Coherent States of Systems with Pure Continuous Energy Spectra
aa r X i v : . [ m a t h - ph ] S e p COHERENT STATES OF SYSTEMS WITHPURE CONTINUOUS ENERGY SPECTRA
Z. MOUAYN AND H. A. YAMANI
Abstract.
While dealing with a Hamiltonian with continuous spectrum we use a tridiagonalmethod involving orthogonal polynomials to construct a set of coherent states obeying a Glauber-type condition. We perform a Bayesian decomposition of the weight function of the orthogonalitymeasure to show that the obtained coherent states can be recast in the Gazeau-Klauder approach.The Hamiltonian of the ℓ -wave free particle is treated as an example to illustrate the method. Introduction
Coherent states (CS) have been introduced by Schr¨odinger as states which behave in manyrespects like classical states [1]. They got this name after that Glauber [2] realized that they wereparticularly convenient to describe optical coherence. In particular, the electromagnetic radiationgenerated by a classical current is a multimode coherent state, and so is the light produced bya laser in certain regimes [3]. Therefore, CS are cornerstones of modern quantum optics [4] andmore recently, CS found applications in quantum information experiments [5].CS also are mathematical tools which provide a close connection between classical and quantumformalisms so they play a central role in the semiclassical analysis [6, 7]. In general, CS are aspecific overcomplete family of vectors in a Hilbert space associated with a quantum mechanicalsystem and can be constructed for that space having either a discrete or continuous basis in differentways [8] : “ `a la Glauber ” as eigenfunctions of an annihilation operator; as states minimizing someuncertainty principle or they can be obtained as orbits of a unitary operator acting on a specificor fiducial state. For the latter one, Weyl defined them for nilpotent groups [9] and this hasbeen extended to Lie groups [10, 11] and further to the continuous spectrum corresponding to theinfinite-dimensional unitary representations of noncompact groups [12, 13].Unlike the case of systems with pure discrete spectrum, constructing CS for a pure continuousspectrum is a challenging problem which may be addressed in different manners but, generallymost can be recast in the Gazeau-Klauder CS [14] which were constructed in terms of the energyeigenstates of a given non-degenerate system without referring to any group structure. In [15] amodification allowing to deal with degenerate systems and to treat discrete states and continuousstates in a unified way was proposed. The problem of building CS from non-normalizable fiducialstates was considered in [16]. In [17] the authors obtained the CS for the continuous spectrum bystarting from the hypergeometric CS for the discrete spectrum, and applying a discrete-continuouslimit. In [18], the notion of ladder operators was introduced for systems with continuous spec-tra together with two kinds of annihilation operators allowing the definition of CS as modifiedeigenvectors of these operators.Here, our purpose is to construct, under a Glauber-type condition, a set of CS for a Hamiltonianwith continuous spectrum by using the tridiagonal method. We show that this procedure also tellsus how to connect the constructed CS with their Gazeau-Klauder version for the Hamiltonianunder consideration. This connection is achieved by making appeal to the Bayesian decompositionof the weight function associated with the orthogonal polynomials arising in this method . Indeed,we use the above connection together with the energy eigenstates of the non-degenerate system under consideration to show that we recover the Gazeau-Klauder CS. We illustrate our methodfor the Hamiltonian of the l -wave free particle.The paper is organized as follows. In section 2, we introduce a set of Glauber-type CS by usinga tridiagonal method. In Section 3, we recover the Gazeau-Klauder CS using a Bayesian approach.In Section 4, we illustrate our method for the Hamiltonian of the ℓ -wave free particle and wediscuss some of its properties. Section 5 is devoted to some concluding remarks.2. Glauber-type CS using the tridiagonal approach
The tridiagonal approach.
Here, we first summarize some needed facts on the tridiagonalmethod. For this, we assume that the matrix representation of the given Hamiltonian H in acomplete orthonormal basis | φ n >, n = 0 , , , ... , is tridiagonal. That is,(2.1) h φ n | H | φ m i = b n − δ n,m +1 + a n δ n,m + b n δ n,m − .We now define the forward-shift operator A by its action on the basis | φ n i as follows(2.2) A | φ n i = c n | φ n i + d n (cid:12)(cid:12) φ n − (cid:11) , n = 1 , , ... .For n = 0, we state that d = 0 . Furthermore, we require from the adjoint operator A † to act onthe ket vectors | φ n > in the following way:(2.3) A † | φ n i = c n | φ n i + d n +1 (cid:12)(cid:12) φ n +1 (cid:11) , n = 0 , , , , ... .The operator A † A now admits the tridiagonal representation(2.4) h φ n (cid:12)(cid:12) A † A (cid:12)(cid:12) φ m i = c m d m +1 δ n,m +1 + ( c m c m + d m d m ) δ n,m + d m c m − δ n,m − in terms of the coefficients ( c n , d n +1 ) , n = 0 , , , ... . We have proved [19] that the coefficientsin (2.1) are connected to those in (2 .
