Coherent states for a system of an electron moving in a plane: case of discrete spectrum

In this work, we construct different classes of coherent states related to a quantum system, recently studied in [1], of an electron moving in a plane in uniform external magnetic and electric fields which possesses both discrete and continuous spectra. The eigenfunctions are realized as an orthonormal basis of a suitable Hilbert space appropriate for building the related coherent states. These latter are achieved in the context where we consider both spectra purely discrete obeying the criteria that a family of coherent states must satisfies.


Introduction
From a generalization of the definition of canonical coherent states, Gazeau and Klauder proposed a method to construct temporally stable coherent states for a quantum system with one degree of freedom [2]. Then, in the literature, the method has been explored for different kinds of quantum systems with several degrees of freedom. See for example [3,4,5] and references therein. Also, in some previous works, motivated by these developments, multidimensional vector coherent states have been performed for Hamiltonians describing the nanoparticle dynamics in terms of a system of interacting bosons and fermions [6]; from a matrix (operator) formulation of the Landau problem and the corresponding Hilbert space, an analysis of various multi-matrix vector coherent states extended to diagonal matrix domains has been performed on the basis of Landau levels [7]. Besides, the motion of an electron in a noncommutative (x, y) plane, in a constant magnetic field background coupled with a harmonic potential has been examined with the relevant vector coherent states constructed and discussed [8].
Following the method developed in [2,4], we investigate in a recent work [1] by considering Landau levels, various classes of coherent states as in [9,5,10] 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. In this work, we consider Consider an electron moving in a plane (x, y) 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 [1] We briefly recall here a summary of results where the details are given in [1]. In the symmetric gauge − → A = B 2 y, − B 2 x with the scalar potentiel given by Φ(x, y) = −Ey, the corresponding classical Hamiltonian, obtained from (1), denoted by H 1 , reads The HamiltonianĤ 1 can be then re-expressed as follows: and splits 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 Therefore, the eigenvectors and the energy spectrum of the HamiltonianĤ 1 are determined by the following formulas: In the symmetric gauge − → A = − B 2 y, B 2 x with the scalar potential given by Φ(x, y) = −Ex, the classical Hamiltonian H in equation (1) becomes The Hamiltonian operatorĤ 2 can be then written aŝ with the harmonic oscillator part is given bŷ and the linear part byT The eigenvectors and the eigenvalues of the HamiltonianĤ 2 , as previously determined for H 1 , are obtained as 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 , since [Ĥ 1 OSC ,Ĥ 2 OSC ] = 0, as follows: so thatĤ 2 OSC lifts the degeneracy ofĤ 1 OSC and vice versa. The present paper is a direct continuation of our work in reference [1], where we construct different classes of coherent states corresponding to the case of discrete spectrum.
The paper is organized as follows. The section 2 is devoted to the construction of coherent states for the quantum Hamiltonian possessing purely discrete spectrum by following the method developped in [2,4]. The section 3 is about the coherent states of the unshifted Hamiltonians H 1 and H 2 defined through multiple summations. In section 4 we construct coherent states related to the Hamiltonian H 1 OSC − H 2 OSC . An outlook is given is section 5.
2 Coherent states for shifted Hamiltonians with more than one degree of freedom In this section, we construct various classes of coherent states for Hamiltonian operators that admit discrete eigenvalues and eigenfunctions in appropriate separable Hilbert spaces as elaborated in [9,4,5]. Let H D be spanned by the eigenvectors |Ψ nl ≡ |n, l of H 1 OSC and H 2 OSC provided by (13). Let us consider I H l D , I H n D the identity operators on the subspaces H n D , Since the Hamiltonians considered are formed by self adjoint operators that act in infinite dimensional Hilbert spaces, from the equations (7) and (12), we set that define families of discrete eigenvalues associated with the eigenvectors {|Ψ nl ⊗|α k } ∞ n,l,k=0 , forming an orthonormal basis of the separable Hilbert spaceĤ = H D ⊗H, withH spanned by the states {|α k } ∞ k=0 .

Coherent states with one degree of freedom
Let l and n fixed. Define the coherent states for the Hamiltonian H 1osc − T 1 − ωc 2 − λ 2 2m IĤ, where α k satisfies (17), with one degree of freedom. Denoting them with the fixed index l, they are given from (23), with K ≥ 0 and 0 ≤ δ < 2π, by With the normalization condition the normalization constant is determined such that we must have Thus if lim k→∞ ǫ k = ǫ, we need to restrict K to 0 ≤ K < L = √ ǫ for the convergence of the above series. In this case we have Proposition 2.1 For fixed l and n, let us write the following measures Then, on the Hilbert subspace H nl D of H D , the coherent states satisfy the resolution of the identity given by Proof. See in the Appendix.

