Pauli equation for joint tomographic probability distribution of spin 1/2 particle
aa r X i v : . [ qu a n t - ph ] D ec Pauli equation for joint tomographic probabilitydistribution of spin 1/2 particle
Ya. A. Korennoy, V. I. Man’ko
P.N. Lebedev Physics Institute,Leninskii prospect 53, 119991, Moscow, Russia
Abstract
The positive vector optical tomogram fully describing the quantum state of spin 1/2 particle withoutany redundancy is introduced. Reciprocally the vector symplectic tomogram and vector quasidistri-butions ~W ( q , p ), ~Q ( q , p ), ~P ( ~α ) are introduced. The evolution equations for proposed vector opticaland symplectic tomograms and vector quasidistributions for arbitrary Hamiltonian are obtained. Thequantum system of charged spin 1/2 particle in arbitrary electro-magnetic field is considered in pro-posed representations and evolution equations which are analogs of Pauli equation are obtained. Thepropagator of evolution equation in the case of homogeneous and stationary magnetic field in Landaugauge is found and the evolution of initial entangled superposition of lower Landau levels in the vec-tor optical representation is considered. The system of linear quantum oscillator with spin in vectoroptical tomography representation is considered and the evolution of initial entangled superpositionof two lower Fock states and spin-up, spin-down states is studied in this representation. Keywords:
Pauli equation, evolution equation, quantum tomography, optical tomogram of quantumstate, vector-portrait of state, spin tomogram, tomographic probability, Landau levels.
The tomographic approach [1, 2] to the quantum state of a system has allowed one to establish a mapbetween the density operator (or any its representation) and a set of probability distributions, oftencalled ‘quantum tomograms’. The latter have all the characteristics of classical probabilities; they arenon-negative, measurable and normalized.Based on this connection, a classical-like description of quantum dynamics by means of ‘symplectictomography’ has been formulated [3, 4], providing a bridge between classical and quantum worlds. Thetomographic distribution for rotated spin variables has been constructed in [5], and the same approachhas been followed in [6]. Different aspects of classical-like description using tomographic probabilitieswere given in [7, 8, 9]. 1he main deficiency of the proposed spin tomograms is a redundancy of information. Attempts toreduce or avoid such a redundancy were made in [10, 11, 12, 13, 14, 15, 16, 17]. The spin tomographywas also studied in [18, 19] and in other papers.The tomographic formulation of quantum evolution equation was suggested in [20] for symplectictomograms. For optical tomograms it was given in [21, 23]. The first attempt of foundation of thePauli equation in tomographic representation was done in [24]. Using the version of spin tomogram withredundancy of information the authors obtained a complicated evolution equation.The aim of our work is the consideration of the special case of spin 1/2 particle quantum statetomography without redundancy of information, constructing the joint vector distribution for spacecoordinates and spin projections, and finally deriving the evolution equation for such distribution, whichwould be an analogue of the Pauli equation. It would also be a simplification of approach attempted in[24].The paper is organized as follows. In Sec. 2 we give basic formulas of tomographic representation ofquantum mechanics and the evolution equation for optical and symplectic tomogram of nonrelativisticspinless quantum system with arbitrary Hamiltonian. In Sec. 3 we introduce a positive four-componentvector probability description of spin 1/2 particle and give the evolution equation for such a vector-portrait of quantum state with arbitrary Hamiltonian. In Sec. 4 charged spin 