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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
Mathematical description of quantum state
Double-slit experiment Faraday wave Fermion Phase-space formulation Schrödinger equation Universal wavefunction Wave function collapse Wave packet The functions
Wave_function
Formulation of classical mechanics in terms of Hilbert spaces
classical probability distributions on phase spaces with complex-valued wavefunctions. This method of classical wavefunctions is conceptually distinct from the
Koopman–von Neumann classical mechanics
Koopman–von_Neumann_classical_mechanics
Wigner distribution function in physics as opposed to in signal processing
The goal was to link the wavefunction that appears in the Schrödinger equation to a probability distribution in phase space. It is a generating function
Wigner quasiprobability distribution
Wigner_quasiprobability_distribution
Concept in physics
through the corresponding parameter space and can acquire a phase that depends on the path taken through that space. Consider a quantum system whose Hamiltonian
Berry connection and curvature
Berry_connection_and_curvature
Interpretation of quantum mechanics
theory emerges from the Bohmian formalism when one considers conditional wavefunctions of subsystems. Pilot wave theory is explicitly nonlocal, which is in
De_Broglie–Bohm_theory
Aspect of theoretical physics
provides a geometric language for how a band wavefunction changes across parameter space and how its phase twists under parallel transport with consequences
Quantum geometry (condensed matter)
Quantum_geometry_(condensed_matter)
State of matter
density. A more concise and experimentally relevant condition involves the phase-space density D = n λ T 3 {\displaystyle {\mathcal {D}}=n\lambda _{T}^{3}}
Bose–Einstein_condensate
Quantum mechanical model
evolution is not a simple shift in wavefunction phase. The time-evolved states are, however, also coherent states but with phase-shifting parameter α instead:
Quantum_harmonic_oscillator
Mathematical function in general imaging
focal plane of objective lens Modify the wavefunction in reciprocal space by a phase factor, also known as the Phase Contrast Transfer Function, to account
Contrast_transfer_function
Area of physical and philosophical debate
physicists' mental arbitrariness. The statistical interpretation of wavefunctions due to Max Born differs sharply from Schrödinger's original intent,
Interpretations of quantum mechanics
Interpretations_of_quantum_mechanics
Notation for quantum states
which have infinite norm, i.e. non-normalizable wavefunctions. Examples include states whose wavefunctions are Dirac delta functions or infinite plane waves
Bra–ket_notation
Mathematical group
phase space of classical mechanics. When one tries to make the same transformations act on the wavefunctions of quantum mechanics, there is a phase ambiguity
Symplectic_group
Mathematical structures that allow quantum mechanics to be explained
regions of space can still be represented using a symmetrized/antisymmetrized wavefunction and that independent treatment of these wavefunctions gives the
Mathematical formulation of quantum mechanics
Mathematical_formulation_of_quantum_mechanics
Specific quantum state of a quantum harmonic oscillator
uncertainty relation: there is no uniquely defined phase operator in quantum mechanics. To find the wavefunction of the coherent state, the minimal uncertainty
Coherent_state
Different states of quantum systems
interactions are often neglected if the spatial overlap of the electron wavefunctions is low. For multi-electron atoms, interactions between electrons cause
Energy_level
Computational quantum mechanical modelling method to investigate electronic structure
forcing the pseudo-wavefunctions to coincide with the true valence wavefunctions beyond a certain distance rℓ. The pseudo-wavefunctions are also forced to
Density_functional_theory
Description of a quantum-mechanical system
the spatial variation of the phase of a wavefunction is said to characterize the probability flux of the wavefunction. Although the factor ∇ S / m {\textstyle
Schrödinger_equation
Reformulation of general relativity
principle; applied to many non-localized wavefunctions spread throughout the curved space to form a localized wavefunction: Ψ = ∑ n c n ψ n , {\displaystyle
Hamilton–Jacobi–Einstein equation
Hamilton–Jacobi–Einstein_equation
Wave-like behavior of an electron in a molecule
