in space
possible to use a Hamiltonian formulation which includes a Symplectic or Poisson manifold structure on the phase space. To be complete there are also
Dynamical_system
The Modulation sphere or M-space formulation is a scheme or theory representing the system of effects of phase modulation and amplitude modulation as
Modulation_sphere
Physical quantity of dimension energy × time
path integral formulation of quantum mechanics makes use of the concept—where a physical system explores all possible paths, with the phase of the probability
Action_(physics)
Hydrodynamic formulation of the Schrödinger equations
equations of quantum hydrodynamics, are Erwin Madelung's alternative formulation of the Schrödinger equation for a spinless non-relativistic particle
Madelung_equations
Mathematical entity to describe the probability of each possible measurement on a system
quantum state. A mixed state for electron spins, in the density-matrix formulation, has the structure of a 2 × 2 {\displaystyle 2\times 2} matrix that is
Quantum_state
Relativistic wave equation in quantum mechanics
and published it in July, motivated by Schrödinger's non-relativistic formulation of his matter wave theory hypothesis. Walter Gordon also derived the
Klein–Gordon_equation
Statistical mechanics hypothesis that all microstates are equiprobable for a given energy
long periods of time, the time spent by a system in some region of the phase space of microstates with the same energy is proportional to the volume of
Ergodic_hypothesis
Interpretation of quantum mechanics
state formulation or the Everett interpretation, after physicist Hugh Everett, who first proposed it in 1957. Bryce DeWitt popularized the formulation and
Many-worlds_interpretation
Set of quantities in accelerator physics
momenta) along that dimension of every particle in a beam are plotted on a phase space diagram, an ellipse enclosing the particles can be given by the equation:
Courant–Snyder_parameters
Force resulting from the quantisation of a field
is a physical force acting on the macroscopic boundaries of a confined space which arises from the quantum fluctuations of a field. The term Casimir
Casimir_effect
Aspect of relativity in physics
unaffected by the opacity of the very early universe. In these early phases, space had not yet become "transparent", so observations based upon light,
Gravitational_wave
Device to determine relative phase shift
Mach–Zehnder interferometer is a device used to determine the relative phase shift variations between two collimated beams derived by splitting light
Mach–Zehnder_interferometer
NASA space mission to Jupiter and Europa
February 2017, the mission moved from Phase A to Phase B (the preliminary design phase). On July 18, 2017, the House Space Subcommittee held hearings on the
Europa_Clipper
Any entity that can be measured
the mathematical formulation of quantum mechanics, up to a phase constant, pure states are given by non-zero vectors in a Hilbert space V. Two vectors v
Observable
Type of quantum state
circle denoting the uncertainty of a coherent state in the quadrature phase space (see right) has been "squeezed" to an ellipse of the same area. Note
Squeezed_coherent_state
Abstract coordinate system
the formulation of many problems in physics employs generalized coordinates, normal modes or eigenvectors, which are only indirectly related to space and
Frame_of_reference
Speed of electromagnetic waves in vacuum
troughs of a plane wave (a wave filling the whole space, with only one frequency) propagate is called the phase velocity vp. A physical signal with a finite
Speed_of_light
Quantum mechanics taking into account particles near or at the speed of light
physics, relativistic quantum mechanics (RQM) is any Poincaré-covariant formulation of quantum mechanics (QM). This theory is applicable to massive particles
Relativistic quantum mechanics
Relativistic_quantum_mechanics
Iranian physicist and university president
mechanics in phase space, and coronal heating, as well as the selection of sites for optical observatories. His work has included a formulation of quantum
Sadollah_Nasiri_Gheydari
Physical spaces representing position and momentum, Fourier-transform duals
either the positions of the constituent particles, or their momenta, both formulations equivalently provide the same information about the system in consideration
Position_and_momentum_spaces
Formulation of general relativity
attempt to quantize the canonical formulation of general relativity (or canonical gravity). It is a Hamiltonian formulation of Einstein's general theory of
Canonical_quantum_gravity
Potential for two waves to interfere
which may be zero (destructive interference), depending on their relative phase. Constructive or destructive interference are limit cases, and two waves
