xxwiⅷ Symbols En =E0Vn+I Rabi frequency ESA coupling between object system and apparatus ESM coupling between object system and meter arbitrary variable,arbitrary (wave)function 5S.5A number of possible configurations of bosons and fermions,respectively arbitrary variable,arbitrary (wave)function arbitrary(continuous)observable 1 eigenkets of n 0 angle,spherical coordinate )9 generic amplitude i(m)=(m0,k amplitude operator connecting a premeasurement ()).a unitary evolution(). and a measurement ( Θm(0) theta component of the spherical harmonics ⊙(8) part of the spherical harmonics depending on the polar coordinate o arbitrary transformation(superoperator) constant,parameter internal state of a system parameter wavelength 入c=h/mc Compton wavelength of the electron 入T=√2mB7 thermal wavelength A=Hg Bext constant used in the Paschen-Bach effect A Lindblad operator classically magnetic dipole momentum 庄=1 orbital magnetic momentum of a massive particle 立，=0端s spin magnetic momentum uB=编 Bohr magneton magnetic permeability frequency random variable,variable (r)=R(r)r change of variable for the radial part of the wave function arbitrary(continuous)observable 15) eigenkets of E(x) Heaviside step function i parity operator ()probability density density matrix (pure state) time-evolved density matrix
xxviii Symbols εn = ε0 √n + 1 Rabi frequency εSA coupling between object system and apparatus εSM coupling between object system and meter ζ arbitrary variable, arbitrary (wave) function ζS, ζA number of possible configurations of bosons and fermions, respectively η arbitrary variable, arbitrary (wave) function ηˆ arbitrary (continuous) observable |η eigenkets of ηˆ θ angle, spherical coordinate ϑ generic amplitude ϑˆk (m) = m Uˆt k amplitude operator connecting a premeasurement (| k), a unitary evolution (Uˆt), and a measurement (|m) lm(θ) theta component of the spherical harmonics (θ) part of the spherical harmonics depending on the polar coordinate θ ˆ , ˆ ˆ arbitrary transformation (superoperator) ι constant, parameter |ι internal state of a system κ parameter λ wavelength λc = h/mc Compton wavelength of the electron λT = √ h¯ 2mkBT thermal wavelength = μB Bext constant used in the Paschen–Bach effect ˆ j Lindblad operator μ classically magnetic dipole momentum μˆl = eh¯ 2m ˆl orbital magnetic momentum of a massive particle μˆ s = Q eh¯ 2m sˆ spin magnetic momentum μB = eh¯ 2m Bohr magneton μ0 magnetic permeability ν frequency ξ random variable, variable ξ (r) = R(r)r change of variable for the radial part of the wave function ξˆ arbitrary (continuous) observable | ξ eigenkets of ξˆ (x) Heaviside step function ˆ parity operator ρ (classical) probability density ρˆ density matrix (pure state) ˆ˙ ρˆ time-evolved density matrix
xxix Symbols mixed density matrix density matrix for the final state of a system g density matrix for the initial state of a system reduced density matrix of the j-th subsystem PSA density matrix of the system plus apparatus DSAE density matrix of the system plus apparatus plus environment 2=2)- variance of O standard deviation (square root of the variance) of元 哈=)-月 variance ofp standard deviation(square root of the variance) of px 6+=le (gl raising operator 6-=1g (el lowering operator =(6x,Gy,) Pauli(two-dimensional)spin matrices wave component of the spin ket of the object system time interval,interaction time between two or more systems 产1() decoherence time angle,spherical coordinate angle operator eigenket of the angle operator 1p）.p1 state vectors p传) eigenfunctions of the observable with eigenvector) 9) plane waves c spherical waves Pp(x) momentum eigenfunctions in the position representation po（x) position eigenfunctions in the position representation Ppo(Px) momentum eigenfunctions in the momentum representation Px(px) position eigenfunctions in the momentum representa E(x) scalar product(x) 4p(E) scalar product(传|nl flux of electric current magnetic flux generic ket for compound systems X()=∫dFx)e classical characteristic function of a random variables