combinations of atomic orbitals, or the sums and differences of the atomic wavefunctions, provide approximate solutions to the Hartree–Fock equations which correspond
Molecular_orbital
Electromagnetic quantum-mechanical effect in regions of zero magnetic and electric field
which the particle passes and the particle's wavefunction being negligible inside the solenoid. This phase shift has been observed experimentally. There
Aharonov–Bohm_effect
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
Probability density of electrons being somewhere
"smeared out" in space. For one-electron systems, the electron density at any point is proportional to the square magnitude of the wavefunction. In molecules
Electron_density
Process by which a quantum system takes on a definitive state
mathematical description of a quantum system; a quantum state vector uses Hilbert space vectors for the description. Reduction of the state vector replaces the
Wave_function_collapse
Foundational principle in quantum physics
and momentum-space wavefunctions for one spinless particle with mass in one dimension. The more localized the position-space wavefunction, the more likely
Uncertainty_principle
Phenomenon in quantum systems
accessible phase space. Thus, it would be natural to expect that the eigenstates of the quantum counterpart would fill the quantum phase space in the uniform
Quantum_scar
Quantum mechanical waves describing matter
wavefunction, a function that assigns a complex number to each point in space. Schrödinger tried to interpret the modulus squared of the wavefunction
Matter_wave
Quantum physics thought experiment
incoming wavefunctions or not, and how to merge the incoming wavefunctions can be controlled by experimenters. There are none of the phase differences
Wheeler's delayed-choice experiment
Wheeler's_delayed-choice_experiment
Type of quantum state
wavefunctions that are covariants under the action of the group formed by multidimensional Linear Canonical Transformations. The quantum phase space (QPS)
Squeezed_coherent_state
Concept in quantum mechanics of perfectly substitutable particles
important property of these wavefunctions is that exchanging any two of the coordinate variables changes the wavefunction by only a plus or minus sign
Indistinguishable_particles
Setting of relativistic physics in geometric algebra
S2CID 119389813 Lasenby, A.N.; Doran, C.J.L. (2002). "Geometric algebra, Dirac wavefunctions and black holes". In Bergmann, P.G.; De Sabbata, Venzo (eds.). Advances
Spacetime_algebra
Excited atomic quantum state with high principal quantum number (n)
response to electric and magnetic fields, long decay periods and electron wavefunctions that approximate, under some conditions, classical orbits of electrons
Rydberg_atom
Short "burst" or "envelope" of restricted wave action that travels as a unit
different wavenumbers, with phases and amplitudes such that they interfere constructively only over a small region of space, and destructively elsewhere
Wave_packet
Quantised attribute of electrons in free space
angular momentum corresponds to helical wavefronts, or, equivalently, a phase proportional to the azimuthal angle. Electron beams with quantized orbital
Orbital angular momentum of free electrons
Orbital_angular_momentum_of_free_electrons
Simple model of topological insulator
distinct kinds of states. For non-zero eigenenergies, the corresponding wavefunctions would be delocalized all along the chain while the zero energy eigenstates
Su–Schrieffer–Heeger_model
Structure of the atomic nucleus
that is, wavefunctions for Z proton variables or N neutron variables, which are antisymmetrized products of single-particle wavefunctions (antisymmetrized
Nuclear_structure
Coefficients in angular momentum eigenstates of quantum systems
Hall 2015 Appendix C Zachos, C K (1992). "Altering the Symmetry of Wavefunctions in Quantum Algebras and Supersymmetry". Modern Physics Letters A. A7
Clebsch–Gordan_coefficients
Particle that is not bound by an external force
normalization condition for the wave function states that if a wavefunction belongs to the quantum state space ψ ∈ L 2 ( R 3 ) , {\displaystyle \psi \in L^{2}(\mathbb
Free_particle
Theorem in quantum mechanics
is not variational. The proof also employs an identity of normalized wavefunctions – that derivatives of the overlap of a wave function with itself must
Hellmann–Feynman_theorem
Function that can be used to build the wave function of a multi-fermionic system