2) by the relations a n = c n c n + d n d n and b n = c n d n +1 , n = 0 , , , ... . The tridiagonal matrix representation of H with respect to the basis | φ n i alsomeans that it acts on the elements of this basis as(2.5) H | φ n i = b n − (cid:12)(cid:12) φ n − (cid:11) + a n | φ n i + b n (cid:12)(cid:12) φ n +1 (cid:11) , n = 0 , , , ... .We may then considered the solutions of the eigenvalue problem H | E i = E | E i by expanding theeigenvector | E i in the basis | φ n i as(2.6) | E i = + ∞ X n =0 C n ( E ) | φ n i . Then, making use of (2 . E C ( E ) = a C ( E ) + b C ( E ) , (2.8) E C n ( E ) = b n − C n − ( E ) + a n C n ( E ) + b n C n +1 ( E ) , n = 1 , , ..., and the orthogonality relations(2.9) δ n,m = Z Ω c C n ( E ) C m ( E ) dE, n, m = 1 , , ... .These relations correspond to the case when the spectrum of the operator H is composed only bya continuous part Ω c . Define(2.10) P n ( E ) := C n ( E ) C ( E ) , n = 0 , , , ... .Then { P n ( E ) } is a set of polynomials that satisfy the three-term recursion relation for n ≥ EP n ( E ) = b n − P n − ( E ) + a n P n ( E ) + b n P n +1 ( E ) OHERENT STATES OF SYSTEMS WITH PURE CONTINUOUS ENERGY SPECTRA 3 with initial conditions P ( E ) = 1 and P ( E ) = ( E − a ) b − . If we now define the density ω ( E ) :=( C ( E )) and assume only existence of continuous spectrum then the relation (2 .
9) reads(2.12) δ n,m = Z Ω c P n ( E ) P m ( E ) ω ( E ) dE. Finally, with the help of the above notations, the coefficients ( c n , d n ) can also be expressed in termsof coefficients b n and the values at zero of consecutive polynomials ( P n ) for n ≥ d n +1 ) = − b n P n (0) P n +1 (0)and(2.14) ( c n ) = − b n P n +1 (0) P n (0) .2.2. Coherent states.
As in our previous paper [19] we here adopt the definition of the Glauber-type CS as the eigenstate of the operator A when the Hamiltonian is written as H = A † A. Notethat A is here playing the role of annihilation operator. Therefore, we first look to the solution ofthe eigenproblem(2.15) A | ϕ z i = z | ϕ z i with z real. It is not hard to show that the state satisfying (2 .
15) has the following representationin the chosen basis | φ n i :(2.16) | ϕ z i = ( N ( z )) − + ∞ X n =0 Q n ( z ) | φ n i where(2.17) Q n ( z ) := n − Y j =0 (cid:18) z − c j d j +1 (cid:19) , Q ( z ) = 1and assuming that(2.18) N ( z ) = + ∞ X n =0 | Q n ( z ) | < + ∞ . For ( z, γ ) ∈ R , the generalized CS associated to H are defined as the orbit of the evolutionsemigroup e − iγH while acting on the fiducial state | ϕ z i . That is,(2.19) | z, γ i = e − iγH | ϕ z i . Note that γ can be interpreted as a time parameter. One observes from (2 .
17) that when z = c k then the series (2 .