Coherent states with two degrees of freedom
Let α k fixed. We obtain infinite component vector coherent states [9], with two degrees of freedom, each component counting the infinite degeneracy of the energy level of the harmonic oscillator shifted eigenvalues E ′ n = ω c n. Taking J ≥ 0, J ′ ≥ 0 and 0 ≤ θ, θ ′ < 2π, the coherent states are given by where with fixed α k , the resolution of the identity can be expressed as follows: where I H l D is defined previously in (14) andH k the subspace ofH,H being spanned by Proof. See in the Appendix.

Proposition 2.3
The coherent states defined in (30) and (24) satisfy the temporal stability property given as follows: Proof. See in the Appendix.
Remark 2.4 Note that since in the equation (16), we have E ′ 0,α k = 0, the coherent states (24) and (30) cannot satisfy the action identity. In this case, we phrase the resulting coherent states as "temporally stable coherent states".
3 Coherent states for unshifted Hamiltonians H 1 and H 2 defined through multiple summations In this paragraph, two types of temporally stable coherent states are constructed in line with the general scheme developed in [4], (see also [5]). The first type is defined as a tensor product of two classes of coherent states with one and two degrees of freedom by setting ρ 1 (n), E n , E α k andρ(k) as independent quantities. The second type, which cannot be considered as a tensor product of vectors with one and two degrees of freedom, is defined by letting that one sum depends on the other through the same quantities.

1-When the summations are independent
Let us set E n, Then, the conditions of positivity required for the eigenvalues E ′ k imposes α k ≤ 0.
Under these considerations, the coherent states for the Hamiltonians H 1 OSC − T 1 and H 2 OSC − T 2 , when taking into account the degeneracies of the Landau levels as before, are defined with three degrees of freedom as a tensor product of two coherent states defined with one and two degrees of freedom, respectively, as follows: Proof. See in the Appendix. For fixed α k , with E ′ n,α k given in (18), let us set From (20), one has ρ(n, α k ) = κ n (γ) n .
Proof. See in the Appendix.
Note that, as previously mentioned, the coherent states (46) are temporally stable.

Coherent states related to the Hamiltonian H 1 OSC − H 2 OSC
• When α k is fixed: the coherent states are defined on H D ⊗H, in an analogous way, by using the 'bi-coherent states' (BCS) [9] as follows: or, by using the complex labels, as They correspond to the multidimensional coherent states [4] of the Hamiltonian The normalization condition is given by Proposition 4.1 They satisfy, on the separable Hilbert spaceH k , the following resolution of the identity: • When α k is not fixed: the coherent states are denoted by |J, θ; J ′ , θ ′ ; α k or |z,z ′ ; α k and given by the same equations (52) and (53).
Here, the normalization condition is given by Proof. See in the Appendix.

Proposition 4.3
The coherent states (52) also satisfy the properties of temporal stability and action identity as stated in [9]. In the situation of the coherent states |J, θ; J ′ , θ ′ ; α k , these properties are given as below Proof. See in the Appendix.

Outlook
The behaviour of an electron moving in a plane in an electromagnetic field background, arising in the quantum Hall effect, has been studied with the related Hamiltonian spectra having both discrete and continuous parts provided. Also, an Hamiltonian in the case of an electric field depending simultaneously on both x and y directions has been discussed with his spectrum provided. The eigenfunctions have been obtained as a countable set realizing an infinite dimensional appropriate Hilbert space. Various coherent states have been constructed by considering shifted and unshifted spectra, respectively. Two kinds of coherent states classes have been obtained. The first kind, with one degree of freedom, is achieved by fixing each index counting the energy levels. The second kind is realized by taking tensor product of two classes of coherent states with one and two degrees of freedom. The discussion can be extended for th case of the potential V = E 1 x + E 2 y. Here, the uniform electric field is defined as − → E = (E 1 , E 2 , 0) with the scalar potential Φ(x, y) = E 1 x + E 2 y = E · r, and the magnetic field given by A = − B 2 y, B 2 x . Then, the Hamiltonian writes as Let us introduce the following pairs of annihilation and creation operators defined by where λ 1 = M cE 1 B and λ 2 = M cE 2 B . They satisfy the following commutation relations The operator HamiltonianĤ is delivered as follows: where we use the following relations The eigenvalue equationĤΨ = EΨ, Ψ(r, θ) = ϕ(r)e ilθ , provides the radial equation with , l 5 (r) = ir(2C) 1 4 , (73) where A 1 and A 2 are constants. Thus, the solutions of the Schrödinger equationĤΨ = EΨ are obtained as the product of (72) by e ilθ .

Acknowlegments
I. Aremua would like to gratefully thank Professor A. S. d'Almeida for some valuable discussions.