aa r X i v : . [ qu a n t - ph ] J a n Continuum States inGeneralized Swanson Models
A. Sinha and P. Roy Physics & Applied Mathematics UnitIndian Statistical InstituteKolkata - 700 108INDIA
Abstract
A one-to-one correspondence is known to exist between the spectra of the discretestates of the non Hermitian Swanson-type Hamiltonian H = A † A + α A + β A † , ( α = β ),and an equivalent Hermitian Schr¨odinger Hamiltonian h , the two Hamiltonians beingrelated through a similarity transformation. In this work we consider the continuumstates of h , and examine the nature of the corresponding states of H . PACS Numbers :
Keywords :
Swanson-type pseudo Hermitian Hamiltonian, continuum states, dampedwaves, progressive waves, P¨oschl-Teller potential, Morse potential e-mail : anjana − [email protected], anjana23@rediffmail.com e-mail : [email protected] H = a † a + αa + βa † ( α = β ),has been found to have a one-to-one correspondence with that of the conventional (Hermitian)Harmonic oscillator Hamiltonian h [1]. Subsequent works [2, 3, 4] have shown that by replacingthe Harmonic oscillator annihilation and creation operators a and a † by generalized annihilationand creation operators A and A † respectively, the Swanson Hamiltonian could be generalizedto include other interactions, and thus have the generalized form H = A † A + α A + β A † ,( α = β ). In these cases too, H can be mapped to an equivalent Hermitian Hamiltonian h , withthe help of a similarity transformation (say ρ ), and the bound state energies were found to bethe same for both the Hamiltonians (i.e., H and h ). However, when h has both discrete as wellas continuous spectra, no correspondence has been established till date between the continuumstates of h and those (if any) of H . It may be mentioned here that two works dealing with one-dimensional scattering in non Hermitian quantum mechanics (and hence, dealing with stateshaving continuous spectra) deserve special mention in this regard. The first of these is thereview article on complex absorbing potentials [5], dealing with one dimensional scatteringin non-hermitian quantum mechanics, with particular emphasis on complex PT -symmetricpotentials. The second work gives a more detailed study of one-dimensional scattering in PT -symmetric potentials in particular, with some explicit examples of solvable potentials [6, 7].However, in this work we shall not investigate scattering in generalized Swanson models.Rather, our aim is to focus our attention on those states of h having continuous energies, andexplore the nature of the corresponding states of H . We shall also find the probability currentdensity and charge density for the η -pseudo Hermitian generalized Swanson Hamiltonian H .Our studies will be based on two well known interactions viz., the P¨oschl-Teller and Morsemodels.It is well known by now that a quantum system described by a η -pseudo Hermitian Hamil-tonian H , can be mapped to an equivalent system described by its corresponding Hermitiancounterpart h , with the help of a similarity transformation ρ [3, 8, 9], h = ρHρ − (1)So we start with the generalized Swanson Hamiltonian H = A † A + α A + β A † , α = β (2)with solutions ψ ( x ) satisfying the eigen value equation Hψ ( x ) = Eψ ( x )where α and β are real, dimensionless constants. Since we are dealing with non Hermitian H ,hence α = β . Here A and A † are generalized annihilation and creation operators given by A = 1 √ − α − β ( ddx + W ( x ) ) , A † = 1 √ − α − β ( − ddx + W ( x ) ) (3)2f we apply to (2) a transformation of the form [10] ψ ( x ) = ρ − φ ( x ) (4)where ρ = e − µ R W ( x ) dx , with µ = α − β − α − β , α + β = 1 (5)(2) reduces to the following Hermitian Schr¨odinger-type Hamiltonian, with the same eigenvalue E (in units ¯ h = 2 m = 1) h φ ( x ) = − d dx + V ( x ) ! φ ( x ) = Eφ ( x ) (6)It may be recalled that the parameters α , β must obey certain constraints, viz., [3, 4] α + β < , αβ < V ( x ) in (6) in the supersymmetric form ( w − w ′ ) [11, 12], W ( x ) and w ( x ) are foundto be inter-related through V ( x ) = w ( x ) − w ′ ( x ) = √ − αβ − α − β W ( x ) ! − − α − β ) W ′ ( x ) (8)Since we are dealing with η -pseudo Hermitian Hamiltonians, H obeys the relationship [13] H † = ηHη − or H † η = ηH (9)where ρ is the positive square root of η ( ρ = √ η ) [3, 8].So, whereas φ should obey the conventional normalization conditions, ψ should follow η innerproduct and hence be normalized according to < ψ m | η | ψ n > = δ m,n [13].Now let us look at the continuity condition for φ , viz., ∂∂t φ ∗ φ + ∇ .j = 0 (10)where φ ∗ φ represents the conventional charge density and j the conventional current density,given by j = i ∂φ ∗ ∂x φ − φ ∗ ∂φ∂x ! (11)in 1-dimension.If we apply the inverse transformation of (4) to φ , then the equivalent continuity equation for ψ in the pseudo Hermitian picture can be cast in the form ∂∂t χ + ∇ . ¯ j = 0 (12)3rovided we identify χ and ¯ j with χ = ψ ∗ η ψ , ¯ j = iη ∂ψ ∗ ∂x ψ − ψ ∗ ∂ψ∂x ! (13)Thus χ plays the role of charge density and ¯ j that of current density for the generalized, pseudoHermitian Swanson Hamiltonian. The states of H with discrete energies have been found tohave a one-to-one correspondence with similar states of h . Our aim in this work is to seewhat happens to those states of the Hermitian Hamiltonian h with continuous spectra whenwe go over to the corresponding non Hermitian Swanson Hamiltonian H , with the help of thesimilarity transformation h = ρHρ − . In particular, we shall do so for a couple of explicitexamples, viz., the P¨oschl-Teller and the Morse models. P¨oschl-Teller interaction :
Following [4], to map the non Hermitian Hamiltonian H to its Hermitian Schr¨odinger equiv-alent h , one needs to solve the highly non trivial Ricatti equation (8) analytically. This ispossible only if we take W ( x ) and w ( x ) to be of the same form. For the P¨oschl-Teller model,taking W ( x ) = λ σ tanh σx , w ( x ) = λ σ tanh σx λ , > h is obtained as V ( x ) = λ σ − λ ( λ + 1) σ sech σx = λ σ − αβ (1 − α − β ) − ζ σ sech σx (15)solving which gives the unknown parameter λ in terms of the known one λ : λ = √ ζ −
12 where ζ = λ (1 − αβ ) + λ (1 − α − β )(1 − α − β ) (16)Introducing a new variable y = cosh σx (17)and writing the solutions of h as φ = y λ u ( y ) (18)reduces equation (6) to the hypergeometric equation y (1 − y ) u ′′ + (cid:26)(cid:18) λ + 32 (cid:19) − ( λ + 2) y (cid:27) u ′ − (cid:26) ( λ + 1) − ǫσ (cid:27) u = 0 (19)with complete solution u = A F (cid:18) a, b,
12 ; 1 − y (cid:19) + A (1 − y ) / F (cid:18) a + 12 , b + 12 ,
32 ; 1 − y (cid:19) (20)4here ǫ = E − λ σ (21)and a, b are given below. Therefore, for ǫ <
0, say ǫ = − κ σ , the complete solution of h isobtained as [14] φ = A y ( λ +1) 2 F (cid:18) a, b,
12 ; 1 − y (cid:19) + A y ( λ +1) (1 − y ) / F (cid:18) a + 12 , b + 12 ,
