Coherent states for a system of an electron moving on plane

In this paper, we construct the coherent states for a system of an electron moving on plane in uniform external magnetic and electric fields. These coherent states are built in the context of both discrete and continuous spectra and satisfy the Gazeau-Klauder coherent states properties [1].


Introduction
The system of charged quantum particles interacting with a constant magnetic field continues to attract intensive studies and is without a doubt one of the most investigated quantum systems, mainly motivated by condensed matter physics and quantum optics. A review article devoted to this quantum system and its related different kind of coherent states (CSs) was recently elaborated by Dodonov (see [2] and the complete reference list therein).
The concept of what is now called coherent states has been of great interest to the scientific community since the work of Schrödinger in 1926 [3] on the quantum harmonic oscillator (HO), where he introduced a specific quantum state that has dynamical behavior that is most similar to that of the classical HO. The conditions any state must fulfill to be coherent were elaborated by Klauder as follows: continuity in complex label, normalization, non orthogonality, unity operator resolution with unique positive weight function of the integration measure, temporal stability and action identity [4]. More details on the CSs and their different generalizations can be found in the literature [5]- [9], the list is not of course exhaustive.
In his study [10], Landau found that the system of electronic motion in a static uniform magnetic field can be assimilated in two dimensions to a harmonic oscillator, with an energy structure of equidistant discrete levels, with a distance ω c (ω c is the cyclotron frequency), each level being highly degenerate. Such a system, more often named Landau model, also provides a natural description for other well known significant phenomena, the so-called integer and fractional quantum Hall effects. In these last years, in the search of understanding the main features of the fractional quantum Hall effect (FQHE) [11,12], many efforts have been done in the literature to find a wave function which minimizes the energy of a two-dimensional system of electrons subjected to a strong constant magnetic field applied perpendicularly to the sample, independently of the electron density. In [13], a system of electrons, essentially a two-dimensional crystal, has been considered. Besides, the wave function introduced has been modified to lower the energy in order to explain the experimental data. From an appropriate quantization of the classical variables of the system Hamiltonian, Bagarello et al (see [13], [14] and references therein), have modified the single electron wave function in view of the study of localization properties. The similar quantization has been also used to investigate the Bohm-Aharonov effect ( [15,16] and references therein) emphasizing the fact that it is not the electric and the magnetic fields but the electromagnetic potentials which are the fundamental quantities in quantum mechanics.
In a previous work [17], a connection has been established between quantum Hall effect and vector coherent states (VCSs) [18,19] by applying the various construction methods developed in the literature. In the same way, the motion of an electron in a noncommutative xy plane, in a constant magnetic field background coupled with a harmonic potential was examined with the relevant VCSs constructed and discussed [20]. The Barut-Girardello CSs have been built for Landau levels of a gas of spinless charged particles, subject to a perpendicular magnetic field confined in a harmonic potential with thermodynamical and the statistical properties have been investigated [21]. See also [2] and references quoted therein. Recently [22], from a matrix (operator) formulation of the Landau problem and the corresponding Hilbert space, an analysis of various VCSs extended to diagonal matrix domains has been performed on the basis of Landau levels.
The construction of CSs for continuous spectrum was first proposed for the Gazeau-Klauder CSs in [1] and later in [23,24,25]. In the present work, we follow the method developed in [1], by considering Landau levels, to built various classes of CSs as in [17,26,27] arising from physical Hamiltonian describing a charged particle in an electromagnetic field, by introducing additional parameters useful for handling discrete and continuous spectra of the Hamiltonian. The eigenvalue problem is presented and the quantum Hamiltonian spectra provided in the two possible orientations of the magnetic field by considering the infinite degeneracies of the Landau levels. The CSs are constructed with relevant properties discussed for both continuous and discrete spectra, and for purely discrete spectrum.
The paper is organized as follows. In section 2, we revisit the model of electron moving on plane where the eigenvalue problems are explicitely set and solved. The position and momentum operators, satisfying canonical commutation relations, established for the considered Hamiltonians are also defined. Section 3 is devoted to the construction of CSs for the quantum Hamiltonian possessing both continuous and discrete spectra by following the method developped in [1,17]. Concluding remarks are given in Section 4.

Electron moving on plane revisited
In this section, we revisit the system of an electron moving on plane as in [16], where we consider different scenarios for the symmetric gauge and the scalar potential.
Consider an electron moving on the plane xy in the uniform external electric field − → E = − − → ∇Φ(x, y) and the uniform external magnetic field − → B which is perpendicular to the plane described by the Hamiltonian

Case of the symmetric gauge
Experimentally, the electric field − → E is oriented according to one of the two possible directions of the plane. Suppose the scalar potential is defined as Substituting the relations (2) and (3) in (1), the corresponding classical Hamiltonian, denoted by H 1 , reads A canonical quantization of this system is obtained by promoting the classical variables x, y, p x , p y , to the operators X, Y, P x , P y which satisfy the nonvanishing canonical commutation relations The Hamiltonian operator is derived from (4) as followŝ In order to solve the eigenvalue problem it is convenient to perform the change of variables as below satisfying the nonvanishing commutations relations and to define two sets of annihilation and creation operators b, b † and d, d † given by with λ = mcE B . These two sets of operators commute each other and satisfy the following commutation relations where ω c = eB mc is known as the cyclotron frequency and 1 I is the unit operator. The Hamil-tonianĤ 1 can be then re-expressed as follows: In order to compute the eigenvalues E and eigenvectors Ψ, we splitĤ 1 in (13) into two commuting parts in the following manner: whereĤ 1 OSC denotes the harmonic oscillator part while the part linear in d and d † is given bŷ The annihilation and creation operators b and b † can be also rewritten as follows: Then, one has leading to and, recurrently, to The harmonic oscillator HamiltonianĤ 1 OSC reduces tô with eigenvalues E n 1 OSC given by corresponding to the eigenvectors defined by (21). The eigenvalue equationT 1 φ = Eφ can be reduced to whose solution is readily found to be Then, the eigenvalues of the operatorT 1 , corresponding to eigenfuctions (26), are given by indicating that this spectrum, labeled by α, is continuous. Therefore, to sum up, the eigenvectors and the energy spectrum of the HamiltonianĤ 1 are determined by the following formulas:

