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Wigner functions, coherent states, one-dimensional marginal probabilities and uncertainty structures of Landau levels

Following an approach based on generating function method phase space characteristics of Landau system are studied in the autonomous framework of deformation quantization. Coherent state property of generating functions is established and marginal probability densities along canonical coordinate lines are derived. Well defined analogs of inner product, Cauchy-Bunyakowsy-Schwarz inequality and state functional have been defined in phase space and they have been used in analyzing the uncertainty structures. The general form of the uncertainty relation for two real-valued functions is derived and uncertainty products are computed in states described by Wigner functions. Minimum uncertainty state property of the standard coherent states is presented and uncertainty structures in the case of phase space generalized coherent states are analyzed.


I. INTRODUCTION AND SUMMARY
Wigner function is the quantum mechanical analogue of probability distribution in a classical phase space and it is the central concept of Weyl-Wigner-Groenewold-Moyal quantization [1]. In a broader sense this is known also as deformation quantization which as an alternative autonomous formulation of quantum mechanics has been successfully used in various fields of physics and mathematics [2,3,4]. Wigner function and associated marginal probability (MP) distributions have also become important source of information in quantum optics, atomic optics and signal processing. In these rapidly developing fields experiments aiming to probe the fundamental structure and predictions of quantum mechanics by observing non-classical behaviors of Wigner functions that are reconstructed from measured MP densities along various directions have also been designed [5,6,7,8].
In this paper we study the phase-space properties of Landau system [9,10], that is the motion of a charged particle under the influence of a vertical uniform magnetic field, in the autonomous framework of deformation quantization without using wavefunctions and operators of the conventional quantization method. Following a constructive approach based on the generating function we will show that basic quantum mechanical phase-space characteristics of the problem can be completely analyzed. The generating function in deformation quantization was firstly introduced in [11] and then used in [10] in generating Wigner functions and two-dimensional (2D) MP densities and in investigating their properties.
Three main contributions of the present study can be summarized as follows. We first analyze the coherent state property of generating function and establish main characteristic properties of coherent states in phase space. Secondly we compute all MP densities along phase space coordinate lines by introducing new generating functions. Remarkable properties of the generating functions and 1D MP densities and the fact that they provide important integral equalities between the classical orthogonal polynomials hardly obtainable by other means are emphasized. The concentration is then on analyzing the uncertainty structures in the phase space. It is shown that all the mathematical structures and tools such as state functional, inner product and related inequalities responsible from the uncertainty relations in the Hilbert space formulation and forming the basis of many quantum mechanical facts, have well defined analogs in deformation quantization. To the best of our knowledge, the first studies addressing the uncertainty structure in the same context are [12,13] (see also the recent study [14]).
Our approach and results can directly be adapted to two mode and extended to multi mode systems of quantum optics and they may provide an autonomous phase space perspective to our current understanding of quantum Hall effects [15] and related issues. Our analysis of the uncertainty structures may also have some relevance in the context of quantum information processing where a class of inequalities for detecting entanglement and for a better understanding of correlations has been given in terms of uncertainty relations [16,17].
Two main composition rules that will be used throughout this study are the ⋆-product and Moyal bracket {f, g} M = f ⋆ g − g ⋆ f where f, g are arbitrary phase-space functions. The ⋆product is bilinear and associative which imply that Moyal bracket is bilinear, antisymmetric and obey the Jacobi identity and Leibnitz rule. All quantum effects are encoded in these (the Planck constant) dependent composition rules and they obey where f g denotes the pointwise product (f g)(x) = f (x)g(x) of f, g and { , } stands for the usual Poisson bracket of the classical mechanics. The above limit relations constitute the principle of quantum-classical correspondence in the most concise form at two algebraic levels of observables (real valued phase space functions) of deformation quantization.
The phase space in our case is R 4 equipped with canonical coordinates q = (q 1 , q 2 ) and conjugate momenta p = (p 1 , p 2 ). The Hamiltonian function describing the motion of a spinless particle of charge q > 0, mass m moving on the q 1 q 2 -plane, reads as is the vector potential of the magnetic field B = ∂ q 1 A 2 − ∂ q 2 A 1 which is perpendicular to the plane of motion, c denotes the speed of light and v k are the velocity components. With the abbreviation ∂ x k ≡ ∂/∂x k and convention that ← ∂ and → ∂ act, respectively, on the left and on the right, the ⋆-product for the present system is Note that for the kth star power ( x is any linear combination of the phase space coordinates.
Some details of the phase space description of Landau system are briefly outlined in the next section where we show how the generating function is determined and used in generating Wigner functions. For additional details of this section we refer to [10,11].
Coherent state property of the generating function is specified and analyzed in section 3. In

