Underlying non-Hermitian character of the Born rule in open quantum systems
UUnderlying non-Hermitian character of the Born rule in open quantum systems
Gast´on Garc´ıa-Calder´on ∗ and Lorea Chaos-Cador † Instituto de F´ısica, Universidad Nacional Aut´onoma de M´exico,Apartado Postal 20 364, 01000 M´exico, Ciudad de M´exico, Mexico Universidad Aut´onoma de la Ciudad de M´exico,Prolongaci´on San Isidro 151, 09790 M´exico, Ciudad de M´exico, Mexico
The absolute value squared of the probability amplitude corresponding to the overlap of an initialstate with a continuum wave solution to the Schr¨odinger equation of the problem, has the physicalinterpretation provided by the Born rule. Here, it is shown that for an open quantum system, theabove probability may be written in an exact analytical fashion as an expansion in terms of thenon-Hermitian resonance (quasinormal) states and complex poles to the problem which provides anunderlying non-Hermitian character of the Born rule.
I. INTRODUCTION
It is well known that Max Born stated the rule thatnow bears his name in the early days of quantum mechan-ics [1]. This rule provides a probabilistic interpretationfor the wave function and establishes a link between theformalism of quantum mechanics and experiment, butmore importantly, it represents the irruption of indeter-minism in the description of matter at microscopic scale.This interpretation has had a profound impact into thenotion of what is real and has initiated a debate that re-mains alive up to the present day as shown by distinctinterpretations of quantum mechanics [2–5], of studieson the classical-quantum transition [6], studies concernedwith topics as the reality of the wave function [7–9], andon the Born rule [10, 11].Here we refer to the Born rule for open quantum sys-tems characterized by a continuous spectrum. For thatpurpose we consider a simple problem, namely, the timehonored problem of a particle confined initially within afinite region of space by a potential from which it escapesto the outside by tunneling. The time-dependent wavefunction may be expanded in terms of the continuumwave functions to the problem, and, as is well known,the absolute value squared of the probability amplitudecorresponding to the overlap of the corresponding initialstate with a continuum wave solution to the Schr¨odingerequation to the problem, has the physical interpretationprovided by the Born rule [12, 13]. The above probabilityinvolves an integral over values of the momentum that ex-tends from zero to infinity, and constitutes a ‘black box’type of numerical calculation from which little physicalinsight may be obtained.In this work we derive an exact analytical expressionfor the above probability by expanding the corresponding continuum wave solutions in terms of resonance (quasi-normal) states that we believe provides a deeper physicalinsight on the Born rule. These states may be definedfrom the residues at the complex poles of the outgoing ∗ gaston@fisica.unam.mx † [email protected] Green’s function to the problem and allow for an exactanalytical non-Hermitian formulation of the descriptionof decay by tunneling in open quantum systems [14, 15].In fact, the present work has been motivated by therecent result that the time evolution of decay by tunnel-ing involving continuum wave functions yields identicalresults to that using resonance (quasinormal) states [16].The reason is that both basis follow from the analyti-cal properties of the Green’s function to the problem.However, whereas the physical meaning of the expansioncoefficients involving continuum wave functions is pro-vided by the Born rule, that does not occur in the caseof resonance (quasinormal) states .We find of interest to elaborate a little bit on the notionof open quantum system that we employ here. If, as timeevolves, a particle initially confined within a region ofspace cannot escape to the outside, the system is said tobe a closed system. In that case, the system possessesa purely discrete energy