16) terminates and the coherent states reduce to(2.20) | c k , γ i = ( N ( c k )) − k X n =0 Q n ( c k ) e − iγH | φ n i . We may also write the | φ n i as(2.21) | φ n i = + ∞ Z | E i p ω ( E ) P n ( E ) dE which leads to Z. MOUAYN AND H. A. YAMANI (2.22) e − iγH | φ n i = + ∞ Z e − iγH | E i p ω ( E ) P n ( E ) dE. We use the fact that(2.23) e − iγH | E i = e − iγE | E i gives(2.24) e − iγH | φ n i = + ∞ Z | E i p ω ( E ) P n ( E ) e − iγE dE. Recall that(2.25) h r | E i = p ω ( E ) + ∞ X j =0 P j ( E ) φ j ( r ) . Therefore,(2.26) h r (cid:12)(cid:12) e − iγH (cid:12)(cid:12) φ n i = + ∞ Z e − iγE " + ∞ X j =0 φ j ( r ) P j ( E ) ω ( E ) P n ( E ) dE (2.27) = + ∞ Z K ( r, y ) P n ( y ) ω ( y ) e − iγy dy where(2.28) K ( x, y ) := + ∞ X j =0 φ j ( x ) P j ( y ) . Finally, summarizing the above calculations, the wave function in Eq.(2 .
20) may also be presentedin an integral form as(2.29) h r | c k , γ i = ( N ( c k )) − + ∞ Z K ( r, y ) S ( c k , y ) ω ( y ) e − iγy dy where(2.30) S ( u, y ) := k X n =0 Q n ( u ) P n ( y ) . Deducing Gazeau-Klauder CS using a Bayesian analysis
Bayesian analysis.
Here, our goal is the deduce the Gazeau-Klauder CS [14] from the aboveconstructed ones (2 . ω λ ( E ) associated withorthogonal polynomials { P n ( E ) } depends on a parameter λ and that it’s a density function fora probability distribution. Now, the question is to determine two functions: E q ( E ) and theother λ τ ( λ ) that may enter in the following decomposition(3.1) ( τ ( λ )) E q ( E ) R ( τ ( λ )) y q ( y ) dy = ω λ ( E ) . OHERENT STATES OF SYSTEMS WITH PURE CONTINUOUS ENERGY SPECTRA 5
For that we may look at this problem from a Bayesian viewpoint by saying that (3 .
1) also meansthat the weight function(3.2) ω λ ( E ) ≡ π ( E ⌋ λ )can be considered as a posterior distribution (or inverse) for an unknown distribution denoted hereby π ( η ⌋ E ) where E may play the role of a parameter and η denotes the variable or the observeddata. We say that π ( η ⌋ E ) is the statistical model . Also from (3 . τ ( λ )) E may play the role of a prior distribution on the parameter E which itself is modeled as a randomvariable. That is,(3.3) ( τ ( λ )) E ≡ π λ ( E )called the prior . According to the general basic definition ([20], pp.8-10) we also say that π ( E ⌋ λ )the posterior are conjugate under π ( η ⌋ E ) the model . Doing so, our problem in (3 . ω λ ( E ) ≡ π ( E ⌋ λ ) as posterior, we may ask under which model π ( η ⌋ E ) ≡ q ( E ) the probability law defined by ω λ ( E ) could be conjugate to some prior π λ ( E ) tobe determined ?.Finally, in concrete situations one will be dealing with the weight function ω λ ( E ) will be givenexplicitly therefore we can find the two quantities q ( E ) and τ ( λ ). The latter ones, can be used toprove that the constructed CS we have introduced via the tridiagonal method procedure agree withthe Gazeau-Klauder CS for the continuous spectrum. Indeed, as we will see below this analysiswill provide us with the factorial function f ( E ) and the re-parametrization formula s = τ ( λ ) thatserve as a bridge linking the two approaches.3.2. Gazeau-Klauder CS.