1/2 particle in arbitraryelectro-magnetic field is considered in proposed representations and evolution equations which are analogsof Pauli equation are obtained. In Sec. 5 vector quasidistributions ~W ( q , p ), ~Q ( q , p ), ~P ( ~α ) are introducedand analogs of Pauli equation for ~W ( q , p ) and ~Q ( q , p ) are obtained. In Sec. 6 the propagator of evolutionequation in the case of homogeneous and stationary magnetic field in Landau gauge is found and theevolution of initial entangled superposition of lower Landau levels in the vector optical representation isconsidered. In Sec. 7 the system of linear quantum oscillator with spin in vector optical tomographyrepresentation is considered and the evolution of initial entangled superposition of two lower Fock statesand spin-up, spin-down states is studied. The conclusion and prospects are presented in Sec. 8. Let us review the constructions of the optical and symplectic tomograms for spinless systems. Therelationships between the density operator ˆ ρ and the optical tomogram w ( ~X , ~θ ) of the system in theinvariant form [25] are written as follows w ( ~X , ~θ ) = Tr { ˆ ρ ˆ U w ( ~X, ~θ ) } , ˆ ρ = Z w ( ~X , ~θ ) ˆ D w ( ~X, ~θ )d n X d n θ, (1)2here dequantizer ˆ U w ( ~X, ~θ ) and quantizer ˆ D w ( ~X, ~θ ) operators equal respectivelyˆ U w ( ~X, ~θ ) = | ~X, ~θ ih ~X, ~θ | = ~ n/ n Y σ =1 ( m σ ω σ ) − n/ δ (cid:18) X σ ˆ1 − ˆ q σ cos θ σ − ˆ p σ sin θ σ m σ ω σ (cid:19) , (2)ˆ D w ( ~X, ~θ ) = √ ~ π ! n Z n Y σ =1 | η |√ m σ ω σ exp (cid:26) iη σ (cid:18) X σ − ˆ q σ cos θ σ − ˆ p σ sin θ σ m σ ω σ (cid:19)(cid:27) d n η, (3)where | ~X, ~θ i is an eigenfunction of the operator ~ ˆ X ( ~θ ) with components ˆ X σ = ˆ q σ cos θ σ + ˆ p σ sin θ σ corresponding to the eigenvalue ~X . Notion of quantizer and dequantizer is related to star productquantization schemes (see recent review [22]).The von-Neumann equation without interaction with the environment i ~ ∂∂t ˆ ρ = [ ˆ H, ˆ ρ ] (4)in the optical tomography representation has the form [23] ∂ t w ( ~X, ~θ, t ) = 2 ~ Z Im h Tr n ˆ H ˆ D ( ~X ′ , ~θ ′ ) ˆ U ( ~X, ~θ ) oi w ( ~X ′ , ~θ ′ , t )d n X ′ d n θ ′ , (5)and for a large class of Hamiltonians ˆ H (ˆ p , ˆ q , t ), when the Hamiltonian is an analytic function of positionˆ q and momentum ˆ p components, it can be written as follows ∂ t w ( ~X, ~θ, t ) = ˜ M w ( ~X, ~θ, t ) w ( ~X , ~θ, t ) . (6)where the operator ˆ M ( ~X, ~θ, t ) is obtained form the Hamiltonianˆ M w ( ~X, ~θ, t ) = 2 ~ Im ˆ H (cid:16) [ˆ p ] w ( ~X, ~θ ) , [ˆ q ] w ( ~X, ~θ ) , t (cid:17) , which is an operator depending on two operators of position ˜ q and momentum ˜ p in the tomographicrepresentation [ˆ q σ ] w ( ~X, ~θ ) = sin θ σ ∂∂θ σ (cid:20) ∂∂X σ (cid:21) − + X σ cos θ σ + i ~ sin θ σ m σ ω σ ∂∂X σ , (7)[ˆ p σ ] w ( ~X, ~θ ) = mω σ − cos θ σ (cid:20) ∂∂X σ (cid:21) − ∂∂θ σ + X σ sin θ σ ! − i ~ θ σ ∂∂X σ . (8)Similarly, for symplectic tomogram M ( X, µ, ν, t ) one can be written M ( ~X, ~µ, ~ν, t ) = Tr { ˆ ρ ˆ U M ( ~X, ~µ, ~ν ) } , ˆ ρ = Z M ( ~X, ~µ, ~ν, t ) ˆ D M ( ~X, ~µ, ~ν )d n X d n µ d n ν, (9)where dequantizer ˆ U M ( ~X, ~µ, ~ν ) and quantizer ˆ D M ( ~X, ~µ, ~ν ) operators are respectively equalˆ U M ( ~X, ~µ, ~ν ) = | ~X, ~µ, ~ν ih ~X, ~µ, ~ν | = ~ n/ n Y σ =1 ( m σ ω σ ) − n/ δ ( X σ ˆ1 − ˆ q σ µ σ − ˆ p σ ν σ ) , (10)3 D M ( ~X, ~µ, ~ν ) = 1(2 π √ ~ ) n n Y σ =1 ( m σ ω σ ) n/ exp (cid:26) i r m σ ω σ ~ ( X σ − ˆ q σ µ σ − ˆ p σ ν σ ) (cid:27) , (11)where | ~X, ~µ, ~ν i is an eigenfunction of the operator ~ ˆ X ( ~µ, ~ν ) with components ˆ X σ = µ σ ˆ q σ + ν σ ˆ p σ corre-sponding to the eigenvalue ~X .For the same Hamiltonians the evolution equation for the symplectic tomogram [23] ∂ t M ( ~X, ~µ, ~ν, t ) = ˆ M M ( ~X, ~µ, ~ν, t ) M ( ~X, ~µ, ~ν, t ) , (12)with notation ˆ M M ( ~X, ~µ, ~ν, t ) = 2 ~ Im ˆ H (cid:16) [ˆ p ] M ( ~X, ~µ, ~ν ) , [ˆ q ] M ( ~X~µ, ~ν ) , t (cid:17) , where [ˆ q ] M and [ˆ p M ] are operators of positions and momentums in the symplectic representation[ˆ p σ ] M = − (cid:20) ∂∂X σ (cid:21) − ∂∂ν σ − i µ σ ~ ∂∂X σ ! , [ˆ q σ ] M = − (cid:20) ∂∂X σ (cid:21) − ∂∂µ σ + i ν σ ~ ∂∂X σ ! . (13) As known, that pure states of quantum spin 1/2 particle are described by two-component spinor wavefunctions ( ψ , ψ ), and mixed states can be described by density matrixes ˆ ρ ij , where i, j = 1 ,