spectrum and exhibits unitarytime evolution. On the contrary, if the particle can escapeby tunneling to the outside, the system constitutes anopen system that has the distinctive feature of exhibitinga continuous energy spectrum. Since energy can escapeto the outside the time evolution is non unitary. However,the continuity equation is fulfilled and hence the total fluxis conserved.Our approach considers the full Hamiltonian H to theproblem and relies on the analytical properties of the out-going Green’s function in the complex momentum plane.It is worth mentioning that there are approaches wherethe full Hamiltonian H to the system is separated intoa part H , corresponding to a closed system, and a part H which couples the closed system to the continuum.This is usually treated to some order of perturbation.This type of approximate approaches has become a stan-dard procedure in the treatment of open systems whereperturbation theory can be justified. It has its roots inthe old work of Weisskopf and Wigner [17]. There arealso related approaches, that have a great deal of at-tention nowdays because of its implications for quantuminformation theory, which are referred to in the literaturealso as open quantum systems. Here, a quantum system S is coupled to another quantum system O called the a r X i v : . [ qu a n t - ph ] F e b environment and hence it represents a subsystem of thetotal system S + O . Usually it is assumed that the com-bined system is closed, however due to the interactionswith the environment the dynamics of the subsystem S does not preserves unitarity and hence it is referred to asan open system. These approaches involve many degreesof freedom and the description of the mixed states ofthe total system is made in terms of the density matrix[18, 19]. The case where the environment may be ne-glected and still energy can escape from the system S ina non-perturbative fashion would essentially correspondto the notion of open system considered here.The paper is organized as follows. In section 2 webriefly review the main aspects of the time evolutionof decay using continuum wave functions and resonance(quasinormal) states and refer to the Born rule. In sec-tion 3 we derive the expansion of the coefficient of thewave solution in terms of resonance (quasinormal) states.In Section 4, we illustrate our findings by considering anexactly solvable model, and finally, Section 5 deals withsome concluding remarks. II. TIME-DEPENDENT SOLUTION
Let us consider, to bring the problem into perspective,the time evolution of decay of a particle that is initiallyconfined by a real spherical potential of arbitrary shape inthree dimensions. Without loss of generality we restrictthe discussion to s waves. Notice that the descriptionholds also on the half-line in one dimension. We consideran interaction potential of arbitrary shape V ( r ) that van-ishes after a finite distance, i.e. V ( r ) = 0 for r > a . Thisis justified on physical grounds for a large class of sys-tems, in particular artificial quantum systems as double-barrier resonant structures [20] or ultracold atoms [21].Also, since we are interested in the continuum, we referto potentials that do not hold bound states. The unitsemployed are (cid:126) = 2 m = 1.The solution to the time-dependent Schr¨odinger equa-tion in the radial variable r , as an initial value problem,may be written at time t > r, t ) = (cid:90) a g ( r, r (cid:48) ; t )Ψ( r (cid:48) , dr (cid:48) , t > , (1)where g ( r, r (cid:48) ; t ) stands for the retarded time-dependentGreen’s function, which may be written in terms of theoutgoing Green’s function as, g ( r, r (cid:48) ; t ) = i π (cid:90) ∞−∞ G + ( r, r (cid:48) ; k )e − ik t kdk, t > . (2) A. The time-dependent solution in terms ofcontinuum wave functions and the Born rule