Case of the second possible symmetric gauge
We consider now the symmetric gauge with the scalar potential given by The classical Hamiltonian H in equation (1) becomes By mean of canonical quantization and proceeding like in the previous section, we define the two sets of annihilation and creation operators defined by with λ defined as in (10) and (11). They also commute each with other and satisfy the commutation relations (12). The corresponding Hamiltonian operatorĤ 2 can be then written asĤ where the following relation is obtained. Here, the harmonic oscillator part is given bŷ and the linear part byT The annihilation and creation operators b and b † become here From (35), it comes Then, the eigenvalue equationT 2 φ = Eφ is equivalent in this case to which leads to Taking again α = mE λ , it follows the equation which can be solved to give the eigenfunctions of the operatorT 2 corresponding to eigenvalues expressed as in (27). Therefore, the eigenvectors and eigenvalues of the HamiltonianĤ 2 , as previously determined forĤ 1 , are obtained as Introduce the position and momentum operators obtained from the annihilation and creation operators (10) and (32) aŝ respectively, where the following commutation relations are satisfied. Then, we respectively have in the gauges Thus, from (28), (45) and (48), the eigenvectors denoted |Ψ nl := |n, l = |n ⊗ |l ofĤ 1 OSC can be so chosen that they are also the eigenvectors ofĤ 2 OSC as follows: |Ψ nl , n, l = 0, 1, 2, . . . , ∞(49) so thatĤ 2 OSC lifts the degeneracy ofĤ 1 OSC and vice versa. From (27), consider the shifted eigenvalues where the states |ǫ α are delta-normalized states and form the orthonormal basis {|ǫ α , α ∈ R}. The satisfy the eigenvalue equation which is the same equation for the operatorT 2 .

Construction of coherent states
In this section, CSs are constructed, considering the two possible orientations of the magnetic field as in [17] as well as additional parameters, originated from discrete and continuous aspects of the Hamiltonian spectrum in line with [1]. As a matter of comparison, we first replace the original Hamiltonian operators by their corresponding shifted counterparts, as done in [1]. Then, we investigate the full operators and analyze the results.

Case of the shifted quantum Hamiltonian
Let H D+C := H D ⊕ H C be the Hilbert space associated to the operator H D ⊕ H C , where H D and H C are associated to discrete and continuous spectra, respectively. Let consider the discrete shifted Hamiltonian H D := H 1osc − ωc 2 1 I H D and the continuous shifted Hamiltonian H C := T 1 − λ 2 2m 1 I H C , where 1 I H D and 1 I H C denote the identity operators on H D and H C , respectively. Let H D spanned by the eigenvectors |Ψ nl ≡ |n, l of H 1 OSC and H 2 OSC provided by (49). Besides, let H C be the Hilbert space associated to the continuous spectrum spanned by the eigenvectors of the operator T 1 denoted |ǫ α in equation (51).
Now, let us investigate the resolution of the identity or the completeness relation which is expressed in terms of the projectors onto the states |J, γ; J ′ , γ ′ ; l; K, θ; β .
dµ B refers to the Bohr measure [17] provided as follows given on the Hilbert space H ns of functions f : R → C, which is complete with respect to the scalar product .|. ns . dλ(K) = σ(K)dK, and σ(K) is a non-negative weight function On the Hilbert spaces H D , H C and H D+C , we have the following essential relations that need to be satisfied, where dµ D and dµ C are the measures associated to the discretespectrum CSs {J, γ, J ′ , γ ′ } and continuous-spectrum CSs {K, θ} labeling parameters, respectively. The identity operator 1 I H D+C is the direct sum of the identity operators 1 I H D and 1 I H C which act on the complementary subspaces H D and H C , respectively, corresponding to discrete and continuous spectra.
Noting that the integration over β, 0 ≤ β < 2π eliminates the third relation above, which is related to the off-diagonal terms, the three conditions (62) are reduced to In view of getting the resolution of the identity, let us take the functions f and g as in [1], such that where the factors N g and N f are chosen so that Proof. See in the Appendix 5.
Proof. See in the Appendix 5.
The action identity as noticed in [1] is difficult to obtain with the combined CSs given in (53).

Case of the unshifted Hamiltonians H 1 and H 2
The eigenvalues E n,α of the Hamiltonian operators H 1 and H 2 , given respectively in Eq. (28) and (45), can be rewritten as E n,α = E n + E α , where The required conditions E n,α ≥ 0 for all n ∈ N and E α ≥ 0 lead to the relations such that where ( 3 2 ) n stands for the Pochhammer symbol [28]. The CSs, related to H 1 , defined in line with (53), are now given, where the function ρ(n) = n! is replaced by the one in (73), by yields with the relation (55) also remaining here valid.
Proposition 3.5 The CSs (74) satisfy, on H D+C , a resolution of the identity given in (58), where the measures dν(J) and dν(J ′ ) are now given by where the quantities are the Laguerre polynomials, and lead to the identities [28] n!
The CSs for the Hamiltonian H 2 , similar to the ones in (74), can be constructed in the same way, with the labeling parameters J, γ playing the role of J ′ , γ ′ and vice versa.