2. In thecase of pure state ˆ ρ = ψ ∗ ψ ψ ∗ ψ ψ ∗ ψ ψ ∗ ψ . (14)The density matrix satisfy Pauli equation (4) in the von-Neumann form with the 2 × w ( ~X, ~θ, t ) = Tr n ˆ ρ ( t ) ˆ U w ( ~X, ~θ ) ⊗ | s = 1 / ih s = 1 / | o ,w ( ~X, ~θ, t ) = Tr n ˆ ρ ( t ) ˆ U w ( ~X, ~θ ) ⊗ | s = 1 / ih s = 1 / | o ,w ( ~X, ~θ, t ) = Tr n ˆ ρ ( t ) ˆ U w ( ~X, ~θ ) ⊗ | s = 1 / ih s = 1 / | o ,w ( ~X, ~θ, t ) = Tr n ˆ ρ ( t ) ˆ U w ( ~X, ~θ ) ⊗ | s = − / ih s = − / | o , (15)where ˆ U w ( ~X, ~θ ) is a spinless dequantizer operator (2) and | s j = ± / i is an eigenfunction of the projectionof spin operator to the direction q j corresponding to the eigenvalue ± /
2. In more compact form formula415) can be written as follows ~w ( ~X, ~θ, t ) = Tr n ˆ ρ ( t ) ~ ˆ U w ( ~X, ~θ ) o , (16)where ~w ( ~X, ~θ, t ) is a four component vector of probability distributions and dequantizer operator ~ ˆ U w ( ~X, ~θ )has the form ~ ˆ U w ( ~X, ~θ ) = ˆ U w ( ~X, ~θ ) ⊗ ~ ˆ U , (17)where ~ ˆ U is a four-component vector of 2 × ~ ˆ U = n ˆ U j ( kl ) o = (cid:18) (cid:20) (cid:21) , (cid:20) − ii (cid:21) , (cid:20) (cid:21) , (cid:20) (cid:21)(cid:19) . (18)Here the first index j = 1 , , , kl )are the indexes of 2 × ~w → ˆ ρ can be written in terms of quantizeroperator [ ~ ˆ D jk ] w ( ~X, ~θ ) as followsˆ ρ jk ( t ) = Z [ ~ ˆ D jk ] w ( ~X, ~θ ) ~w ( ~X, ~θ, t ) d X d θ , j, k = 1 , . (19)Quantizer operator [ ~ ˆ D jk ] w ( ~X, ~θ ) in these notations is defined as a direct product[ ~ ˆ D jk ] w ( ~X, ~θ ) = ˆ D w ( ~X, ~θ ) ⊗ ~ ˆ D jk , (20)where ˆ D w ( ~X, ~θ ) is a spinless quantizer operator (3) and ~ ˆ D is a 2 × ~ ˆ D = n ˆ D ( jk ) l o = (0 , , , (cid:0) , − i, − i , − i (cid:1)(cid:0) , i, − i , − i (cid:1) (0 , , , , (21)where ( jk ) are the indexes of 2 × l = 1 , , , w ( ~X, ~θ ), w ( ~X, ~θ ), w ( ~X, ~θ ) are the probability distributions of theoperator ˆ ~X ( ~θ ) at time t under the conditions that the particle has the value of spin projection equal 1 / q , q , or q directions respectively, and the function w ( ~X, ~θ ) is the probability distribution of thisoperator under the condition that it has the spin projection − / q direction. Obviously that thetwo components of the vector ~w ( ~X, ~θ, t ) are normalized by the condition Z w ( ~X, ~θ, t )d X + Z w ( ~X, ~θ, t )d X = 1 . (22)The other two components must be integrable over X and must satisfy the inequalities0 ≤ w j ( ~X, ~θ, t ) ≤ , ≤ Z w j ( ~X, ~θ, t )d X ≤ , j = 1 , . We see, that the four-component vector ~w ( ~X, ~θ ) completely define the density matrix ˆ ρ and consequentlyit contains all accessible information about the quantum state.5imilarly, for symplectic vector tomography we can write ~M ( ~X, ~µ, ~ν, t ) = Tr { ˆ ρ ~ ˆ U M ( ~X, ~µ, ~ν ) } , ˆ ρ jk = Z [ ~ ˆ D jk ] M ( ~X, ~µ, ~ν ) ~M ( ~X, ~µ, ~ν, t )d X d µ d ν, (23)where dequantizer ~ ˆ U M ( ~X, ~µ, ~ν ) and quantizer [ ~ ˆ D jk ] M ( ~X, ~µ, ~ν ) are defined by the similar formulas (17)and (20) as in the case of optical tomography but the spinless optical dequantizer and quantizer must bereplaced with the corresponding symplectic operators (10) and (11) ~ ˆ U M ( ~X, ~θ ) = ˆ U M ( ~X, ~θ ) ⊗ ~ ˆ U , [ ~ ˆ D jk ] M ( ~X, ~θ ) = ˆ D M ( ~X, ~θ ) ⊗ ~ ˆ D jk . (24)Generalizing equation (5) to the case of spin particles we can write the evolution equation for thevector tomogram in spin optical tomography representation ∂ t ~w j ( ~X, ~θ, t ) = 2 ~ X k =1 Z Im Tr X l,m =1 [ ˆ U j ( lm ) ] w ( ~X, ~θ ) ˆ H [ ˆ D ( ml ) k ] w ( ~X ′ , ~θ ′ ) ~w k ( ~X ′ , ~θ ′ , t )d n X ′ d n θ ′ , (25)or in spin symplectic tomography representation ∂ t ~w j ( ~X, ~µ, ~ν, t ) =2 ~ X k =1 Z Im Tr X l,m =1 [ ˆ U j ( lm ) ] M ( ~X, ~µ, ~ν ) ˆ H [ ˆ D ( ml ) k ] M ( ~X ′ , ~µ ′ , ~ν ′ ) ~w k ( ~X ′ , ~µ ′ , ~ν ′ , t )d n X ′ d n µ ′ d n ν ′ . (26) Let’s consider the quantum system of a charged spin 1/2 particle with charge e , mass m in electro-magnetic field with potentials A ( q , t ), ϕ ( q , t ). As well known, the Hamiltonian of this system has theform ˆ H = 12 m (cid:16) ˆ p − ec A (cid:17) + eϕ − κ s ˆ s H = ˆ H − κ s ˆ s H , (27)where ˆ H is an independent on spin part of Hamiltonian, H = rot A is a magnetic field, and κ is amagnetic moment of the particle.Making the transformation of the Pauli equation (4) with the Hamiltonian (27) with the help of trans-forms (16) and (19) we find the evolution equation of the four-component function ~w ( ~X, ~θ, t ) dependenton time t and three component vectors ~X and ~θ∂ t ~w ( ~X, ~θ, t ) = ˆ M w ( ~X, ~θ, t ) ~w ( ~X, ~θ, t ) + ˆ S w ( ~X, ~θ, t ) ~w ( ~X, ~θ, t ) , (28)where ˆ M w ( ~X, ~θ, t ) = 2 ~ Im ˆ H (cid:16) [ˆ q ] w ( ~X, ~θ ) , [ˆ p ] w ( ~X, ~θ ) , t (cid:17)
6s an operator depending on two operators of position [ˆ q ] w and momentum [ˆ p ] w defined by (7) and (8) inthe tomographic representation, and ˆ S w ( ~X, ~θ, t ) is a 4 × A j ] w = A j (cid:16) [ˆ q ] w ( ~X, ~θ ) , t (cid:17) , ˜H j = [ ˆH j ] w = H j (cid:16) [ˆ q ] w ( ~X, ~θ ) , t (cid:17) , h ∇ q ˆ A i w = ∇ q A (cid:16) q → [ˆ q ] w ( ~X, ~θ ) , t (cid:17) the explicit forms of ˆ M w and ˆ S w in general case of timedependent and nonhomogeneous electromagneticfield are written asˆ M w ( ~X, ~θ, t ) = X n =1 ω n (cid:20) cos θ n ∂∂θ n −
12 sin 2 θ n (cid:26) X n ∂∂X n (cid:27)(cid:21) + 2 e ~ Im [ ˆ ϕ ] w + e mc ~ Im[ ˆ A ] w − emc ~ Im h ˆ A ˆ p i w + emc Re [ ∇ q A ] w , (29)[ ˜ S ] w = − κ Im ˜H , [ ˜ S ] w = 2 κ n − Im ˜H + Re ˜H o , [ ˜ S ] w = − κ n + Re ˜H − Im ˜H + Re ˜H + Im ˜H o , [ ˜ S ] w = κ n Re ˜H + Im ˜H − Re ˜H + Im ˜H o , [ ˜ S ] w = − κ n Im ˜H + Re ˜H o , [ ˜ S ] w = − κ Im ˜H , [ ˜ S ] w = κ n Re ˜H + Im ˜H + Re ˜H − Im ˜H o , [ ˜ S ] w = κ n − Re ˜H + Im ˜H + Re ˜H + Im ˜H o , [ ˜ S ] w = − κ n Im ˜H − Re ˜H o , [ ˜ S w ] w = − κ n Re ˜H + Im ˜H o , [ ˜ S ] w = κ n Re ˜H + Im ˜H − Re ˜H + Im ˜H − o , [ ˜ S ] w = κ n Re ˜H + Im ˜H − Re ˜H + Im ˜H o , [ ˜ S ] w = − κ n Im ˜H + Re ˜H o , [ ˜ S ] w = 2 κ n Re ˜H − Im ˜H o , [ ˜ S ] w = κ n − Re ˜H + Im ˜H + Re ˜H + Im ˜H o , [ ˜ S ] w = κ n − Re ˜H + Im ˜H + Re ˜H + Im ˜H + 2 Im ˜H o . (30)Making the similar procedure with symplectic vector tomography (23) we can find the evolutionequation ∂ t ~M ( ~X, ~µ, ~ν, t ) = ˆ M M ( ~X, ~µ, ~ν, t ) ~M ( ~X, ~µ, ~ν, t ) + ˆ S M ( ~X, ~µ, ~ν, t ) ~M ( ~X, ~µ, ~ν, t ) , (31)where operator ˆ M M ( ~X, ~µ, ~ν, t ) corresponds to spinless part ˆ H of the Hamiltonian (27)ˆ M M ( ~X, ~µ, ~ν, t ) = 2 ~ Im ˆ H (cid:16) [ˆ p ] M ( ~X, ~µ, ~ν ) , [ˆ q ] M ( ~X~µ, ~ν ) , t (cid:17) = ~µ ∂∂~ν + 2 e ~ Im [ ˆ ϕ ] M + e mc ~ Im[ ˆ A ] M − emc ~ Im h ˆ A ˆ p i M + emc Re [ ∇ q A ] M , (32)where [ ˆ A j ] M = A j (cid:16) [ˆ q ] M ( ~X, ~µ, ~ν ) , t (cid:17) , [ ˆ ϕ ] M = ϕ (cid:16) [ˆ q ] M ( ~X, ~µ, ~ν ) , t (cid:17) , ∇ q A ] M = ∇ q A (cid:16) q → [ˆ q ] M ( ~X, ~µ, ~ν ) , t (cid:17) , and [ˆ q ] M , [ˆ p ] M are position and momentum operators (13) in the symplectic representation. The 4 × S M ( ~X, ~µ.