Equation (2) may be used to derive the well knownexpression of the time-dependent wave evolution in terms of the continuum wave solutions ψ + ( k, r ) [22],Ψ( r, t ) = (cid:90) ∞ C ( k ) ψ + ( k, r )e − ik t dk , (3)where the expansion coefficient C ( k ) is given by C ( k ) = (cid:90) a [ ψ + ( k, r (cid:48) )] ∗ Ψ( r (cid:48) , dr (cid:48) . (4)The continuum wave functions ψ + ( k, r ) are solutions tothe Schr¨odinger equation of the problem[ k − H ] ψ + ( k, r ) = 0 , (5)satisfying [22], ψ + ( k,
0) = 0 (6) ψ + ( k, r ) = (cid:114) π i (cid:2) e − ikr − S ( k )e ikr (cid:3) r ≥ a , (7)where S ( k ) is the S -matrix of the problem.The factor (cid:112) /π in Eq. (7) arises from the Dirac delta normaliza-tion.Using Eq. (3) one may calculate, in particular, twoquantities that are of interest in decay problems. Oneof them is the survival amplitude A ( t ), which gives theprobability amplitude that at time t the decaying particleis still described by the initial state Ψ( r, A ( t ) = (cid:90) a Ψ ∗ ( r, r, t ) dr . (8)Notice that if the initial state Ψ( r,
0) is normalized tounity, then A (0) = 1, which is a probability. Substitutionof Eq. (3) into Eq. (8), using (4), gives A ( t ) = (cid:90) ∞ | C ( k ) | e − ik t dk, (9)which corresponds to a probability amplitude due to theeffect of the time evolving factor exp( − ik t ). The sur-vival probability is defined as S ( t ) = | A ( t ) | . The otherquantity of interest is the non-escape probability P ( t ),that provides the probability that at time t , the decayingparticle is found within the the interaction region r < a , P ( t ) = (cid:90) a | Ψ( r, t ) | dr = (cid:90) ∞ dk (cid:48) (cid:90) ∞ dk C ∗ ( k (cid:48) ) C ( k ) × (cid:90) a dr [ ψ + ( k (cid:48) , r )] ∗ ψ + ( k, r ) e − i ( k − k (cid:48) ) t . (10)The Eqs. (9) and (10), depend on the expansion coeffi-cient C ( k ) which, therefore, plays a relevant role in stud-ies on quantum decay involving the basis of continuumwave functions .It is well known, that the Born rule establishes that theprobability density for a measurement of the momentum k gives a result in the range dk is [12, 13] d P ( k ) = | C ( k ) | dk , (11)where C ( k ) is given by Eq. (4). According to the usualinterpretation of quantum mechanics, upon measurementthe wave function Ψ( r,
0) “collapses” around a narrowrange of continuum wave functions ψ + ( k, r ) of the mea-sured value. If the initial state is normalized to unity, itthen follows that (cid:90) ∞ | C ( k ) | dk = 1 . (12) B. Relationship between ψ + ( k, r ) and the outgoingGreen’s function The analytical properties of the outgoing Green’s func-tion to the problem are the relevant quantity in thederivation of Eq. (3). This expression is given by anintegral that involves only real values of the momentum k [22].The outgoing Green’s function obeys the equation,[ k − H ] G + ( r, r (cid:48) ; k ) = δ ( r − r (cid:48) ) , (13)with boundary conditions, G + (0 , r (cid:48) ; k ) = 0; (cid:20) ∂∂r G + ( r, r (cid:48) ; k ) (cid:21) r = a = ikG + ( a, r (cid:48) ; k ) . (14)Using the Green’s theorem between Eqs. (5) and (13)together with the conditions given by Eqs. (6), (7) and(14) yields [23], ψ + ( k, r ) = − (cid:114) π kG + ( r, a ; k )e − ika , r ≤ a . (15)The above expression relates, for a given value of the mo-mentum k , the continuum wave function with the out-going Green’s function to the problem along the internalinteraction region. As shown below, Eq. (15) consti-tutes in our analysis the relevant expression to relate the continuum wave functions with the the resonance (quasi-normal) states of the system.It is of interest to recall that the outgoing Green’s func-tion may be written in terms of the regular , φ ( k, r ), and irregular , f ± ( k, r ), solutions to the Schr¨odinger equation,obeying respectively, boundary conditions: φ ( k,
0) = 0,[d φ ( k, r ) / d r ] r =0 = 1 and f ± ( k, r ) = e ± ikr for r ≥ a , andthe Jost function J ± ( k ) = f ± ( k,