32 ; 1 − y (cid:19) (22)with a = 12 (cid:18) λ + 1 − κσ (cid:19) , b = 12 (cid:18) λ + 1 + κσ (cid:19) (23)Thus for negative energies, bound state normalizable solutions exist only if either a = − n or a + 12 = − n , giving E n = − ( λ − n ) σ , n = 0 , , · · · ≤ λ (24)Consequently, the bound state energies E for the generalized Swanson Hamiltonian H , for thisparticular model, are also given by (24). Since the form of W ( x ) in (14) yields ρ − = (cosh σx ) λ µ (25)the bound state solutions of H , obtained from (22) by applying (4), are found to be ψ ( x ) = A cosh λ µ + λ +1 σx F (cid:18) a, b,
12 ; − sinh σx (cid:19) + A cosh λ µ + λ +1 σx sinh σx F (cid:18) a + 12 , b + 12 ,
32 ; − sinh σx (cid:19) (26)with either A = 0 or A = 0, for well behaved, normalizable solutions; i.e., the eigenstates areeither even ( A = 0) or odd ( A = 0). However, in this work, our interest lies in those statesof H which correspond to the continuum states (i.e., positive energy states, with ǫ = k σ .) of h , rather than the bound states. If φ ( x ) be the continuum states of h , with [7, 14, 16] : φ ( x ) = a φ ( x ) + a φ ( x ) (27)then φ and φ are of the form φ = (cosh σx ) λ +1 2 F (cid:18) a, b,
12 ; − sinh σx (cid:19) (28) φ = (cosh σx ) λ +1 sinh σx F (cid:18) a + 12 , b + 12 ,
32 ; − sinh σx (cid:19) (29)where a = 12 λ + 1 + i kσ ! , b = 12 λ + 1 − i kσ ! (30)5he continuum states of H are obtained (by applying eq (4) to φ ( x )) in the same form as(26), with a, b as defined in (30). Using the asymptotic limit of the Hypergeometric Functions F ( a, b, c ; z ) for large | z | [15], viz., F ( a, b, c ; z ) ∼ Γ ( c ) Γ ( b − a )Γ ( b ) Γ ( c − a ) ( − z ) − a F (cid:18) a, − c + a, − b + a ; 1 z (cid:19) + Γ ( c ) Γ ( a − b )Γ ( a ) Γ ( c − b ) ( − z ) − b F (cid:18) b, − c + b, − a + b ; 1 z (cid:19) (31)along with the fact that when x → ±∞ , − sinh σx → − − e σ | x | , and cosh σx → − e σ | x | ,it can be shown by simple straightforward algebra that the solutions φ have asymptotic be-haviour of the form [14] φ ( x ) = e ikx + Re − ikx , for x < T e ikx , for x > a , a obey certain restrictions, giving the reflection and transmissionamplitudes in (32) by : R = 12 (cid:16) e iϕ e + e iϕ o (cid:17) , T = 12 (cid:16) e iϕ e − e iϕ o (cid:17) (33)with ϕ e = arg Γ( ik/α ) e − i kα log 2 Γ (cid:16) λ +12 + i k α (cid:17) Γ (cid:16) − λ + i k α (cid:17) (34) ϕ o = arg Γ( ik/α ) e − i kα log 2 Γ (cid:16) λ + i k α (cid:17) Γ (cid:16) − λ + i k α (cid:17) (35)Thus the conventional conservation law | R | + | T | = 1 is obeyed.We shall now use the relationship ψ ( x ) = ρ − φ ( x ), to obtain those states of H which correspondto the continuum states of h , and hence possess continuous energies. If these solutions arewritten as ψ ( x ) = B ψ ( x ) + B ψ ( x ) (36)then, applying (4) to (32) yields the following asymptotic behaviour of ψ : ψ ( x ) ∼ e λ µ | x | n e ikx + Re − ikx o , for x < e λ µ | x | T e ikx , for x > R and T as defined in (33). However, unlike in the Hermitian case, here R and T canno longer be identified as the reflection and transmission amplitudes. Hence the plane wavesolutions of the Hermitian Hamiltonian h are replaced in case of the pseudo Hermitian Swansontype Hamiltonian H , by progressive waves for µ >
0, i.e., α > β , or damped waves for µ < α < β . Thus the role played by µ , i.e. the parameters α, β is quite significant in caseof the continuum states. However, for the bound states no significant change is introducedin the nature of the solutions of h and H , depending on the sign of µ . For instance, for α = 18 , β = 14 , λ = 5, we obtain ζ = 384 , µ = − , λ = 9 .