~ν, t ) is defined by the similar formulae (30) where the operators of componentsof the magnetic field ˜H j must be replaced with corresponding operators in the symplectic tomographyrepresentation [ ˆH j ] M = H j (cid:16) [ˆ q ] M ( ~X, ~µ, ~ν ) , t (cid:17) . Quasiprobability distributions such as Wigner function or Husimi function are powerful tools of descrip-tion of quantum systems. For spinless particles their definitions from the density matrix can be writtenas follows W ( q , p , t ) = Tr n ˆ ρ ( t ) ˆ U W ( q , p ) o , (33) Q ( q , p , t ) = Q ( ~α ) = h ~α | ˆ ρ ( t ) | ~α i = Tr n ˆ ρ ( t ) ˆ U Q ( q , p ) o , (34)with corresponding ”dequantizers” having the formsˆ U W ( q , p ) = 1(2 π ) N Z | q − u / i exp( − i pu / ~ ) h q + u / | d N u, (35)ˆ U Q ( q , p ) = | ~α ih ~α | , ~α = 1 √ (cid:18)r mω ~ q + i √ ~ mω p (cid:19) , (36)where | q i is an eigenvalue of the position operator, | ~α i is a coherent state. Inverse maps W → ˆ ρ and Q → ˆ ρ are expressed with corresponding ”quantizers”ˆ ρ = Z ˆ D W ( q , p ) W ( q , p ) d q d p = Z ˆ D Q ( q , p ) Q ( q , p ) d q d p (37)where quantizers ˆ D W and ˆ D Q are given by (see [26, 27, 28])ˆ D W ( q , p ) = 2 N Z d N u exp(2 i pu / ~ ) | q + u ih q − u | , (38)ˆ D Q ( q , p ) = (cid:16) mωπ ~ (cid:17) / Z d x d y n | x ih y | exp (cid:16) mω ~ ( x − y ) (cid:17) × exp " − mω ~ (cid:18) q − x + y (cid:19) − mω ~ ( x − y ) + i ~ p ( x − y ) × Y σ =1 " ∞ X n =0 ( − n n !2 n H n (cid:18)r mω ~ q σ − mω ~ ( x σ + y σ ) (cid:19) (39)8uch definitions provide that the quasidistributions in spinless case are real functions contrary to densitymatrices, whose nondiagonal elements may be complex. If the particle has spin, the density matrixˆ ρ jk additionally depend on spin indexes, and usually in literature many authors make generalizations ofdefinitions (33), (34) handling the trace operations as a partial trace over all of the variables exceptingspin indexes. In such definitions Wigner function W jk ( q , p , t ) and Husimi function Qjk ( q , p , t ) become(2 s + 1) × (2 s + 1) matrices dependent on position and momentum. But their nondiagonal elements overthe spin indexes are not surely real. So, the main advantage of such quasidistributions with respect todensity matrix disappears.To decide this problem let us expand our approach to the quasidistributions and define four-componentWigner ~W ( q , p , t ) and Husimi ~Q ( q , p , t ) vector-functions as follows ~W ( q , p , t ) = Tr n ˆ ρ ( t ) ~ ˆ U W ( q , p ) o , (40) ~Q ( q , p , t ) = Tr n ˆ ρ ( t ) ~ ˆ U Q ( q , p ) o , (41)where dequatizers ~ ˆ U W and ~ ˆ U Q are defined by the same formulae as (17) with replacement of ~ ˆ U ( ~X, ~θ ) byˆ U W ( q , p ) or ˆ U Q ( q , p ) ~ ˆ U W ( ~X, ~θ ) = ˆ U W ( ~X, ~θ ) ⊗ ~ ˆ U , ~ ˆ U Q ( ~X, ~θ ) = ˆ U Q ( ~X, ~θ ) ⊗ ~ ˆ U . (42)Such definitions guarantee that all of the components of ~W ( q , p , t ) and ~Q ( q , p , t ) are real, more over,all of the components of ~Q ( q , p , t ) are nonnegative. Here W j ( q , p , t ) and Q j ( q , p , t ) are components ofWigner and Husimi vector quasiprobability corresponding definite spin projection along q , q , or q direction.Inverse mappings of (40) and (41) are obviousˆ ρ jk ( t ) = Z [ ~ ˆ D jk ] W ( q , p ) ~W ( q , p , t )d q d p = Z [ ~ ˆ D jk ] Q ( q , p ) ~Q ( q , p , t )d q d p , (43)where quantizers [ ~ ˆ D jk ] W and [ ~ ˆ D jk ] Q are defined by the similar formulas as in the case of optical tomog-raphy (17), (20) but the spinless optical quantizers must be replaced with the corresponding operators(38) and (39) [ ~ ˆ D jk ] W ( q , p ) = ˆ D W ( q , p ) ⊗ ~ ˆ D jk , [ ~ ˆ D jk ] Q ( q , p ) = ˆ D Q ( q , p ) ⊗ ~ ˆ D jk . (44)Let us give the expression for Wigner function in terms of the Husimi function ~W ( q , p ) = exp (cid:18) − ~ mω △ q − mω ~ △ p (cid:19) ~Q ( q , p ) , (45)where △ q and △ p are Laplace operators in 3D spaces { q j } and { p j } . This formula is a trivial general-ization of the corresponding formula [29] for spinless W and Q .9ikewise we can introduce the vector Glauber-Sudarshan P-function [30, 31] ~P ( ~α, t ) ~P ( ~α, t ) = Tr n ˆ ρ ( t ) ~ ˆ U P ( ~α ) o , ˆ ρ ( t ) = Z [ ~ ˆ D ] P ( ~α ) ~P ( α, t )d n α, where ~ ˆ U P ( ~α ) = e | ~α | π n Z | ~β ih β | e | ~β | − ~β ∗ ~α + ~β~α ∗ d n β ! ⊗ ~ ˆ U , [ ~ ˆ D ] P ( ~α ) = | α ih α | ⊗ ~ ˆ D . In previous sections we have found the evolution equation of charged spin 1/2 particle in electro-magnetic field in optical and symplectic tomographic representations. Making similar calculation we canobtain such evolution equation for our vector Wigner function, which will be a generalization of the Moyalequation [32] ∂∂t ~W ( q , p , t ) = (cid:20) − p m ∂∂ q + 2 e ~ Im ϕ (cid:18) q + i ~ ∂∂ p , t (cid:19) + e mc ~ Im A (cid:18) q + i ~ ∂∂ p , t (cid:19) + − emc ~ Im (cid:26) A (cid:18) q + i ~ ∂∂ p , t (cid:19) (cid:18) p − i ~ ∂∂ q (cid:19)(cid:27) + emc Re ∇ q A (cid:18) q → q + i ~ ∂∂ p , t (cid:19) + ˆ S W ( q , p , t ) (cid:21) ~W ( q , p , t ) , (46)where 4 × S W ( q , p , t ) is defined by the same formulae (30) where the operators ofcomponents of the magnetic field ˜H j must be replaced with corresponding operators in the Wignerrepresentation H j (cid:16) q + i ~ ∂∂ p , t (cid:17) .The corresponding equation for Husimi function is obtained from (46) with the help of expression(45) (see also [28]). For simplicity we choose the system of measurements so that m = ω = ~ = 1 ∂∂t ~Q ( q , p , t ) = (cid:20) − p ∂∂ q − ∂∂ q ∂∂ p + 2 e ~ Im ϕ (cid:18) q + 12 ∂∂ q + i ∂∂ p , t (cid:19) + e c Im A (cid:18) q + 12 ∂∂ q + i ∂∂ p , t (cid:19) − ec Im (cid:26) A (cid:18) q + 12 ∂∂ q + i ∂∂ p , t (cid:19) (cid:18) p + 12 ∂∂ p − i ∂∂ q (cid:19)(cid:27) + ec Re ∇ q A (cid:18) q → q + 12 ∂∂ q + i ∂∂ p , t (cid:19) + ˆ S Q ( q , p , t ) (cid:21) ~Q ( q , p , t ) , (47)where 4 × S Q ( q , p , t ) is defined by (30) in which components of the magnetic field ˜H j are replaced with H j (cid:16) q + ∂∂ q + i ∂∂ p , t (cid:17) . Choose the vector and scalar potentials as follows (Landau gauge) A = ( − q H , , , ϕ = 0 , H = (0 , , H) , (48)10uppose ω = ω = ω = | ω | , where ω = e H mc , and choose the system of measurements so that | ω | = m = ~ = 1. In these units ω = +1 when the charge is positive and ω = − ω = κ H ~ . For electrons ω = ω , but for other particlesthese two frequencies may be different.Then the Hamiltonian will have the formˆ H = 12 ˆ p + 12 ˆ p + 12 ˆ p + 12 ˆ q + ω ˆ p ˆ q − ω (cid:18) − (cid:19) . (49)Using the general formulae (28) – (30) we find the evolution equation ∂∂t ~w ( ~X, ~θ, t ) = X σ =1 , (cid:18) cos θ σ ∂∂θ σ −
12 sin 2 θ σ (cid:26) X σ ∂∂X σ (cid:27)(cid:19) + ∂∂θ − ω cos θ (cid:20) ∂∂X (cid:21) − − X sin θ ! sin θ ∂∂X − ω sin θ (cid:20) ∂∂X (cid:21) − + X cos θ ! cos θ ∂∂X + ˆ S w ~w ( ~X, ~θ, t ) , (50)where upper signs correspond to positive charge and lower signs correspond to negative charge, and thematrix ˆ S w is given by the expressionˆ S w = ω − / − / − / /
20 0 0 00 0 0 0 , (51)or in symplectic tomography representation ∂∂t ~M ( ~X, ~µ, ~ν ) = " ~µ ∂∂~ν − ων (cid:20) ∂∂X (cid:21) − ∂∂X ∂∂ν + ωµ (cid:20) ∂∂X (cid:21) − ∂∂X ∂∂ν + ˆ S M ~M ( ~X, ~µ, ~ν ) . (52)In Wigner representation this equation has the form ∂∂t ~W ( q , p , t ) = (cid:20) − p ∂∂ q + q ∂∂p + ωp ∂∂p − ωq ∂∂q + ˆ S w (cid:21) ~W ( q , p , t ) . (53)It is obvious that for homogeneous magnetic field ˆ S M = ˆ S w = ˆ S W = ˆ S Q .If we integrate equation (50) over d X and introduce the notation R ~w ( ~X, ~θ, t )d X = ~P ( t ), thenequation (50) take the form ∂ t ~P ( t ) = ˆ S w ~P ( t ) . (54)11his equation corresponds to the case when we are interesting in only spin dynamic. After some cal-culations we get the propagator of this equation and, consequently, the solution ~P ( t ) = ˆΠ s ( t ) ~P (0) forarbitrary initial condition ~P (0)ˆΠ s ( t ) = cos ω t sin ω t (1 − cos ω t − sin ω t ) / − cos ω t − sin ω t ) / − sin ω t cos ω t (1 − cos ω t + sin ω t ) / − cos ω t + sin ω t ) /