Let
H > {| E i} stands for a basis of eigenstates in some Hilbert space H , forwhich(3.4) H | E i = E | E i , < E < E so that the energy support is (cid:2) , E (cid:1) . Here E = + ∞ could be considered. We also can choose anormalized basis of eigenvectors of H :(3.5) h E | E ′ i = δ ( E − E ′ )and(3.6) E Z | E i h E | dE = H . For s ≥ γ ∈ R , the Gazeau-Klauder CS [14] are defined by(3.7) | s, γ i = ( N ( s )) − E Z s E p f ( E ) e − iγE | E i dE. These states are normalized(3.8) h s, γ | s, γ i = 1and(3.9) N ( s ) = Z E s E f ( E ) dE Z. MOUAYN AND H. A. YAMANI is a normalization factor. The function E f ( E ) is determined by a suitable non-negative weightfunction σ ( s ) ≥ f ( E ) = E Z s E σ ( s ) ds. With the measure(3.11) dµ ( s, γ ) = 12 π N ( s ) σ ( s ) dsdγ, the resolution of the identity reads(3.12) Z | s, γ i h s, γ | dµ ( s, γ ) = H . Finally, from the above Bayesian decomposition of ω λ ( E ) , we choose f ( E ) to be the inverse of q ( E ), i.e.,(3.13) f ( E ) ≡ q ( E )and we take(3.14) s ≡ τ ( λ )as a new parametrization.4. Coherent states associated with the ℓ -wave free particle We start with separating the angular part of the wavefunction of the free particle in terms of thespherical harmonics that are eigenfunctions of the angular momentum which is conserved for thiskind of potentials. That leaves for the radial part of the wavefunction the Schr¨odinger operator(4.1) H ℓ := − d dr + 12 ℓ ( ℓ + 1) r which acts on the Hilbert space H := L ( R + , dr ) and admits a continuous spectrum E ∈ [0 , + ∞ ).Hence it is positive semi-definite. Here, the oscillator space H is endowed with the orthonormalbasis whose elements are given by(4.2) φ ( ℓ,λ ) n ( r ) := s λn !Γ (cid:0) n + ℓ + (cid:1) ( λr ) ℓ +1 exp (cid:18) − λ r (cid:19) L ( ℓ + ) n (cid:0) λ r (cid:1) , n = 0 , , , ...,r ∈ R + where λ denotes a real free parameter, ℓ is the angular momentum number and L ( σ ) n ( . ) is theLaguerre polynomial ([21], p.1000). Using differential recurrence relations for these polynomials,one finds by direct calculations that the matrix elements defined by(4.1) h φ ( ℓ,λ ) n | H ℓ | φ ( ℓ,λ ) m i = + ∞ Z φ ( ℓ,λ ) n ( r ) H h φ ( ℓ,λ ) m i ( r ) dr have the following expression(4.3) h φ ( ℓ,λ ) n | H ℓ | φ ( ℓ,λ ) m i = λ (cid:18) n + ℓ + 32 (cid:19) δ n,m + λ s n (cid:18) n + ℓ + 32 (cid:19) δ n,m +1 + λ s ( n + 1) (cid:18) n + ℓ + 32 (cid:19) δ n,m − . OHERENT STATES OF SYSTEMS WITH PURE CONTINUOUS ENERGY SPECTRA 7
Therefore, we can identify the coefficients ( a n ) and ( b n ) in (2 .
1) as follows(4.4a) a n = λ (cid:18) n + ℓ + 32 (cid:19) , (4.4b) b n = λ s ( n + 1) (cid:18) n + ℓ + 32 (cid:19) . According to equation (4 . P n ( E ) = ( − n s n !Γ (cid:0) ℓ + (cid:1) Γ (cid:0) n + ℓ + (cid:1) L ( ℓ + ) n (cid:18) λ E (cid:19) . These polynomials satisfy the orthogonality relations(4.6) + ∞ Z P j ( E ) P k ( E ) ω ℓ,λ ( E ) dE = δ j,k with respect to the weight function(4.7) ω ℓ,λ ( E ) = 2 λ Γ (cid:0) ℓ + (cid:1) (cid:18) λ E (cid:19) ℓ + exp (cid:18) − λ E (cid:19) . Note that ω ℓ,λ ( E ) is the continuous density function of the Gamma probability distribution G ( α, β ) with the shape parameter α = ℓ + and the scale parameter β = 2 λ − . Finally, with(4.8) P n (0) = ( − n s Γ (cid:0) n + ℓ + (cid:1) n !Γ (cid:0) ℓ + (cid:1) equations (2 . , (2 .
14) together with (3 .
20) yield(4.9) d n +1 = λ √ √ n + 1 , c n = λ √ r n + ℓ + 32 . The kernel function K in (2 .
28) has the form(4.10) K ( r, E ) = + ∞ X j =0 φ ( ℓ,λ ) j ( r ) P j ( E )(4.2) = ( λr ) ℓ +1 e − ( λr ) s λ Γ (cid:18) ℓ + 12 (cid:19) + ∞ X j =0 j ! ( − j Γ (cid:0) j + ℓ + (cid:1) L ( ℓ + ) j (cid:0) ( λr ) (cid:1) L ( ℓ + ) j (cid:18) Eλ (cid:19) . We now make use of the formula(4.11) + ∞ X j =0 j ! ( − h ) j Γ ( j + ν + 1) ( − h ) j L ( ν ) j ( x ) L ( ν ) j ( y ) = e ( x + y ) h h h ( xyh ) − ν J ν √ xy h h ! (see [23], p.139, (12a) and p.140 for h →
1) for ν = ℓ + , x = λ r and y = 2 λ − E, we get that(4.12) K ( r, E ) = 12 λ ℓ +1 r s λ Γ (cid:18) ℓ + 12 (cid:19) e λ − E (2 E ) − ( ℓ + ) J ℓ + (cid:16) √ r E (cid:17) . Z. MOUAYN AND H. A. YAMANI
We also need to specify the kernel function S defined in (2 .