xxix Symbols ˆ ρ˜ mixed density matrix ρˆf density matrix for the final state of a system ρˆi density matrix for the initial state of a system !ˆ j reduced density matrix of the j-th subsystem ρˆSA density matrix of the system plus apparatus ρˆSAE density matrix of the system plus apparatus plus environment σ2 x = xˆ2 − xˆ 2 variance of xˆ σx standard deviation (square root of the variance) of xˆ σ2 p = pˆ2 x − pˆx 2 variance of pˆx σp standard deviation (square root of the variance) of pˆx σˆ+ = |e g | raising operator σˆ− = |g e | lowering operator σˆ = (σˆx , σˆ y , σˆz) Pauli (two-dimensional) spin matrices ς(s) wave component of the spin |ς ket of the object system τ time interval, interaction time between two or more systems τd γ −1 λT x 2 decoherence time φ angle, spherical coordinate φˆ angle operator |φ eigenket of the angle operator |ϕ, ϕ state vectors ϕ(ξ ) eigenfunctions of the observable with eigenvector | ξ ϕk (x) plane waves ϕk(r) spherical waves ϕp(x) momentum eigenfunctions in the position representation ϕx0 (x) position eigenfunctions in the position representation ϕ˜ p0 (px ) momentum eigenfunctions in the momentum representation ϕ˜x (px ) position eigenfunctions in the momentum representation ϕξ (x) scalar product x | ξ ϕη(ξ ) scalar product ξ | η % flux of electric current %M magnetic flux |% generic ket for compound systems χξ (η) = dF(x)eıηx classical characteristic function of a random variable ξ
XXX Symbols dhfgorioa characteristic function xw(n.)=ex() Wigner characteristic function 1),） state vectors |t》 time-evolved or time-dependent state vector Eigenket of energy corresponding to eigenvalue E(in the continuous case) 1Ψe) quantum state of the electromagnetic field 1山.) n-th stationary state H state vector in the Heisenberg picture |1 state vector in the Dirac picture I山)s state vector in the Schrodinger picture (x),(r) wave functions in the position representation (,(传) (px),(p) Fourier transform of the wave functions (r,s) wave function with a spinor component eigenfunctions ofin spherical coordinates x,p(r,(r,吹(r) momentum eigenfunctions in the positior representation Ve(x) energy eigenfucntion in the position represer ation Vs.VA symmetric and antisymmetric wavefucntions, respectively 1业) )SA ket desc ing an objects system plus apparatu compound system )sM ket describing an objects system plus meter compound system |Ψ)sAE ket describing an objects system plus apparatu plus environment compound system Ψ(x),业(r) wave function of a compound system @=2Tv angular frequency ®jk ratio between energy levels Ek-Ej and h 2 space Other Symbols Nabla operator 1 scalar product (j.j2,mi,m2l j,m) Clebsch-Gordan coefficient external product ) mean value
xxx Symbols χ(η, η∗) = e|η| 2 d2αeηα∗−αη∗ Q(α, α∗) characteristic function χW (η, η∗) = e− 1 2 |η| 2 χ(η, η∗) Wigner characteristic function |ψ, ψ state vectors |ψ(t) time-evolved or time-dependent state vector ψE Eigenket of energy corresponding to eigenvalue E (in the continuous case) |&F quantum state of the electromagnetic field |ψn n-th stationary state |ψ H state vector in the Heisenberg picture |ψI state vector in the Dirac picture |ψ S state vector in the Schrödinger picture ψ(x), ψ(r) wave functions in the position representation ψ(η), ψ(ξ ) wave functions of two arbitrary continuous observables, η and ξ , respectively ψ˜ (px ), ψ˜ (p) Fourier transform of the wave functions ψ(r,s) wave function with a spinor component ψ(r, θ, φ) eigenfunctions of ˆlz in spherical coordinates ψp(x), ψp(r), ψk (x), ψk(r) momentum eigenfunctions in the position representation ψE (x) energy eigenfucntion in the position representation ψS, ψA symmetric and antisymmetric wavefucntions, respectively |& ket of a compund system |&SA ket describing an objects system plus apparatus compound system |&SM ket describing an objects system plus meter compound system |&SAE ket describing an objects system plus apparatus plus environment compound system &(x), &(r) wave function of a compound system ω = 2πν angular frequency ωB = eB m electron cyclotron frequency ωjk ratio between energy levels Ek − E j and h¯ ' space Other Symbols ∇ Nabla operator · | · · scalar product j1, j2, m1, m2 | j, m Clebsch–Gordan coefficient |· ·| external product · mean value