20 0 1 00 0 0 1 , (55)We can see, that the probability of finding the particle in the state with the spin projection ± / q direction remains constant during evolution.The spinless part of the propagator for eq. (50) or (52) corresponds to the free motion along q direction and to the evolution of ordinary quadratic system with respect to q and q degrees of freedom.For free motion the propagator was found in [33] for optical tomography and in [20] for symplectic one[Π f ] w ( X , θ , X ′ , θ ′ , t ) = δ ( X cos θ ′ − X ′ cos θ ) δ (cos θ ′ ( t + tan θ ) − sin θ ′ ) , (56)[Π f ] M ( X , µ , ν , X ′ , µ ′ , ν ′ ) = δ ( X − X ′ ) δ ( ν ′ − ν − µ t ) δ ( µ − µ ′ ) . (57)Free motion propagator for the Wigner function, obviously, equals[Π f ] W ( q , p , q ′ , p ′ , t ) = δ (cid:18) q − q ′ − p ′ m t (cid:19) δ ( p − p ′ ) . (58)For quadratic subsystem the propagator can be found by the method of motion integrals, or it can be ob-tained from the known propagator for the wave function by means of transformation to the correspondingrepresentation (see [33, 34, 35]). After some calculations we have[Π ] w ( X , X , θ , θ , X ′ , X ′ , θ ′ , θ ′ , t ) == δ (cid:26) cos θ tan (cid:18) ωt (cid:19) [2 cos θ ′ − ω sin θ ′ ] + sin( θ − θ ′ ) (cid:27) × δ (cid:26) cos θ tan (cid:18) ωt (cid:19) (cid:20) θ ′ + ω sin( θ − θ ′ )cos θ sin θ (cid:21) + sin( θ − θ ′ ) + ω sin θ sin( θ − θ ′ )cos θ sin θ (cid:27) × δ (cid:26) X cos θ ′ cos θ + ωX sin( θ − θ ′ )cos θ cos θ sin θ − X ′ (cid:27) δ (cid:26) X cos θ ′ cos θ − X ′ (cid:27) , (59)12r in symplectic representation[Π ] M ( X , X , µ , µ , ν , ν , X ′ , X ′ , µ ′ , µ ′ , ν ′ , ν ′ , t ) == (2 π ) − exp (cid:26) i (cid:20) − X µ ′ µ − X µ ′ µ + X ′ + X ′ − ωX µ (cid:18) µ ′ µ − ν ′ ν (cid:19)(cid:21)(cid:27) × δ (cid:26) µ tan (cid:18) ωt (cid:19) (2 µ ′ − ων ′ ) − µ ν ′ + µ ′ ν (cid:27) × δ (cid:26) µ tan (cid:18) ωt (cid:19) (cid:20) µ ′ + ω (cid:18) µ ′ µ − ν ′ ν (cid:19)(cid:21) + ων (cid:18) µ ′ µ − ν ′ ν (cid:19) − µ ν ′ + µ ′ ν (cid:27) . (60)In Wigner representation this propagator has the form[Π ] W ( q , q , p , p , q ′ , q ′ , p ′ , p ′ , t ) = m ω ~ sin ( ωt/ × δ (cid:26) mω ~ cot (cid:18) ωt (cid:19) ( q − q ′ ) − mω ~ ( q − q ′ ) − p ′ (cid:27) × δ (cid:26) mω ~ cot (cid:18) ωt (cid:19) ( q − q ′ ) + mω ~ ( q − q ′ ) − p ′ (cid:27) × δ n mω ~ ( q − q ′ ) + p − p ′ o δ (cid:8) p − p ′ (cid:9) . (61)The total propagator of eq.(50) equals to the product of corresponding propagators (55), (56), and (59)ˆΠ w ( ~X, ~θ, ~X ′ , ~θ ′ , t ) = ˆΠ s ⊗ [Π f ] w ⊗ [Π ] w . (62)Reciprocally for symplectic or Wigner representation we haveˆΠ M ( ~X, ~µ, ~ν, ~X ′ , ~µ ′ , ~ν ′ , t ) = ˆΠ s ⊗ [Π f ] M ⊗ [Π ] M , (63)ˆΠ W ( q , p , q ′ , p ′ , t ) = ˆΠ s ⊗ [Π f ] W ⊗ [Π ] W . (64)Let us average the evolution of the system over free motion along q direction, i.e. integrate the evolutionequation and initial condition over X and consider an initial condition which is the entangled superposi-tion of lower Landau levels of electron in Landau gauge (48) (the charge is negative and in our notations ω = ω = − a , ˆ a † and ˆ b , ˆ b † (see[36, 37]) ˆ a = (ˆ p − ˆ q − i ˆ p ) / √ , ˆ a † = (ˆ p − ˆ q + i ˆ p ) / √ , (65)ˆ b = (ˆ q − ˆ p + i ˆ p ) / √ , ˆ b † = (ˆ q − ˆ p − i ˆ p ) / √ . (66)These operators have the following properties[ˆ a, ˆ a † ] = 1 , [ˆ b, ˆ b † ] = 1 , [ˆ a, ˆ b ] = [ˆ a, ˆ b † ] = 0 , q degree of freedom is expressed in terms of ˆ a and ˆ a † asˆ H = ˆ a † ˆ a + 12 + 12 (cid:18) − (cid:19) . The Landau levels correspond to the states with wave functions | nm i = (ˆ a † ) n (ˆ b † ) m √ n ! m ! | i , where | i is a vacuum state of the systemˆ a | i = 0 , ˆ b | i = 0 , h | i = 1 , h q , q | i = 1 √ π exp (cid:18) − q − q i q q (cid:19) . The first exited state | i has the wave function h q , q | i = h q , q | ˆ a † | i = iq − q √ π exp (cid:18) − q − q i q q (cid:19) . Consider an initial condition which is the entangled superposition of lower Landau levels | Ψ(0) i = 1 √ | i ⊗ | − / i + | i ⊗ | / i ) . It corresponds to our vector optical tomogram ~w ( X , X , θ , θ ,