constant is implied by noting the number N, and only the one-particle wavefunctions (first shorthand) or the indices for the fermion coordinates (second
Slater_determinant
Point with minimum wave amplitude
equally spaced intervals where the wave amplitude (motion) is zero (see animation above). At these points the two waves add with opposite phase and cancel
Node_(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
the classical phase space. This is consistent with the intuition that the flows of ergodic systems are equidistributed in phase space. By contrast, classical
Quantum_ergodicity
as the reduced Planck constant or Dirac constant. The general form of wavefunction for a system of particles, each with position ri and z-component of spin
List of equations in quantum mechanics
List_of_equations_in_quantum_mechanics
Type of two-dimensional quasiparticle
e i θ {\displaystyle e^{i\theta }} is the phase factor. In space of three or more dimensions, the phase factor is 1 {\displaystyle 1} or − 1 {\displaystyle
Anyon
Physics software package
Bloch's Theorem which means a wavefunction of a periodic system has a cell-periodic factor and a phase factor. The phase factor is represented by a plane
CASTEP
Quantum state with all observables independent of time
Hamiltonian is unchanging in time.) The wavefunction itself is not stationary: It continually changes its overall complex phase factor, so as to form a standing
Stationary_state
Equation of statistical mechanics
the spatial extension of the wavefunction can affect the dynamics, making it questionable whether the classical phase space distribution f that appears
Boltzmann_equation
Physics experiment
configuration space or 'phase space'. It is difficult to visualize a reality comprising imaginary functions in an abstract, multi-dimensional space. No difficulty
Double-slit_experiment
Branch of physics seeking to explain chaotic dynamical systems in terms of quantum theory
systems in different regions of space, minimizing the non-separable part of the Hamiltonian in each region. Wavefunctions are obtained in these regions
Quantum_chaos
Mechanism that explains the generation of mass for gauge bosons
is zero when the phase change along any path from parallel transport is equal to the phase difference in the condensate wavefunction. The condensate value
Higgs_mechanism
spin-1 superfluids or Bose condensates, the condensate wavefunction is invariant if the superfluid phase changes by π {\displaystyle \pi } , along with a π
Fractional_vortices
Solid form of the 7th element
wavefunctions for N2 have infinite extent. The quoted dimensions correspond to an arbitrary cutoff at electron density 0.0135 (e−)/Å3. The ε-δ phase transition
Solid_nitrogen
Superconducting circuit element
{\displaystyle \varphi } is the phase difference between the two superconductor's wavefunctions. Because Cooper pairs tunnel phase-coherently, the supercurrent
Josephson_junction
Description of physical properties at the atomic and subatomic scale
classical mechanics and quantum mechanics Macroscopic quantum phenomena Phase-space formulation Regularization (physics) Two-state quantum system A momentum
Quantum_mechanics
Quantum mechanical state change
stationary quantum states of an atom are orthogonal: the overlap of the wavefunctions between an excited state and the ground state of the atom is zero. Thus
Spontaneous_emission
Calculation rule in quantum mechanics
position is proportional to the square of the amplitude of the system's wavefunction at that position. It was formulated and published by German physicist
Born_rule
Loss of quantum coherence
Quantum states are either pure or mixed; pure states are also known as wavefunctions. Assigning a pure state to a quantum system implies certainty about
Quantum_decoherence
Formulation of quantum mechanics
P which obey the commutation relations can be made to act on a space of wavefunctions, with P a derivative operator. This implies that a Schrödinger picture
Matrix_mechanics
Symmetry of spatially mirrored systems
transformations have some eigenvalues which are phases other than ± 1 {\displaystyle \pm 1} . For electronic wavefunctions, even states are usually indicated by
Parity_(physics)
Sensor implementation technique
detection is a method of extracting information encoded as modulation of the phase and/or frequency of an oscillating signal, by comparing that signal with
Homodyne_detection
Formulation of quantum mechanics