30) :(4.13) S ( c k , y ) := k X n =0 Q n ( c k ) P n ( y ) = k X n =0 n − Y j =0 (cid:18) c k − c j d j +1 (cid:19) P n ( y ) . Therefore,(4.14) h r | c k , γ i = ( N ( c k )) − k X n =0 n − Y j =0 (cid:18) c k − c j d j +1 (cid:19) + ∞ Z K ( r, y ) P n ( y ) ω λ,ℓ ( y ) e − iγy dy. The integral in (4 .
14) reads + ∞ Z K ( r, y ) P n ( y ) ̺ ( y ) e − iγy dy = ( − n λ − ℓ − (cid:0) ℓ + (cid:1) s n !Γ (cid:0) n + ℓ + (cid:1) r ( ℓ + )(4.15) × + ∞ Z e − λ y y ( ℓ + ) J ℓ + (cid:16)p r y (cid:17) L ( ℓ + ) n (cid:18) λ y (cid:19) e − iγy dy. For k = 0 formula (4 .
14) reduces to(4.16) h r | c , γ i = ( N ( c )) − + ∞ Z K ( r, y ) ω λ,ℓ ( y ) e − iγy dy = ( N ( c )) − λ ℓ +1 r s λ Γ (cid:18) ℓ + 12 (cid:19) λ (cid:0) ℓ + (cid:1) − ( ℓ + ) (cid:18) λ (cid:19) + ℓ (4.17) × + ∞ Z y ( ℓ + ) J ℓ + (cid:16)p r y (cid:17) e − λ y e − iγy dy. By the variable change y = x , the last integral becomes(4.18) 2 + ∞ Z x ( ℓ + ) +1 J ℓ + (cid:16) √ r x (cid:17) e − ( λ + iγ ) x dx. Next, by using the identity ([21], p.706)(4.19) + ∞ Z x ν +1 e − αx J ν ( βx ) dx = β ν (2 α ) ν +1 exp (cid:18) − β α (cid:19) , Re α > , Re ν > − , for β = √ r , ν = (cid:0) ℓ + (cid:1) and α = ( λ + iγ ), we arrive at the expression(4.20) h r | λ, γ i = √ (cid:18) Γ (cid:18) ℓ + 32 (cid:19)(cid:19) − (cid:18) λ (cid:19) ( ℓ + ) r ℓ +1 (cid:0) λ + iγ (cid:1) ℓ + exp − r λ + iγ ) ! . Note that with respect to the basis (4 .
2) one can observe that the coefficient c in (4 .
16) coincideswith the labeling parameter z according to calculations that start by the formula (2 . . So theabove equation (4 .
20) represents in fact the wave function in r − coordinate of a coherent state withthe given z provided we choose the value λ = √ l +3 z. OHERENT STATES OF SYSTEMS WITH PURE CONTINUOUS ENERGY SPECTRA 9
Now, for a Bayesian decomposition of the weight function purpose, we first observe that ω λ,ℓ ( E )as given by (4 .