II. PHASE-SPACE DESCRIPTION AND GENERATING FUNCTION
For the phase-space quantization of the problem we shall use, when B is constant, H and as a set of Moyal-commuting functions. Here ω = qB/mc is the cyclotron frequency and Higher level Wigner functions can be generated by successive application of the creation functions (and of the annihilation functions from the right) to the ground state Wigner function W 0 . An efficient way of achieving this goal is to introduce a generating function which can be inferred from W 0 . Defining equations of W 0 are a ⋆ W 0 = 0 = b ⋆ W 0 whose real solution, normalized with the volume element dV = dq 1 dq 2 dp 1 dp 2 , is We now introduce, in terms of complex parameters α k , β k the phase-space functions and and a similar relation for G 2 by recalling the definition of the generalized Laguerre polynomials: (1 + y) k e −xy = n=0 L k−n n (x)y n [18]. As the generating function we take G = G 1 G 2 . Finally in this section let us consider the phase-space functions where n k , l k are positive integers and N n = (n 1 !n 2 !) −1/2 , N ℓ = (ℓ 1 !ℓ 2 !) −1/2 . Then all Wigner functions can be constructed from W n 1 n 2 ℓ 1 ℓ 2 = 4w n 1 n 2 w ℓ 1 ℓ 2 whose special cases for n 1 = n 2 = n and ℓ 1 = ℓ 2 = ℓ correspond to the diagonal (or pure state) Wigner functions It is now easy to check the ⋆-ladder structures; a ⋆ W nℓ = W n−1,ℓ ⋆ a and a ⋆ W nℓ = W n+1,ℓ ⋆ā which implies n a ⋆ W nℓ = W nℓ ⋆ n a = nW nℓ for real number function n a =ā ⋆ a. Similar relations hold for b,b and n b =b ⋆ b. These justify the fact that {W nℓ ; n, ℓ = 0, 1, 2, ...} is the set of simultaneous ⋆-eigenfunctions of the so-called two-sided ⋆-eigenvalue equations H ⋆ W nℓ = W nℓ ⋆ H = E n W nℓ for H = ω(2n a + 1)/2 and the similar one for J = (n b − n a ) with eigenvalues E n are the well-known infinitely degenerate (for they are independent from ℓ) Landau levels.

III. PHASE-SPACE COHERENT STATES
In this section we shall establish another essential property of generating function where α · β = α 1 β 1 + α 2 β 2 . This is the fact that G with its complex parameters represents the standard phase-space coherent states of Landau system. The standard coherent states can be defined by three interrelated ways [19,20] (and for a recent review [21]) which, by adopting them for the deformation quantization, can be stated as follows.
Similar relations hold for b andb. Thus G behaves as a left/right coherent state with (onesided) ⋆-eigenvalues α k and β k . For real G we take β =ᾱ. In such a case G corresponds to the Glauber-Perelomov standard coherent state and reads from (2.4) and (3.1) as Note that among all Wigner functions only W 0 is a (normalized) coherent state.
We will now show that G ′ can be defined by application of a phase space displacement function to the ground state Wigner function W 0 . For this purpose we first observe that Similar calculations show that e −2aā is also a right ⋆-eigenfunction of e κā , hence This may be inferred from the definition of W 0 . We now introduce the complex-valued For the factorizations in the second line we made use of e ηa ⋆ e κā = e ηκ/2 e ηa+κā = e ηκ e κā ⋆ e ηa .
D 1 being the complex conjugate of D 1 , from Eqs. (3.4) and (3.5) we obtain By defining D 2 = D 2 (b,b, α 2 ,ᾱ 2 ) and taking the displacement function as the desired result is achieved as follow That is, up to a positive constant, G ′ can be defined by left-right action of the phase- To emphasize the displacement property of D α 1 α 2 , we first observe that and then we obtain (complex conjugation gives similar relations forā andb) For generalization suppose that f is a phase space function that can be expanded in a star power series of creation and annihilation functions such as where c jj ′ kk ′ 's are some constants and all the indices take positive integer values. Then the displaced function f ′ of f defined by is of the form Up to now two essential properties of G have been established. The first employed in the previous section was that all (diagonal and off-diagonal) Wigner functions can be generated from G. The second presented above emphasize the phase space coherent state property of G. As has been shown to each point (α 1 , α 2 ) of a 2D complex space G ′ assigns a real-valued, phase space coherent state function of the Landau system. In the next section another basic property of G is established. This is the fact that integrated forms of G serve as generating functions for MP densities.