0) as [22], G + ( r, r (cid:48) ; k ) = − φ ( k, r < ) f + ( k, r > ) J + ( k ) , (16)where r < and r > stand respectively for the smaller andlarger of r and r (cid:48) . The functions f ± ( k, r ) are linearly independent and hence φ ( k, r ) may be written as φ ( k, r ) = 12 ik [ J − ( k ) f + ( k, r ) − J + ( k ) f − ( k, r )] . (17)The continuum wave functions may also be written as[22] ψ + ( k, r ) = (cid:114) π kφ ( k, r ) J + ( k ) . (18)In fact, using Eq. (17) into (18) yields, for r ≥ a , Eq.(7) with S ( k ) = J − ( k ) /J + ( k ). Since flux conservationrequires that S ( k ) S ∗ ( k ) = 1, it follows that J ∗− ( k ) = J + ( k ). C. Relationship between G + ( r, r (cid:48) ; k ) and resonance(quasinormal) states The expression for the outgoing Green’s function givenby Eq. (16) has been used to study in a rigorous formits analytical properties away from real values of k intothe complex momentum plane [22]. For potentials of ar-bitrary shape vanishing exactly after a distance, as con-sidered here, the function G + ( r, r (cid:48) ; k ) may be extendedanalytically to the entire complex k plane, where it hasan infinite number of poles, distributed in a well knownfashion, corresponding to the zeros of the Jost function J + ( k ). In fact, a finite number of them lie on the positiveand the negative imaginary k -axis, corresponding respec-tively to bound and antibound states, and the rest, aninfinite number of poles, are located in the lower half ofthe k plane, where due to time-reversal considerations,they are distributed symmetrically with respect to theimaginary k -axis. Thus, for pole at κ n = α n − iβ n lo-cated on the fourth quadrant of the k plane, there cor-responds a pole κ − n = − κ ∗ n . As discussed below, thesepoles correspond to the resonant (quasinormal) states ofthe problem. In fact, all states arise from the residues ofthe outgoing Green’s function at these poles.The residues at the poles κ n of the outging Green’sfunction G + ( r, r (cid:48) ; k ) are proportional to the resonance(quasinormal) states to the problem. They may be ob-tain, as discussed in Ref. [24], by adapting to the k planethe derivation in the energy plane given in Ref. [25],namely, ρ n ( r, r (cid:48) ) = u n ( r ) u n ( r (cid:48) )2 κ n (cid:8)(cid:82) a u n ( r ) dr + iu n ( a ) / κ n (cid:9) , (19)which yields the normalization condition for resonant(quasinormal) states , (cid:90) a u n ( r ) dr + i u n ( a )2 κ n = 1 . (20)Notice that for bound states, where κ n = iη n , with η > resonance (quasinormal) state formalism reduces tothe usual formalism.The Resonant (quasinormal) states are solutions to theSchr¨odinger equation to the problem,[ κ n − H ] u n ( r ) = 0 , (21)with outgoing boundary conditions: u n (0) = 0 , (cid:20) dd r u n ( r ) (cid:21) r = a = iκ n u n ( a ) . (22)The second of the above conditions implies that for r > a , u n ( r ) = D n exp( iκ n r ), which leads to complex energyeigenvalues, as first discussed by Gamow [26], that is, κ n = E n = E n − i Γ n .An interesting expression follows by using Green’s the-orem between equations for u n ( r ) and u ∗ n ( r ), and its cor-responding boundary conditions, provided α n (cid:54) = 0, β n = 12 | u n ( a ) | I n , (23)where I n = (cid:90) a | u n ( r ) | dr . (24)As discussed in detail in Refs. [14, 15], the expansion ofthe outgoing Green’s function in terms of the resonance(quasinormal) states of the problem may be obtained byconsidering the integral I = i π (cid:90) C G + ( r, r (cid:48) ; k (cid:48) ) k (cid:48) − k dk (cid:48) , (25)where C corresponds to a large closed contour of radius R about the origin in the complex momentum k (cid:48) plane,which excludes all the poles κ n and the real value k (cid:48) = k located inside, that is, C = C S + (cid:80) n c n + c k . SinceCauchy’s integral theorem establishes that I = 0, onemay use the theorem of residues to evaluate the distinctcontours, in view of (19) and (20), to write G + ( r, r (cid:48) ; k ) = N (cid:88) n = − N u n ( r ) u n ( r (cid:48) )2 κ n ( k − κ n )+ i π (cid:90) C S G + ( r, r (cid:48) ; k (cid:48) ) k (cid:48) − k dk (cid:48) . (26)The number of poles appearing in the sum of (26) maybe increased by considering successively larger values ofthe radius R . This follows because the poles are