Coherence_(physics)
Branch of mathematics that studies dynamical systems
recurrence theorem, which claims that almost all points in any subset of the phase space eventually revisit the set. Systems for which the Poincaré recurrence
Ergodic_theory
Physics experiment
relative phases of the contributions) produces the interference pattern observed by the double-slit experiment. Feynman stressed that his formulation is merely
Double-slit_experiment
French mathematician, physicist and engineer (1854–1912)
perfect invariance of all of Maxwell's equations, an important step in the formulation of the theory of special relativity, for which he is also credited with
Henri_Poincaré
Mathematical description of spacetime used in relativity
applied to the frame in motion and shifts the phase of light. Minkowski spacetime is a pseudo-Euclidean space equipped with an isotropic quadratic form called
Minkowski_spacetime
Formulation of geometrical optics
Hamiltonian optics and Lagrangian optics are two formulations of geometrical optics which share much of the mathematical formalism with Hamiltonian mechanics
Hamiltonian_optics
Idealization of a large number of atomic-sized systems
written as a probability distribution in phase space; the microstates are the result of partitioning phase space into equal-sized units, although the size
Ensemble (mathematical physics)
Ensemble_(mathematical_physics)
Loss of quantum coherence
each xi is a point in 3-dimensional space. This has analogies with the classical phase space. A classical phase space contains a real-valued function in
Quantum_decoherence
Trace radiation from the early universe
compelling evidence that it is out of phase with the T-mode spectrum. In June 2001, NASA launched a second CMB space mission, WMAP, to make much more precise
Cosmic_microwave_background
PHASE SPACE-FORMULATION
PHASE SPACE-FORMULATION
Male
English
Middle English surname (of Norman French origin) transferred to forename use, CHASE means "hunter."Â
Girl/Female
Tamil
Antariksha | அஂதரிகà¯à®·
Space, Sky
Antariksha | அஂதரிகà¯à®·
Male
French
French form of Latin Stephanus, STÉPHANE means "crown."
Boy/Male
Tamil
Antrix | அஂதà¯à®°à¯€à®•à¯à®·
Space
Antrix | அஂதà¯à®°à¯€à®•à¯à®·
Boy/Male
Hindu
Space
Boy/Male
Tamil
Antariksh | அஂதரிகà¯à®·
Space
Antariksh | அஂதரிகà¯à®·
Surname or Lastname
English
English : from a vernacular short form of the Latin personal name Paschalis (see Pascal, Italian Pasquale).nickname for a mild-mannered and peaceable person, from Middle English pace, pece ‘peace’, ‘concord’, ‘amity’ (via Anglo-Norman French from Latin pax, genitive pacis).Italian : from the medieval personal name Pace, used for both men and women, from the word pace ‘peace’ (see 1).
Surname or Lastname
German
German : nickname for a swift runner or a timorous person, from Middle High German, Middle Low German hase ‘hare’.Jewish (Ashkenazic) : ornamental name from German Hase ‘hare’.English : from a Middle English nickname, Hase, from Old English hÄs ‘harsh, raucous, or hoarse voice’.Japanese : usually written with characters meaning ‘long valley’; habitational name from a place in Yamato (now Nara prefecture). Listed in the Shinsen shÅjiroku. Some bearers are descended from the Taira clan; they are found mainly in eastern Japan. Also pronounced Nagaya and Nagatani; the original pronunciation was Hatsuse, meaning ‘beginning of the strait’.
Surname or Lastname
English or Scottish
English or Scottish : unexplained.
Surname or Lastname
English
English : nickname for a frugal person, from Middle English spare ‘sparing’, ‘frugal’.
Girl/Female
Indian, Japanese, Tamil
Space; Star
Girl/Female
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Sanskrit, Tamil, Telugu
Space; Sky
Boy/Male
Hindu
Space
Boy/Male
Tamil
Antareeksh | அஂதரீகà¯à®·
Space
Antareeksh | அஂதரீகà¯à®·
Surname or Lastname
English
English : metonymic occupational name for a huntsman, or rather a nickname for an exceptionally skilled huntsman, from Middle English chase ‘hunt’ (Old French chasse, from chasser ‘to hunt’, Latin captare).Southern French : topographic name for someone who lived in or by a house, probably the occupier of the most distinguished house in the village, from a southern derivative of Latin casa ‘hut’, ‘cottage’, ‘cabin’.Thomas Chase came to MA from Chesham, Buckinghamshire, England, in the 1640s, and had many prominent descendants. Samuel Chase, born in Somerset Co., MD, in 1741, was one of the first members of the U.S. Supreme Court; Philander Chase, born in Cornish, NH, in 1741 was a prominent Episcopal clergyman, and his nephew Salmon Portland Chase (1808–73), also born in Cornish, was governor of OH, a U.S. senator, and secretary of the U.S. Treasury during the Civil War.
