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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
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
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
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
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
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
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
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)
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
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
Mathematical description of quantum state
Broglie–Bohm theory Double-slit experiment Faraday wave Fermion Phase-space formulation Schrödinger equation Universal wavefunction Wave function collapse
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
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
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)
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
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
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
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
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
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
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 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
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
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 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
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
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
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
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
Branch of mathematics
algebraic geometry Noncommutative torus Noncommutative topology Phase space formulation Quasi-free algebra Spectral action principle Spectral triple Connes
Noncommutative_geometry
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
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
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
Property of a mathematical function
space. Directional derivative Partial derivative Gradient Gateaux derivative Fréchet derivative Derivative (generalizations) Phase space formulation § Star
Semi-differentiability
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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)
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
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
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
Generalized Euclidean space in mathematics
Liu, Wei-Yin (2021). "Noncommutative Coordinate Picture of the Quantum Phase Space". Chinese Journal of Physics. 71. Elsevier BV: 418. arXiv:1903.11962
Projective_Hilbert_space
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
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
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
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
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 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
Space exploration using nuclear energy
and information sharing in the event of space nuclear accidents. It also led to the intergovernmental formulation of emergency protocols, such as Operation
Nuclear_power_in_space
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
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
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)
Concept in mathematics
In mathematics, a configuration space is a construction closely related to state spaces or phase spaces in physics. In physics, these are used to describe
Configuration space (mathematics)
Configuration_space_(mathematics)
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
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
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
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
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
Multi particle state space
arXiv:math-ph/0007040. Folland, Gerald B. (1989). Harmonic Analysis in Phase Space. Annals of Mathematics Studies. Vol. 122. Princeton University Press
Fock_space
Exactly solvable model of coupled oscillators
a model for the behavior of a large set of coupled oscillators. Its formulation was motivated by the behavior of systems of chemical and biological oscillators
Kuramoto_model
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
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
Interpretation of quantum mechanics
w)} is the local Hermitian inner product on the value space of the wavefunction. This formulation allows for stochastic theories such as the creation and
De_Broglie–Bohm_theory
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
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
Formalism in classical field theory based on Hamiltonian mechanics
derivatives. The fields φi and conjugates πi form an infinite dimensional phase space, because fields have an infinite number of degrees of freedom. For two
Hamiltonian_field_theory
Physics phenomenon
mathematical formulation of quantum mechanics. Consider two arbitrary quantum systems A and B, with respective Hilbert spaces HA and HB. The Hilbert space of the
Quantum_entanglement
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
Mathematical transformation in physics
observed in 2017, break discrete time-translation symmetry. Absolute time and space Mach's principle Spacetime Time reversal symmetry Portals: Astronomy Mathematics
Time-translation_symmetry
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
Fringe hypothesis
due to the wide range of length scales in the fluid filled extracellular space. Nevertheless, University of Oxford quantum physicist Vlatko Vedral told
Quantum_mind
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
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
PHASE SPACE-FORMULATION
PHASE SPACE-FORMULATION
Girl/Female
Tamil
Antariksha | அஂதரிகà¯à®·
Space, Sky
Antariksha | 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
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.
Male
French
French form of Latin Stephanus, STÉPHANE means "crown."
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.
Male
English
English surname transferred to forename use, derived from the French personal name Pascal, PACE means "Passover; Easter."
Girl/Female
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Sanskrit, Tamil, Telugu
Space; Sky
Male
English
Middle English surname (of Norman French origin) transferred to forename use, CHASE means "hunter."Â
Girl/Female
Indian, Telugu
Space
Boy/Male
Hindu
Space
Surname or Lastname
English or Scottish
English or Scottish : unexplained.
Boy/Male
Tamil
Antareeksh | அஂதரீகà¯à®·
Space
Antareeksh | அஂதரீகà¯à®·
Boy/Male
Hindu
Space
Boy/Male
Hindu, Indian
Space; Outer Space; Sky
Boy/Male
Tamil
Antariksh | அஂதரிகà¯à®·
Space
Antariksh | அஂதரிகà¯à®·
Girl/Female
Indian, Japanese, Tamil
Space; Star
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
English : nickname for a frugal person, from Middle English spare ‘sparing’, ‘frugal’.
Boy/Male
Tamil
Antrix | அஂதà¯à®°à¯€à®•à¯à®·
Space
PHASE SPACE-FORMULATION
PHASE SPACE-FORMULATION
Boy/Male
Tamil
Pearl
Boy/Male
English
Famed; famous.
Girl/Female
American, British, Christian, English, Latin
Bird; Believed to have been Introduced During the Norman Conquest; Like a Bird
Girl/Female
British, English, German, Scottish
Bright Fame; Robin
Boy/Male
British, English
From the Priest's Meadow
Boy/Male
Welsh
Old face.
Boy/Male
Muslim
Garden of flowers
Girl/Female
American, Australian, Danish, Finnish, French, German, Greek, Latin, Portuguese, Swedish
Rose Garlands; Form of Rose; Flower Name; Horse; Fame; Combination of Rose and Lily
Boy/Male
Australian, Chinese, Gaelic, Irish
Ancient; Archaic
Girl/Female
Hindu, Indian
Full Moon
PHASE SPACE-FORMULATION
PHASE SPACE-FORMULATION
PHASE SPACE-FORMULATION
PHASE SPACE-FORMULATION
PHASE SPACE-FORMULATION
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.
v. i.
To give chase; to hunt; as, to chase around after a doctor.
v. t.
To dig with a spade; to pare off the sward of, as land, with a spade.
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.
adv.
With a quick pace; quick; fast; speedily.
v. i.
To group notes into phrases; as, he phrases well. See Phrase, n., 4.
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.
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.
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.
v. t.
Scanty; not abundant or plentiful; as, a spare diet.
n.
To arrange or adjust the spaces in or between; as, to space words, lines, or letters.
v. t.
To develop, guide, or control the pace or paces of; to teach the pace; to break in.
imp. & p. p.
of Space
pl.
of Phase
n.
One of that suit of cards each of which bears one or more figures resembling a spade.
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.
Space.
n.
Any appearance or aspect of an object of mental apprehension or view; as, the problem has many phases.