simpleand are ordered as | κ | ≤ | κ | ≤ | κ | ≤ ... [27]. In thelimit as R → ∞ , there will be an infinite number ofterms in the sum. In that limit, however, G + ( r, r (cid:48) ; k )diverges unless r and r (cid:48) are smaller than the radius a of the interaction potential, or r = a with r (cid:48) < a andviceversa, but not both of them. We denote the above conditions as ( r, r (cid:48) ) † ≤ a . In this case, G + ( r, r (cid:48) ; k (cid:48) ) → | k | → ∞ along all directions in the complex k planeand hence the integral term in (26) vanishes exactly asshown rigorously in Refs. [28, 29]. As a result one maywrite G + ( r, r (cid:48) ; k ) = ∞ (cid:88) n = −∞ u n ( r ) u n ( r (cid:48) )2 κ n ( k − κ n ) , ( r, r (cid:48) ) † ≤ a. (27)Substitution of (27) into (13) yields, after straightforwardmanipulations, the closure relationship,12 ∞ (cid:88) n = −∞ u n ( r ) u n ( r (cid:48) ) = δ ( r − r (cid:48) ); ( r, r (cid:48) ) † ≤ a , (28)and the sum rule12 ∞ (cid:88) n = −∞ u n ( r ) u n ( r (cid:48) ) κ n = 0 , ( r, r (cid:48) ) † ≤ a . (29)Noticing that 1 / [2 κ n ( k − κ n )] = 1 / k [1 / ( k − κ n ) + 1 /κ n ],one may write (27), in view of (29), as [14, 15] G + ( r, r (cid:48) ; k ) = 12 k ∞ (cid:88) n = −∞ u n ( r ) u n ( r (cid:48) ) k − κ n , ( r, r (cid:48) ) † ≤ a . (30) D. The time-dependent solution in terms ofresonance (quasinormal) states
One may obtain the time-dependent solution Ψ( r, t ) interm of resonant (quasinormal states) by substitution ofEq. (30) into Eq. (2) and the resulting expression intoEq. (1) to obtain [14, 15],Ψ( r, t ) = ∞ (cid:88) n = −∞ (cid:40) C n u n ( r ) M ( y ◦ n ) , r ≤ aC n u n ( a ) M ( y n ) , r ≥ a , (31)where C n = (cid:90) a Ψ( r, u n ( r ) dr , (32)and the functions M ( y n ) are defined as [14, 15] M ( y n ) = i π (cid:90) ∞−∞ e ik ( r − a ) e − ik t k − κ n dk = 12 e ( imr / t ) w ( iy n ) , (33)where y n = e − iπ/ (1 / t ) / [( r − a ) − κ n t ], y ◦ n is identicalto y n with r = a , and w ( z ) = exp( − z )erfc( − iz) standsfor the Faddeyeva or complex error function [30] for whichthere exist efficient computational tools [31].Assuming that the initial state Ψ( r,
0) is normalizedto unity, it follows from the closure relationship given byEq. (29) that, Re ∞ (cid:88) n =1 { C n ¯ C n } = 1 , (34)where ¯ C n is given by¯ C n = (cid:90) a Ψ ∗ ( r, u n ( r ) dr . (35)Equation (34) shows that the coefficients C n cannot beinterpreted as probability amplitudes, since the sum oftheir square moduli does not add up to the norm ofΨ( r, { C n ¯ C n } may be seen to rep-resent the ‘strength’ or ‘weight’ of the initial state in thecorresponding resonance (quasinormal) state [32, 33].Using Eq. (31), one may obtain resonance (quasinor-mal) state expansions of the survival amplitude A ( t ),and hence of the survival probability S ( t ), and of thenonescape probability P ( t ), namely, A ( t ) = ∞ (cid:88) n = −∞ C n ¯ C n M ( y ◦ n ) , S ( t ) = | A ( t ) | , (36)and P ( t ) = ∞ (cid:88) n = −∞ ∞ (cid:88) l = −∞ C n C ∗ l I nl M ( y ◦ n ) M ∗ ( y ◦ l ) , (37)where I nl = (cid:82) a u ∗ l ( r ) u n ( r ) dr .In particular, using some properties of the function M ( y ◦ n ), the survival amplitude may be written for theexponential and long times regimes as [14, 15, 32], A ( t ) ≈ ∞ (cid:88) n =1 C n ¯ C n e − i E n t e − Γ n t/ − i πi ) / Im (cid:40) ∞ (cid:88) n =1 C n ¯ C n κ n (cid:41) t / , (38)or to discuss the ultimate fate of a decaying quantumstate [34].Equations (36), (37), and (38) should be contrastedwith the ‘black-box’ type of calculations that provideEqs. (9) and (10) in terms of continuum wave functions .However, as pointed out above, both formulations giveidentical numerical results.It is worth mentioning that the formalism outlinedabove differs from the so called rigged Hilbert space for-mulation in many respects, as discussed in Refs. [14, 35].For example, since in that approach the poles located onthe third quadrant of the k plane are not taken explicitlyinto consideration, there is no analytical description asthat given by Eqs. (36), (37) and Eq. (38). III. RESONANCE (QUASINORMAL)EXPANSION OF | C ( k ) | Substitution of Eq. (30) into Eq. (15) yields the expan-sion of the continuum wave function ψ + ( k, r ) in terms of resonant (quasinormal) states , ψ + ( k, r ) = − (cid:114) π