31, implying that thereare ten bound states ( n = 0 , , , · · · ,
9) in both h as well as H [3]. The density function ψ ∗ ψ = τ ( x ) (say) for such a damped wave solution for ψ as given in (37) is plotted in Fig. 1,for the parameter values mentioned above, i.e., α = 18 , β = 14 , λ = 5 , µ = − / τ ( x ) for a progressive wave solution for the same ψ for theparameter values α = 12 , β = 14 , λ = 1 , µ = 1 , so that ζ = 12 and λ = 3. -30 -20 -10 0 10 20 30-0.75-0.5-0.2500.250.50.751 Damped wave solution of H Figure 1:
A plot of Re ψ vs x for the damped wave solutions with continuous spectra, for − ve µ
30 -20 -10 0 10 20 30-15-10-50510 Progressive wave solution of H
Figure 2:
A plot of Re ψ vs x for the progressive solutions with continuous spectra, for +ve µ Morse interaction :
To check whether the results obtained so far are peculiar to the particular interaction studied,we consider a second example, viz., the Morse model. Taking the following ansatz for W ( x )and w ( x ) : W ( x ) = a σ − b σ e − σx , w ( x ) = a σ − b σ e − σx a , , b , > V ( x ) in equation (8) : V ( x ) = a σ + b σ e − σx − b (cid:18) a + 12 (cid:19) σ e − σx = 1 − αβ (1 − α − β ) a σ + 1 − αβ (1 − α − β ) b σ e − σx − (1 − αβ )2 a + (1 − α − β )(1 − α − β ) b σ e − σx (39)with a = √ − αβ − α − β a + 12 √ − αβ − , b = √ − αβ − α − β b (40)Substituting z = 2 b e − σx , φ = e − z/ z s u ( z ) (41)8o that −∞ < x < ∞ transforms over to 0 ≤ z < ∞ , and eq. (6) reduces to u ′′ + (cid:18) s + 1 z − (cid:19) u ′ + a − sz + ǫ/σ + s z ! u = 0 (42)where ǫ = E − a σ . For bound states, ǫ = − κ giving s = ± κ/σ . Thus the bound statesolutions of the Hermitian Hamiltonian h and its corresponding non Hermitian one H arerespectively given by [11], φ n = e − z/ z s L sn ( z ) (43) ψ n = ρ − φ n ∼ e − (1+ µb /b ) z/ z s − µa L sn ( z ) (44)where L sn are associated Laguerre polynomials [15], s = a − n and normalization requirementrestricts s to positive values only. The bound state energies of both the Hermitian as well asnon Hermitian systems are obtained as ǫ n = − ( a − n ) σ giving E n = (2 a − n ) nσ , n = 0 , , , · · · < a (45)However, our aim in this work is to explore those states of H whose Hermitian equivalents arethe continuum states (i.e. positive energies) of h . We proceed in a manner analogous to theprevious section, with the following form of ρ for this model ρ = (cid:18) z b (cid:19) µa e − µb b z , where µb b = α − β √ − αβ (46)Writing ǫ = E − a σ = k , so that s = ± ikσ , the continuum state solutions of h are found tobe [17] φ = A e − z/ z ik/σ F ikσ − a , ikσ + 1 , z ! + A e − z/ z − ik/σ F − ikσ − a , − ikσ + 1 , z ! (47)where F ( a, b, z ) are the Kummer confluent Hypergeometric functions [15]. Since the potential V ( x ) goes to infinity for large negative values of x (i.e., z → φ ( x ) shouldvanish in this region : φ ( x → −∞ ) →