7) is a Gamma distribution G (cid:0) ℓ + , λ (cid:1) . It is also well known that for the Poissonmodel X ∼ P ( κ ) with κ > X = j ) = κ j j ! e − κ , j = 0 , , , ..., if the prior distribution on the parameter κ is a Gamma distribution G ( α, β ) then the posteriordistribution is also a Gamma distribution G ( α + j, β + 1) . Thus, in terms of our notations,(4.22) Pr ( X = ℓ ) = E ℓ ℓ ! e − E ≡ p E ( l ) , l = 0 , , , ..., E > X ∼ P ( E ) , is a convenient statistical model. This also indicates that the angular momen-tum integer number l may in fact play the role of an observed data of a discrete random variable X ∼ P ( E ) with the energy E > l and proceed toreverse X by fixing ”`a priori” a law π λ ( E ) that E is supposed to follow. The prior law on theprameter can be obtained just by writting our weight function ω λ,ℓ ( E ) ≡ G (cid:0) ℓ + , λ (cid:1) as a gammadistribution G ( α + ℓ, β + 1) . This gives us(4.23) π λ ( E ) := G (cid:18) , λ − (cid:19) and therefor we can rewrite the weight function as a posterior distribution as(4.24) [ π λ ( E )] [ p E ( ℓ )] R [ π λ ( y )] [ p y ( ℓ )] dy = ω λ,l ( E ) , which, after simplification, reduces to(4.25) h e − λ E i h E + ℓ i + ∞ R y + ℓ e − λ y dy = ω λ,ℓ ( E ) . In other words,(4.26) ω λ,l ( E ) ∝ ( τ ( λ )) E q ( E )where(4.27) τ ( λ ) = e − λ and(4.27) q ( E ) = E + l . Now, in order to recover the constructed CS in (4 .
20) by the Gazeau-Klauder formalism let usrecall that the operator H ℓ acts on the Hilbert space L ( R + , dr ) and admits a continous spectrum E ∈ [0 , + ∞ ). The Schr¨odinger equation H ℓ ϕ = Eϕ has a regular solution given by(4.28) ˆ ℓ ( kr ) = √ krJ ℓ + ( kr ) , where J ν denotes the Bessel function of the first kind and of order ν ([21], p.910) and k = √ E .The function ˆ ℓ is regular for r → ℓ > . Therefore, eigenstates are those given by(4.29) h r | E i = √ krJ ℓ + ( kr ) . From the above Bayesian decomposition of the weight function ω ℓ,λ ( E ) we choose the factorialfunction to be defined by(4.30) f ℓ ( E ) := 1 q ( E ) = (2 E ) − ( + ℓ ) . Therefore, the corresponding Steiljes moment problem(4.31) f l ( E ) = + ∞ Z s E σ ( s ) ds can be solved by the weight function(4.32) σ ℓ ( s ) = 1Γ (cid:0) + ℓ (cid:1) s (cid:18) Log s (cid:19) − + ℓ , s < σ ℓ ( s ) = 0 for s ≥ , by making appeal to the Mellin transform ([22], p.343) :(4.33) + ∞ Z φ α,ν ( x ) x p − dx = Γ ( ν ) ( p + 1) − ν , Re ν > , Re p > − Re α, where φ α,ν ( x ) = x α ( − Logx ) ν − , 0 < x < φ α,ν ( x ) = 0, x ∈ [1 , + ∞ ) , for p = 2 E +1 , ν = + ℓ and α = −
1. Therefore, the normalization factor (2 . N ℓ ( s ) = 12 Γ (cid:18)
32 + ℓ (cid:19) (cid:18) Log s (cid:19) − ( + ℓ ) . With these ingredients, the CS (3 .
7) take the form(4.35) | s, γ i = ( N ℓ ( s )) − + ∞ Z dE s E e − iγE q (2 E ) − ( + ℓ ) | E i . Now, from the above Bayesian decomposition of ω ℓ,λ ( E ) we choose the following reparametrizationfor the labeling parameter s according to (4 .
27) as(4.36) s = τ ( λ ) = exp (cid:18) − λ (cid:19) , ≤ s < , λ ∈ R , then (4 .
18) takes the form(4.37) | λ, γ i = ( N l ( s )) − + ∞ Z e − ( λ + iγ ) E q (2 E ) − ( + ℓ ) | E i dE. Next, making use of (4 . h r | λ, γ i = ( N l ( s )) − + ∞ Z (2 E ) ( + l ) e − ( λ + iγ ) E h √ rJ l + ( kr ) i dE (4.39) = √ r ( N l ( s )) − + ∞ Z √ E ( + l ) e − ( λ + iγ ) ( √ E ) J l + (cid:16) √ Er (cid:17) dE (4.40) = √ r ( N l ( s )) − + ∞ Z x ( + l )+1 e − ( λ + iγ ) x J l + ( xr ) dx. OHERENT STATES OF SYSTEMS WITH PURE CONTINUOUS ENERGY SPECTRA 11
By applying the formula ([21], p.706):(4.41) + ∞ Z x ν +1 e − αx J ν ( βx ) dx = β ν (2 α ) ν +1 exp (cid:18) − β α (cid:19) , Re α > , Re ν > − , for parameters ν = l + , α = ( λ + iγ ) and β = r , Eq. (4 .