12 e − ika ∞ (cid:88) n = −∞ u n ( a ) u n ( r ) k − κ n , r < a . (39)We may then substitute (39) into the expression of C ( k )given by (4), using (32), to obtain C ( k ) = − (cid:34)(cid:114) π
12 e − ika ∞ (cid:88) n = −∞ ¯ C n u n ( a ) k − κ n (cid:35) ∗ . (40)One may run the above sum from n = 1 up to infinity, bynoticing that κ − n = − κ ∗ n and u − n ( r ) = u ∗ n ( r ) [14, 15].Hence, this allow us to write the expression for | C ( k ) | as, | C ( k ) | = 1 π ∞ (cid:88) n =1 | C n | I n β n ( k − α n ) + β n + 1 π ∞ (cid:88) n =1 | C n | I n β n ( k + α n ) + β n + 1 π Re (cid:40) ∞ (cid:88) n ± s C n C s u n ( a ) u s ( a )[( k − α n ) + iβ n ][( k + α s ) − iβ s ] (cid:41) , (41)where we have used Eqs. (23) and (24). Using (41) allowsus also to calculate (cid:90) ∞ | C ( k ) | dk = ∞ (cid:88) n =1 | C n | (cid:90) a | u n ( r ) | dr − Re (cid:40) ∞ (cid:88) n ± s C n C s i u n ( a ) u s ( a ) κ n + κ s (cid:41) = 1 . (42)It is worth mentioning that Eq. (41) for | C ( k ) | , is givenby a sum of resonance peaks having a Lorentzian shapethat depends on the resonance terms α n and β n , eachpeak multiplied by the coefficients | C n | , formed by theoverlap of the initial state Ψ( r,
0) and the correspondingresonance (quasinormal) state u n ( r ) and I n , given by theintegral of | u n ( r ) | along the internal interaction regionplus an interference term. One sees that the contributionof each resonance peak depends on the value attained bythe corresponding product | C n | I n , which in view of Eq.(42) does not add up to a unity value, precisely due to thecontribution of the interference term. Hence, in general, ∞ (cid:88) n =1 | C n | I n > . (43)The role of the initial state Ψ( r,
0) is crucial. Inthe case of an initial state that overlaps strongly withone of the resonance (quasinormal) states of the system,say, u r ( r ), as in the example below, one may see that |C(k)|2 k FIG. 1. Plot of | C ( k ) | using continuum wave functions (fullline) and resonance (quasinormal) states (dotted line) versus k for the delta -shell potential with parameters λ = 100 and a = 1. See text. Re C r ¯ C r ≈
1, and also | C r | I r ≈
1. In that case Eq. (42)may be written as (cid:90) ∞ | C ( k ) | dk ≈ | C r | I r ≈ . (44)In general, however, Eq. (42) suggests that the coef-ficients | C n | and I n may possess a quasi-probabilisticnature [36].It is worth commenting that it has been a commonpractice in the literature to approximate | C ( k ) | by justa single Lorentzian, which therefore implies that the co-efficient involving the initial state has a unity value, asin the work by Khalfin, who showed that the exponen-tial decay law cannot hold at all times [37]. In general,however, this is not justified. When more resonance lev-els are involved in the decay process and the initial stateoverlaps with several resonance (quasinormal) levels, amore complex decaying behavior arises [38–40]. IV. MODEL
In order to illustrate our findings, we consider the ex-actly solvable model given by a δ -shell potential of inten-sity λ and radius a , for zero angular momentum, V ( r ) = λδ ( r − a ) , (45)and an initial state, the infinite box state,Ψ( r,