integrals well-defined. Regardless of whether one works in configuration space or phase space, when equating the operator formalism and the path integral formulation
Path-integral_formulation
Dynamic disturbance in a medium or field
waves, the phase velocity and the group velocity. Phase velocity is the rate at which the phase of the wave propagates in space: any given phase of the wave
Wave
Atoms with a single valence electron, so they behave like hydrogen
Numerical methods must be applied in order to obtain (approximate) wavefunctions or other properties from quantum mechanical calculations. Due to the
Hydrogen-like_atom
Group in mathematical representation theory
momentum. When one tries to make those same transformations act on wavefunctions, one is naturally led not to the symplectic group itself but to a closely
Metaplectic_group
Scientific subjects
of a dynamic system—and is a wave equation that is used to solve for wavefunctions. For example, the light, or electromagnetic radiation emitted or absorbed
Branches_of_physics
Relativistic quantum mechanical wave equation
Lorentz group. This is because the states in a Hilbert space are defined only up to a complex phase, so particles belong to projective representations rather
Dirac_equation
Complex number whose squared absolute value is a probability
and potential, the Schrödinger equation fully determines subsequent wavefunctions. The above then gives probabilities of locations of the particle at
Probability_amplitude
Subfield of materials science
molecular dynamics simulations, continuum dislocation dynamics, and phase field models. Phase field methods are focused on phenomena dependent on interfaces
Computational materials science
Computational_materials_science
Geometric transformation which produces an identical image
of molecular symmetry, quantum wavefunctions need not be invariant, because the operation can multiply them by a phase or mix states within a degenerate
Symmetry_operation
Electromagnetic effect in physics
^{2}}{2m^{*}L^{2}}}} , n z = 1 , 2 , 3... {\displaystyle n_{z}=1,2,3...} and the wavefunctions are sinusoidal. For the x {\displaystyle x} and y {\displaystyle y}
Quantum_Hall_effect
Mathematical transform that expresses a function of time as a function of frequency
real vector space with a p-axis and a q-axis called the phase space. In contrast, quantum mechanics chooses a polarisation of this space in the sense
Fourier_transform
Quantum mechanics principle
exclusion principle with a single-valued many-particle wavefunction is equivalent to requiring the wavefunction to be antisymmetric with respect to exchange. If
Pauli_exclusion_principle
Quantum field of electrons
electronic wave function, and is created by combining electron states, or wavefunctions, of opposite momenta. The effect is somewhat analogous to the standing
Charge_density_wave
Mathematical tool in quantum physics
on physical systems. It is a generalization of the state vectors or wavefunctions: while those can only represent pure states, density matrices can also
Density_matrix
Mathematical entity to describe the probability of each possible measurement on a system
position or of momentum. Historically, definitions of quantum states used wavefunctions before the more formal methods were developed. The wave function is
Quantum_state
Electrical conductivity with exactly zero resistance
and Ginzburg. This theory, which combined Landau's theory of second-order phase transitions with a Schrödinger-like wave equation, had great success in
Superconductivity
Set of functions used to represent the electronic wave function
adding local orbitals to the basis set. This allows representations of wavefunctions beyond the linearized description. The plane waves in the interstitial
Basis_set_(chemistry)
State of matter with insulating bulk but conductive boundary
and trivial insulators are separate regions in the phase diagram, connected only by conducting phases. In this way, topological insulators provide an example
Topological_insulator
=|m_{s}|+m_{s}^{2}=S(S+1)} making ROHF wavefunctions eigenfunctions of Ŝ². For multi-configurational wavefunctions expressed as | Ψ ⟩ = ∑ I c I | Φ I ⟩
Spin_contamination
Model of an energy potential in quantum mechanics
simulate situations where a particle is free to move in two regions of space with a barrier between the two regions. For example, an electron can move
Delta_potential
Elementary particles with a spin of 1/2