IV. 1D MARGINAL PROBABILITY DENSITIES
For the third property of G stated above we take multiple integral of G on various phasespace regions. As an example let us consider the function of q 1 , q 2 defined by where Z = (q 1 + iq 2 )/γ. We derive in terms of it Combining these two relations and comparing the result with the definition of Wigner functions given by (2.7) we obtain P nℓ (q 1 , q 2 ) = R 2 W nl dp 1 dp 2 . That is, P nℓ (q 1 , q 2 ) is indeed the MP density in the q 1 q 2 -plane and M αβ plays the role of generating function for it.

A. Generating Functions For 1D Marginal Probability Densities
Integrating M αβ (x i , x j ) on one of its coordinate gives the generating function depending on the remaining coordinate. There are three possibilities in each case that can be utilized for checking the calculations. As an example, the generating function for the MP densities in q 1 -direction can be computed as Gdp 1 dp 2 dq 2 , This can equivalently be obtained from the integral of M αβ (q 1 , p 1 ) or of M αβ (q 1 , p 2 ) over p 1 and p 2 , respectively. All generating functions have been computed in this way and they are found to be where N q = π 3/2 2 γ , N p = π 3/2 γ . In the next subsection, 1D MP densities in x i -direction will be found from Now by comparing this equation with W nl dp 1 dp 2 dq 2 , (4.6) one of the advantages of the generating function method can easily be recognized. Instead of taking multiple integral of special functions in deriving MPs from (4.6) and (2.7) one can obtain them more easily by taking derivatives of a exponential function given by one of (4.3). 2D probability densities for the phase-space planes were derived [10] in a similar way.

B. Derivation of 1D Marginal Probability Densities
By defining u = exp(α · β) and v = exp(−z 2 ) with, we can write Q αβ (q 1 ) = N q uv. We then obtain P nℓ (q 1 ) in two steps. Firstly we compute and then the result will be read, in view of (4.5), from Using the Leibnitz rule and ∂ j x x n = n!x n−j /(n − j)!, for j ≤ n, we get by direct computations where prime over s−summation indicates the restriction s ≤ j and In the first and last line of (4.10), ∂ α 1 and ∂ β 1 are taken, in view of (4.7), as ∂ α 1 = −i∂ z /2 and ∂ β 1 = i∂ z /2 when they are acting on v. Evaluating (4.10) at α 1 = 0 = β 1 amounts to taking k = 0 and s = j. Hence I 1 = e α 2 β 2 g(z 1 ) where z 1 = 1 γ q 1 + i 2 (α 2 − β 2 ) and In the second step we compute, in a similar way For α 2 = 0 this transforms to where z 2 = z 1 | α 2 =0 . Evaluating for β 2 = 0 it simplifies to where y = z 2 | β 2 =0 = q 1 /γ. Finally using the Rodriguez formula for Hermite polynomials H n (y) = (−1) n e y 2 ∂ n y e −y 2 , (4.14) MP density in q 1 -direction are found from (4.9) to be P nℓ (q 1 ) = N q e −q 2 where we have defined The calculations for other coordinates have been performed as well and the results are given altogether as follows (i = 1, 2) The rest of the section is devoted to investigation of some properties of these 1D MPs.  It follows from Eqs. (2.6) that the quantum number n determines the energy levels and hence the radius of cyclotron while ℓ determines, together with n, the angular momentum states with (ℓ − n). On the other hand, from the content of section II it is not hard to verify that the distance of cyclotron center from the origin is specified by ℓ itself. That is, Eq.
(4.19) simply says that 1D position and momentum probability distributions are symmetric with respect to these two distances and one can not distinguish them from a given P nℓ (x).
As a consistency check we can easily show that P nℓ (x i ) are all normalized as follow Noting that H 0 (x) = 1, this follows from the orthogonality relation More explicitly, for y = q 1 /γ and for non-zero positive integer m, we get for odd-parity Hermite polynomials vanishes at the origin. Hence only j = n, k = ℓ term in (4.20) contributes to the normalization. Therefore A n ℓ n ℓ = 4 and the well-known equality ∞ −∞ e −y 2 dq 1 = γπ 1/2 prove the result. Note that (4.20) proves the normalization of all Wigner functions and this is a direct result of normalization of W 0 .