40) reads(4.42) h r | λ, γ i = ( N l ( s )) − r l +1 (cid:0) λ + iγ (cid:1) l + +1 exp − r λ + iγ ) ! . Finally, we replace N l ( s ) by is expression (4 .
34) to arrive at the expression(4.43) h r | λ, γ i = √ (cid:18) Γ (cid:18) l + 32 (cid:19)(cid:19) − (cid:18) λ (cid:19) ( l + ) r l +1 (cid:0) λ + iγ (cid:1) l + exp − r λ + iγ ) ! . The above expression of coherent states (4 .
42) is a major result. It has the following properties.For γ = 0 , the corresponding expression of CS reduces to(4.44) h r | λ, i = s λ Γ (cid:0) ℓ + (cid:1) ( λr ) l +1 exp (cid:18) − λ r (cid:19) . Recall that for the basis vectors φ ( ℓ,λ ) n ( r ) in (4 .
2) we have for n = 0(4.45) φ ( ℓ,λ )0 ( r ) := s λ Γ (cid:0) ℓ + (cid:1) ( λr ) ℓ +1 exp (cid:18) − λ r (cid:19) . So we may rewrite (4 .
44) as h r | λ, i = φ ( ℓ,λ )0 ( r ) as expected. The combined energy exponential inthe integral in (4 .
37) is now(4.46) e − β E , β = 1 λ + iγ. We therefore have the result(4.47) h r | λ, γ i = (cid:18) βλ (cid:19) ℓ + φ ( ℓ,β )0 ( r ) . Explicitly, we have the density function(4.48) ρ ( r ; λ, γ ) := |h r | λ, γ i| = 2 λ l +3 Γ (cid:0) l + (cid:1) (1 + γ λ ) l +3 / r l +2 e − r λ λ γ . In figure 1, we show the behavior of the function r ρ ( r ; λ, γ ) for several discrete values of γ. We now can calculate the average position(4.49) r ( γ ) := + ∞ Z rρ ( r ; λ, γ ) dr = 2 λ l +3 Γ (cid:0) l + (cid:1) (1 + γ λ ) l +3 / ∞ Z e r λ γ − λ − r l +3 dr. Applying the integral ([21], p.337):(4.50) + ∞ Z x m e − βx n dx = 1 nβ m +1 n Γ( m + 1 n ) , Re m > , Re n > , Re β > m = 2 l + 3, n = 2 and β = λ λ γ , Eq.(4 .
49) takes the form
Progress of the density ρ r ρ ( r ; λ , γ ) γ =0, λ =1,l=0 γ =1, λ =1,l=0 γ =2, λ =1,l=0 γ =3, λ =1,l=0 Progress of the density ρ r ρ ( r ; λ , γ ) γ =0, λ =1,l=5 γ =1, λ =1,l=5 γ =2, λ =1,l=5 γ =3, λ =1,l=5 Progress of the density ρ r ρ ( r ; λ , γ ) γ =0, λ =1,l=10 γ =1, λ =1,l=10 γ =2, λ =1,l=10 γ =3, λ =1,l=10 Figure 1. r ρ ( r ; λ, γ ) for several discrete values of γ. (4.51) r ( γ ) = C ( l + 1)!Γ ( l + 3 / (cid:18) λ + λ γ (cid:19) / . OHERENT STATES OF SYSTEMS WITH PURE CONTINUOUS ENERGY SPECTRA 13
Progress of the velocity ν γ ν λ =1,l=0 λ =1,l=5 λ =1,l=10 λ =1,l=30 Figure 2.
Note that when l = 0, the average position reduces to √ π (cid:0) λ + λ γ (cid:1) / . On other hand thevelocity (with respect to γ ) is υ ( γ ) := ∂ γ ( r ( γ )) = ( l + 1)!Γ ( l + 3 / λ γ q λ + λ γ . Figure 2 shows how quickly this velocity reaches its asymptotic value as γ goes to infinity:lim γ → + ∞ υ ( γ ) = λ Γ( l + 2)Γ ( l + 3 / . Concluding remarks
We have constructed a set of CS obeying a Glauber-type condition for a Hamiltonian withcontinuous spectrum by using a tridiagonal method involving orthogonal polynomials. The basicquantities in our procedure are the parameters ( c n , d n ) which are related to the matrix elements( a n , b n ) of the tridiagonal Hamiltonian by (2 .