{C} ^{\times }} . This is because the overall phase is physically irrelevant. Equivalently, the space of pure spin-1/2 states is the complex projective
Spin_1/2
Mathematical model in quantum mechanics
in space, but ψ n ( x , t ) {\displaystyle \psi _{n}(x,t)} changes. Notice that x c − L 2 {\displaystyle x_{c}-{\tfrac {L}{2}}} represents a phase shift
Particle_in_a_box
Propagation of information or matter faster than the speed of light
loopholes around general relativity, such as by expanding or contracting space to make the object appear to be travelling greater than c. Such proposals
Faster-than-light
Equations that describe the behavior of a physical system
quantum operators and the classical Poisson bracket by the commutator, the phase space formulation closely follows classical Hamiltonian mechanics, placing
Equations_of_motion
Properties underlying modern physics
these quantities. In what follows, transformations on only one-particle wavefunctions in the form: Ω ^ ψ ( r , t ) = ψ ( r ′ , t ′ ) {\displaystyle {\widehat
Symmetry_in_quantum_mechanics
Operators useful in quantum mechanics
subfields of physics and chemistry, the use of these operators instead of wavefunctions is known as second quantization. They were introduced by Paul Dirac
Creation and annihilation operators
Creation_and_annihilation_operators
Theory of stochastic partial differential equations
system's past, much like wavefunctions in quantum theory. STS uses generalized probability distributions, or "wavefunctions", that depend not only on
Supersymmetric theory of stochastic dynamics
Supersymmetric_theory_of_stochastic_dynamics
Description of the ground state of a quantum system
The solution for condensate wavefunction Ψ ( r , t ) {\displaystyle \Psi (\mathbf {r} ,t)} is a superposition of two phase-conjugated matter–wave vortices:
Gross–Pitaevskii_equation
Physical function
where the sum is over each lattice vector R in the crystal. The set of wavefunctions ϕ R {\displaystyle \phi _{\mathbf {R} }} is an orthonormal basis for
Wannier_function
Hypothetical inflationary universe model
decay of the meta-stable false vacuum in one region led to the inflationary phase of the universe's expansion. In surrounding regions, the false vacuum continues
Eternal_inflation
Formulation of general relativity
kinds of phase space: the unrestricted (also called kinematic) phase space on which constraint functions are defined and the reduced phase space on which
Canonical_quantum_gravity
Quantization of cyclotron orbits
0\rangle .} One may verify that the above states correspond to choosing wavefunctions proportional to ψ n , m z ( x , y ) = ( ∂ ∂ w − w ¯ 4 ) n w n + m z
Landau_levels
Graph representation in quantum mechanics
Y.; Chan, G. K.-L.; Yanai, T. (2013). "Entangled quantum electronic wavefunctions of the Mn4CaO5 cluster in photosystem II". Nature Chemistry. 5 (8):
Tensor_network
Isotope of hydrogen with one neutron
wavefunction must be antisymmetric if the isospin representation is used (since a proton and a neutron are not identical particles, the wavefunction need
Deuterium
Process in quantum mechanical theories
variables on the 2 n {\displaystyle 2n} -dimensional phase space. The quantum Hilbert space is then the space of sections that depend only on the n {\displaystyle
Canonical_quantization
Type of electric dipole moment
transition dipole moment is a complex vector quantity that includes the phase factors associated with the two states. Its direction gives the polarization
Transition_dipole_moment
time-dependent, spatially independent, function, c(t), give rise to wavefunctions differing only by a phase factor exp(-i α(t)), with dα(t)/dt = c(t), and therefore
Runge–Gross_theorem
Austrian mathematician and mathematical physicist
of a particle in phase space. To demonstrate this, they picked up on "Fermi's trick" which allows identifying an arbitrary wavefunction as a stationary
Maurice_A._de_Gosson
Statistical model in quantum mechanics of magnetic materials
for finite-length anisotropic Heisenberg chains (the XXZ model), the wavefunctions obtained from Bethe's ansatz are indeed eigenstates of the Hamiltonian
Quantum_Heisenberg_model
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PHASE SPACE-WAVEFUNCTIONS
PHASE SPACE-WAVEFUNCTIONS
PHASE SPACE-WAVEFUNCTIONS
PHASE SPACE-WAVEFUNCTIONS
PHASE SPACE-WAVEFUNCTIONS
PHASE SPACE-WAVEFUNCTIONS
PHASE SPACE-WAVEFUNCTIONS
PHASE SPACE-WAVEFUNCTIONS
PHASE SPACE-WAVEFUNCTIONS
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