0) = 14 w + w + 2Re w w + w + 2Im w w w , where we introduce the designation w nmn ′ m ′ = h X , X , θ , θ | nm ih n ′ m ′ | X , X , θ , θ i . It is easy to see that the corresponding solution of evolution equation will be ~w ( X , X , θ , θ , t ) = 14 w + w + 2Re { w exp( i t ) } w + w + 2Im { w exp( i t ) } w w . Double frequency here is the result of interference of simultaneous spin rotation and cyclotron quantummotion in the plane perpendicular to the magnetic field.14
Linear harmonic oscillator with spin
As another example we consider a system with following Hamiltonian:ˆ H = 12 ( p + q ) + (cid:18) − (cid:19) . It could describe one vibrational degree of a trapped electron plus its spin [38]. The measurability oftomograms in this system was investigated in [39].The evolution equation of the vector tomogram for this system has the simple form ∂∂t ~w ( X, θ, t ) = ∂∂θ ~w ( X, θ, t ) + ˆ S w ~w ( X, θ, t ) , (67)where ˆ S w is defined by (51) with ω = −
2, and the propagator for this equation beˆΠ(
X, θ, t ) = δ ( θ − t − θ ′ ) ⊗ ˆΠ s . If we take an initial entangled state | Ψ(0) i = 1 √ | i ⊗ | − / i + | i ⊗ | / i )with initial vector optical tomogram ~w ( X, θ,
0) = 14 w + w + 2Re w w + w + 2Im w w w , where w = h X, θ | ih | X, θ i = 1 √ π e − X ,w = h X, θ | ih | X, θ i = 2 √ π X e − X ,w = h X, θ | ih | X, θ i = √ √ π Xe iθ e − X , then, the solution of equation (67) be ~w ( X, θ, t ) = 14 w + w + 2Re { w exp( i t ) } w + w + 2Im { w exp( i t ) } w w , and we can see again the addition of two frequencies: the harmonic oscillator frequency and the frequencyof spin rotation. 15 Conclusion
To resume we point out the main results of our paper. We suggested to describe the state of charged spin1/2 particle by a new four-component positive vector of joint probability distributions, that is the vectoroptical tomogram. Such approach of construction of positive vector-portrait of quantum state eliminatesthe redundancy, which is the main difficulty of schemes proposed by another authors. Reciprocally weintroduce the vector symplectic tomogram and vector quasidistributions ~W ( q , p ), ~Q ( q , p ), ~P ( ~α ).We obtained the evolution equations for such vector optical and symplectic tomograms and vectorquasidistributions for arbitrary Hamiltonian. We considered in proposed representations the quantumsystem of charged spin 1/2 particle in arbitrary electro-magnetic field and obtained evolution equations,which are analogs of Pauli equation in appropriate representations.As an example we found the propagator of evolution equation in the case of homogeneous and sta-tionary magnetic field in Landau gauge, we considered the evolution of initial entangled superposition oflower Landau levels in the vector optical representation and illustrated the addition of the frequency ofsimultaneous spin rotation and the frequency of cyclotron quantum motion in the plane perpendicularto the magnetic field.Also as an example we considered the system of linear quantum oscillator with spin in vector opticaltomography representation and studied the evolution of initial entangled superposition of two lower Fockstates and spin-up spin-down states.A possible disadvantage of the approach proposed is a relatively complicated evolution equations, butthis is the price one ought to pay for the possibility of describing quantum objects in term of classicalprobabilities. In addition, the equations obtained in this paper are much more easier than in the previousattempt [24] of description of evolution of spin particles in terms of probabilities.The generalization of the results of this paper to the higher spin particles will be given in furtherpublications. References [1] J. Bertrand and P. Bertrand,
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