13) and (2 . . More specifically, these CS are labeledby the sequence z = c n . But the general form (2 .
16) is still to be exploited. Connecting thesestates with the Gazeau-Klauder CS was not straightforward and bridge the gap between thetwo approaches requires the idea of a Bayesian decomposition for the weight function in theorthogonality measure of polynomials arising from the tridiagonal method. As an example, wehave the ℓ -wave free particle for which the statistical model given by the Poisson probabilitydistribution Pr ( X = l ) = e − E E l /l ! , l = 0 , , , ..., X ∼ P ( E ), has played a central role in writingdown the convenient Bayesian decomposition for the corresponding weight function. Therefore,there should be an explanation for the appearance of the Poisson distribution having the energy E > l as its observed data in thephysics of this system. References [1] E. Schr¨odinger, Die Naturwissenschaften, (1926), 664.[2] R. J. Glauber, The Quantum Theory of Optical Coherence, Phys. Rev . (1963), 2529-2539.[3] M. Schlosshauer: Decoherence and the Quantum-to-Classical Transition, The Frontiers Collection, SpringerVerlag 2007.[4] J. R. Klauder, B. S. Skagerstam, Coherent States Applications in Physics and Mathematics. Singapore: WorldScientific. 1985. [5] F. Grosshans, G. Van Assche, J. Wenger, R. Brouri, N. J. Cerg, Ph. Grangier: “Quantum key distributionusing gaussian-modulated coherent states”, Letters to Nature, Nature (2003), 238-241.[6] S. T. Ali, J. P. Antoine, J.P. Gazeau, Coherent States, Wavelets, and Their Generalizations. Springer, NewYork, 2014.[7] J.P. Gazeau, Coherent states in quantum physics, WILEY-VCH Verlag GMBH & Co. KGaA Weinheim 2009.[8] V. V. Dodonov, ”Nonclassical” states in quantum optics: a ”squeezed” review of the first 75 years. J. Opt. BQuantum Semiclass. Opt. (2002), R1-R33.[9] H. Weyl. Gruppentheorie und Quantenmechanik. (German) Reprint of the second edition. WissenschaftlicheBuchgesellschaft, Darmstadt, 1977. xi+366 pp.[10] A. M. Perelomov, Coherent states for arbitrary Lie group. Comm. Math. Phys . (1972), 222-236.[11] E. Onofri, A note on coherent state representations of Lie groups. J. Mathematical Phys . (1975), 1087-1089.[12] M, Hongoh, Coherent states associated with the continuous spectrum of noncompact groups. J. MathematicalPhys . (1977), 2081-2084.[13] A. Perelomov, Generalized coherent states and their applications. Springer-Verlag, Berlin. 1986.[14] J. P. Gazeau and J. R. Klauder, Coherent states for systems with discrete and continuous spectrum, J. Phys.A . (1999), 123.[15] A. Inomata, M. Sadiq, Modification of Klauder’s coherent states. 8th International Conference on Path Inte-grals: From Quantum Information to Cosmology, PI 2005.[16] J. Ben Geloun, J. Hnybida, J. R. Klauder, Coherent states for continuous spectrum operators with non-normalizable fiducial states. J. Phys. A . (2012), 085301, 14 pp.[17] D. Popov, M. Popov, Coherent states for continuous spectrum as limiting case of hypergeometric coherentstates, Romanian Reports in Physics . (2016), 1335-1348.[18] J. J. Ben Geloun, J. R. Klauder, Ladder operators and coherent states for continuous spectra. J. Phys. A . (2009), 375209, 9 pp.[19] H. A. Yamani, Z. Mouayn, Properties of shape-invariant tridiagonal Hamiltonians, Theor. Math. Phys . (2020), 761.[20] C. P. Robert, The Bayesian Choice, Second Edition, Springer Science+Business Media, LLC, 2007.[21] I. S. Gradshteyn, I. M. Ryzhik. Table of integrals, series, and products. Elsevier/Academic Press, Amsterdam,seventh edition, 2007.[22] H. Bateman, Tables of Integral Transforms, Vol I, McGraw-Hill Book Compagny, Inc., 1954.[23] H. Buchholz, The Confluente Hypergeometric Function, Springer-Verlag Berlin Heidelberg GmbH 1969. E-mail address : [email protected] E-mail address : [email protected]@gmail.com