Boy/Male
Hindu
Space
Surname or Lastname
English
English : from Middle English pese ‘pea’, hence a metonymic occupational name for a grower or seller of peas, or a nickname for a small and insignificant person. The word was originally a collective singular (Old English peose, pise, from Latin pisa) from which the modern English vocabulary word pea is derived by folk etymology, the singular having been taken as a plural.Robert and John Pease came from Great Baddow, Essex, England, to Salem, MA, in 1634. In 1644 Robert died, leaving a son (also called Robert) who was apprenticed as a weaver in Salem. By 1646 John Pease was living on Martha’s Vineyard.
Boy/Male
Hindu, Indian
Space; Outer Space; Sky
Girl/Female
Indian, Telugu
Space
Male
English
English surname transferred to forename use, derived from the French personal name Pascal, PACE means "Passover; Easter."
PHASE SPACE-FORMULATION
PHASE SPACE-FORMULATION
Girl/Female
Hindu
Similar
Girl/Female
German, Teutonic
Firm Defender
Boy/Male
Muslim
Servant of the Guide.
Boy/Male
Indian, Telugu
Lord Siva
Boy/Male
British, English
From the Ox Ford
Boy/Male
British, English, Finnish, French, Greek
Hearkening
Boy/Male
Bengali, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Sanskrit, Sindhi, Telugu
Made of Gold
Male
Spanish
Spanish form of Middle Latin Venceslaus, VENCESLÃS means "more glory."Â
Boy/Male
Arabic
One who Serves the God
Boy/Male
Hindu, Indian, Malayalam, Marathi
A Diety; Lord Vishnu
PHASE SPACE-FORMULATION
PHASE SPACE-FORMULATION
PHASE SPACE-FORMULATION
PHASE SPACE-FORMULATION
PHASE SPACE-FORMULATION
adv.
With a quick pace; quick; fast; speedily.
v. t.
Scanty; not abundant or plentiful; as, a spare diet.
v. t.
To season with spice, or as with spice; to mix aromatic or pungent substances with; to flavor; to season; as, to spice wine; to spice one's words with wit.
n.
The right of bowling again at a full set of pins, after having knocked all the pins down in less than three bowls. If all the pins are knocked down in one bowl it is a double spare; in two bowls, a single spare.
n.
A quantity or portion of extension; distance from one thing to another; an interval between any two or more objects; as, the space between two stars or two hills; the sound was heard for the space of a mile.
v. t.
Held in reserve, to be used in an emergency; as, a spare anchor; a spare bed or room.
n.
The liberty or franchise of having a chase; free chase.
n.
Manner of stepping or moving; gait; walk; as, the walk, trot, canter, gallop, and amble are paces of the horse; a swaggering pace; a quick pace.
v. t.
To develop, guide, or control the pace or paces of; to teach the pace; to break in.
n.
To arrange or adjust the spaces in or between; as, to space words, lines, or letters.
v. i.
To give chase; to hunt; as, to chase around after a doctor.
n.
Space.
pl.
of Phase
n.
Any appearance or aspect of an object of mental apprehension or view; as, the problem has many phases.
v. i.
To group notes into phrases; as, he phrases well. See Phrase, n., 4.
n.
One of that suit of cards each of which bears one or more figures resembling a spade.
v. t.
To dig with a spade; to pare off the sward of, as land, with a spade.
v. t.
To measure by steps or paces; as, to pace a piece of ground.
n.
A particular appearance or state in a regularly recurring cycle of changes with respect to quantity of illumination or form of enlightened disk; as, the phases of the moon or planets. See Illust. under Moon.
imp. & p. p.
of Space