D. Integral Equalities
Having a complete list of Wigner functions and of associated 1D MPs at hand a generic type of integral equality in the context of the theory of orthogonal polynomials can be written from Eq. (4.6). Similarly P nℓ (q 1 , q 2 ) = R 2 W nl dp 1 dp 2 gives different type of equality by using 2D MPs P nl (x i , x j ) found in [10]. In fact by using where ρ 2 = ZZ, N nl = 4πl!/n! , N ′ nl = 4π/n!l!2 n+l and two additional type of equalities can easily be obtained from Indeed, when (4.17) and (4.22) are substituted in (4.23) we get, after canceling the exp(−q 2 1 /γ 2 ) from both sides, the following generic type of integral equalities which can hardly be obtained by other means.

V. UNCERTAINTY STRUCTURES IN THE PHASE SPACE
In this section we show that all the uncertainty structures of the quantum mechanics can be realized in a classical phase space without using wavefunctions and operators. In this regard, the projection property W nℓ ⋆ W nℓ = W nℓ of Wigner functions and the associativity and trace (or the so-called closedness) properties of the star-product stand out. The latter reflects the fact that the integral of f ⋆ g all over the phase space is equal to the integral of f g (and therefore of g ⋆ f ), where f and g are two arbitrary phase space functions.

A. Inner Product, CBS Inequality and State Functional in a Phase Space
In view of the above remarks we define a positive semidefinite Hermitian inner product on the (cartesian product of) linear space of all phase space functions as follows This in particular implies the phase space analogue of the Cauchy-Bunyakowsky-Schwartz The basic mathematical structure and tool that lead to these two important relations are the associative ⋆-algebra structure of the phase space functions which admits the usual complex conjugation as an (anti)involution and the expectation (or mean) value function that can be defined as In algebraic terms the expectation value is a state functional s = s nℓ on the ⋆-algebra of the phase space functions which in our case simply reads as s(f ) =< f > nℓ . Defining properties of s are that it is a complex linear function and obey the relations [13,22] The first relation is guarantied by the normalization of Wigner function and making use of the above mentioned properties the second can be verified as follows (see also [12]) Note that s(f ⋆ f ) = 0 implies f ⋆ W nℓ = 0 instead of f = 0. That is why the inner product is, in the case of fixed W nℓ , positive semidefinite.
It should be stressed that instead of Wigner functions any normalized and real-valued projection function (or a set of such functions) that describes state space of a given system can equally well be used in all these constructions. Two instances of this fact will appear in the last two subsections.

B. General Form of Uncertainty Relation For Two Functions
Adapting the inequality (5.2) to the phase space functions we have is the variance of f in a state described by the Wigner function W whose quantum numbers are, for simplicity, suppressed. Since {δf, δg} M = {f, g} M we also have where {, } +M denotes the anti-Moyal bracket. Provided that f and g are real-valued, at the right hand side of (5.5) the first term is a pure imaginary-valued and the second is a real-valued function. Analyzing the right hand side of (5.4) in view of (5.5) we obtain This is the most general form of the phase space uncertainty relation for two real-valued phase space functions which corresponds to the well-known Robertson-Schrödinger uncertainty relation.

D. Uncertainty Products for Phase Space Standard Coherent States
The real and normalized coherent states of the Landau system defined by satisfy the same normalization and (by Eq. (3.8)) projection properties of W 0 We can therefore define the inner product and analyze the uncertainty structures in a coherent state as well. In that case expectation value will be defined as where the subscripts cs stand for coherent state.
As a result G s represents phase space coherent state with the minimum uncertainty for all values of α 1 and α 2 and the variances of coordinates in these states are equal to their values in the ground state.

E. The Case of Generalized Coherent States
Had we defined, in the sense of Perelomov, the generalized coherent states by applying the displacement function to Wigner function W nℓ such that G g = D α 1 α 2 ⋆ W nℓ ⋆D α 1 α 2 , (5.15) the variances of coordinates in such a state would have been the same as that calculated in the state W nℓ . Finally in this section we will prove that this claim is a special case of a more general fact.
In view of Eqs. (3.8) and (3.9) we can write where f is a smooth arbitrary phase space function and f ′ is its displaced function (see Eqs. (3.9) and (3.10)). Similarly for any kth star power of f we have where k is any positive integer. By integrating both sides of (5.16) and by defining we immediately get We conclude as follows. The expectation value of any integer star power of a smooth phase space function in a generalized phase space coherent state is equal to the expectation value of the same star power of the corresponding displaced function in the state corresponding to the Wigner function used in defining the generalized coherent state.