0. Using the properties of confluent Hypergeometricfunctions [15], φ ( z → → A Γ (cid:16) ikσ + 1 (cid:17) Γ (cid:16) ikσ − a (cid:17) + A Γ (cid:16) − ikσ + 1 (cid:17) Γ (cid:16) − ikσ − a (cid:17) → A : A as A A = − Γ (cid:16) ikσ + 1 (cid:17) Γ (cid:16) − ikσ − a (cid:17) Γ (cid:16) − ikσ + 1 (cid:17) Γ (cid:16) ikσ − a (cid:17) (49)9imilarly for large values of x in the positive direction, φ ( x → ∞ ) → A (2 b ) ik/σ e − ikx + A (2 b ) − ik/σ e ikx (50)Thus φ ( x ) can be written in the form (33), with R = (2 b ) − ik/σ Γ (cid:16) ikσ + 1 (cid:17) Γ (cid:16) − ikσ − a (cid:17) Γ (cid:16) − ikσ + 1 (cid:17) Γ (cid:16) ikσ − a (cid:17) (51)and T = 0, which is expected as the potential goes to ∞ at x → −∞ . Instead of going intodetailed calculations, we just quote the solutions of H having a one-to-one correspondence withthose states of h having continuous spectra : ψ ( x → ∞ ) ∼ (cid:18) z b (cid:19) − µa e µb b z φ = e µa σx (cid:16) c e − ikx + c e ikx (cid:17) (52)Once again, we obtain a result identical to that obtained in the previous case viz., dependingon the sign of µ , states with continuous spectra are either progressive waves ( µ >
0) or dampedwaves ( µ <
0) in the non Hermitian generalized Swanson Hamiltonian.To conclude, we have studied states with continuous spectra in a class of η -pseudo HermitianHamiltonians, which are of the generalized Swanson type, viz., H = A † A + α A + β A † , α = β ,and hence can be mapped to an equivalent Hermitian Schr¨odinger-type Hamiltonian h with thehelp of a similarity transformation ρ . We have also obtained the modified continuity equationthe solutions of such non Hermitian Hamiltonians should obey, and obtained new definitionsfor the charge density and current density. In particular, we have analyzed the positive energysolutions for two one-dimensional pseudo Hermitian models, whose Hermitian equivalents arethe P¨oschl-Teller and the Morse interactions. In both the cases it is observed that the relativestrength of the parameters α and β plays a crucial role in determining the nature of the solutionswith continuous spectra. If α > β implying µ >
0, then the continuum states of h representedby plane wave solutions are replaced by progressive waves in case of the corresponding nonHermitian Hamiltonian H . Similarly, for α < β implying µ <
0, the continuum energy states ofthe generalized Swanson Hamiltonian H are represented by damped waves. This asymmetricalnature of the parameters α, β is quite interesting. For µ <
0, since the solutions ψ ( x ) aredamped at ±∞ , the density function for these states rapidly goes to zero as | x | increases.Thus these solutions may be interpreted as bound states in the continuum. In other words, thescattering states of the hermitian problem get transformed to bound states in the continuum forthe corresponding non Hermitian problem. It is worth noting here that with modified definitionsfor current density and charge density for non Hermitian generalized Swanson models, there isno violation in the equation of continuity in either case.10e have plotted the density function, viz., τ = ψ ∗ ψ for the damped wave solution ( µ <
0) forthe P¨oschl-Teller model in Fig. 1, while the density function for a progressive solution ( µ <
The authors would like to thank the referee for instructive criticism. This work was partlysupported by SERC, DST, Govt. of India, through the Fast Track Scheme for Young Scientists(SR/FTP/PS-07/2004), to one of the authors (AS).
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