0) = (cid:18) a (cid:19) / sin (cid:16) πra (cid:17) . (46)This model has also been used in Ref. [16] to illustratethat the formulations or the probability density Ψ( r, t ) in terms of continuum wave functions and resonance (quasi-normal) states yield identical results for the time evolu-tion of decay.Since the initial state given by (46) is confined withinthe interaction region, the continuum wave function reads, using Eq. (18), ψ + ( k, r ) = (cid:114) π sin( kr ) J + ( k ) , r < a , (47)where the Jost function J + ( k ) reads, J + ( k ) = 2 ik + λ (e ika − . (48)Using Eqs. (46), (47) and (48) into Eq. (4) yields C ( k ) = 2 ik √ πa J + ( k ) (cid:18) sin[( s − k ) a ] s − k − sin[( s + k ) a ] s + k (cid:19) , (49)with s = π/a , from which one may calculate | C ( k ) | . - 6 - 5 - 4 - 3 - 2 - 1 log10 [ |Cn| 2 In ] n FIG. 2. Plot of log | C n | I n as a function of the resonance(quasinormal) states n used in the calculation of Fig. 1. Seetext. Similarly, the resonance (quasinormal) states along theinternal interaction region are given by u n ( r ) = A n sin( κ n r ) , r ≤ a , (50)where, using Eq. (20), the normalization A n reads, A n = (cid:20) λλa + e − iκ n a (cid:21) / . (51)The set of complex poles { κ n } follows from the zeros ofthe Jost function given by Eq. (48), namely, J + ( κ n ) = 2 iκ n + λ (e iκ n a −
1) = 0 . (52)The solutions to the above equation has been discussedelsewhere [14]. For example, for λ (cid:29)
1, they admit theapproximate analytical solution, κ n ≈ nπa (cid:18) − λa (cid:19) − i a (cid:16) nπλa (cid:17) . (53)One may then use iterative methods as the Newton-Raphson method to get the solution with the desireddegree of approximation.Using Eqs. (46), (50), and (51) into Eq. (32) yields, C n = A n √ a (cid:18) sin[( s − κ n ) a ] s − κ n − sin[( s + κ n ) a ] s + κ n (cid:19) , (54)with s = π/a , from which | C n | can be calculated. Ina similar fashion, using (50) and (51) into Eq. (24) oneobtains, I n = | A n | (cid:20) sinh(2 β n a ) β n − sin(2 α n a ) α n (cid:21) . (55)Figure 1 provides a plot of the coefficient | C ( k ) | asa function of k around the first resonance level κ = α − iβ of the delta -shell potential with intensity λ = 100and radius a = 1. The coefficient | C ( k ) | is evaluatedusing the basis of continuum wave functions given byEq. (49) (solid line) and by using the basis of resonance(quasinormal) states corresponding to Eq. (40), which isidentical to Eq. (41), with C n given by Eq. (54). Onesees that they yield identical results. In this case, asshown in Fig. 2 which exhibits a plot of log | C n | I n forthe first 10 resonance levels, the term n = 1 dominates,and hence this term is sufficient to obtain an excellent de-scription of | C ( k ) | . Our calculations show, for this andother cases, the I n (cid:39)
1, and therefore this indicate thatthe coefficient | C n | is the main ingredient to determinethe value of a given resonance term to the probability | C ( k ) | . Figure 2 shows also that the value of | C ( k ) | fora value of k around any of the other resonance values ofthe system is very small. V. CONCLUDING REMARKS
Equations (41) and (42) constitute the main result ofthis work. They refer to a non-Hermitian analytical for-mulation that lies outside the conventional Hilbert space,which however yields identical numerical results as a for-mulation based on continumm wave functions . The dis-tinct resonance (quasinormal) contributions, given by Eq.(41), provide a deeper insight in the way that | C ( k ) | at-tains a given value. In particular, the role of the coeffi-cients | C n | which involve the overlap of the initial statewith the corresponding resonance (quasinormal) state .However, as shown by Eq. (42), the sum over the coeffi-cients | C n | I n does not add up to unity, and hence in gen-eral they cannot be interpreted as probabilities, althoughthey may be seen to represent the ‘strength’ or ‘weight’of the initial state in the corresponding resonance (quasi-normal) state . Presumably, they might be considered asquasi-probabilities [36], but this requires further study.Our formulation clearly shows the relevant role playedby initial states. The recent developments on artificialquantum systems may lead to their control and manipu-lation [21]. Our results might be of interest in the recentdiscussions on the reality of the wave function.We would like to end by quoting Shakespeare’s Hamlet: There are more things in Heaven and Earth, Horatio,than are dreamt of in your philosophy . ACKNOWLEDGMENTS
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