xxxi Symbols Tr(O) trace of the operator direct product ⊕ direct sum for all. 3 there is at least one...such that a∈X the element a pertains to the set X XCY X is a proper subset of Y a is sufficient condition of b inclusive disjunction(OR) conjunction(AND) a→b a maps to b tends to. 10),11) arbitrary basis for a two-level system.qubits 11),12）,13),140 set of eigenstates of a path observable 10)=10,0,0) vacuum state arbitrary basis for a two-level system eigenstate of the spin observable (in the z-direction) 1) state of horizontal polarization state of vertical polarization state of 135 polarization Ie.)e living-and dead-cat states,respectively 小-= commutator anti comm ator Poisson brackets a= partial derivatives = with j=x.y
xxxi Symbols Tr(Oˆ) trace of the operator Oˆ ⊗ direct product ⊕ direct sum ∀ for all . . . ∃ there is at least one . . . such that a ∈ X the element a pertains to the set X X ⊂ Y X is a proper subset of Y a ⇒ b a is sufficient condition of b ∨ inclusive disjunction (OR) ∧ conjunction (AND) a → b a maps to b → tends to . . . |0, |1 arbitrary basis for a two-level system, qubits |1, |2, |3, |4 set of eigenstates of a path observable |0 = |0, 0, 0 vacuum state |↑, |↓ arbitrary basis for a two-level system, eigenstates of the spin observable (in the z-direction) |↔ state of horizontal polarization | state of vertical polarization |! state of 45◦ polarization |" state of 135◦ polarization |* c , |+ c living- and dead-cat states, respectively [·, ··]− = [·, ··] commutator [·, ··]+ anticommutator {·, ··} Poisson brackets ∂t = ∂ ∂t partial derivatives ∂ j = ∂ ∂ j , with j = x, y,z
Abbreviations AB Aharonov-Bohm BS beam splitter chapter CH Clauser and Hore CHSH Clauser,Horne,Shimony,and Holt Cor. cw continuous wave Def. definition EPR Finstein Podoloski and Rosen EPRB Einstein,Podoloski,Rosen,and Bohm GHSZ Greenberger,Horne,Shimony,and Zeilinger GHZ Greenberger.,Horne,and Zeilinger if and only if HV hidden variable LASER light amplification by stimulated emission of radiation LCAO linear combination of atomic orbitals left-hand side MWI many world interpretatior p. page PBS polarization beam splitter POSet partially ordered set Post postulate POVM positive operator valued measure Pr. principle Prob problem PVM projector valued measure rhs right-hand side Sec. section SGM Stern-Gerlach magnet spontaneous parametric down conversion SOUID superconducting quantum interference device Subsec.subsection Tab table theorem VBM valence bond method
Abbreviations AB Aharonov–Bohm BS beam splitter Ch. chapter CH Clauser and Horne CHSH Clauser, Horne, Shimony, and Holt Cor. corollary cw continuous wave Def. definition EPR Einstein, Podoloski, and Rosen EPRB Einstein, Podoloski, Rosen, and Bohm Fig. figure GHSZ Greenberger, Horne, Shimony, and Zeilinger GHZ Greenberger, Horne, and Zeilinger iff if and only if HV hidden variable LASER light amplification by stimulated emission of radiation LCAO linear combination of atomic orbitals lhs left-hand side MWI many world interpretation p. page PBS polarization beam splitter POSet partially ordered set Post. postulate POVM positive operator valued measure Pr. principle Prob. problem PVM projector valued measure rhs right-hand side Sec. section SGM Stern–Gerlach magnet SPDC spontaneous parametric down conversion SQUID superconducting quantum interference device Subsec. subsection Tab. table Th. theorem VBM valence bond method