wykresx.eps Acta Polytechnica Vol. 51 No. 1/2011 Path Integral Solution of PT-/non-PT-Symmetric and non-Hermitian Hulthen Potential N. Kandirmaz, R. Sever Abstract The wave functions and the energy spectrum of PT-/non-PT-Symmetric and non-Hermitian Hulthen potential are of an exponential type and are obtained via the path integral. The path integral is constructed using parametric time and point transformation. Keywords: PT-symmetry, coherent states, path integral, Hulthen Potential. 1 Introduction A suggestion by Bender and Boetcher on PT- symmetric quantummechanics has put forwardadif- ferent point of view from standard quantummechan- ics. For a quantum mechanical system have to a real energy spectrum, the Hamiltonian must be Hermi- tian. Bender and his co-workers showed that even if a Hamiltonian is not Hermitian, it has a real en- ergy spectrum [1]. PT-symmetric andnonHermitian potentials have been studied to prove they have a real energy spectrum, using numerical and analytical techniques. The energy spectrum corresponding to the wave functions is also calculated [2–9]. In this work, we have used Feynman’s path inte- gralmethod to get the energy spectrumand thewave functions of the PT-/Non-PT-Symmetric and non- Hermitian exponential potential. TheFeynmanPath Integral is a given kernel which has transition ampli- tudes between the initial and final positions of the energy dependent Green function. A Feynman Path Integral formalism for deriving the kernel of various potentials was developed in [10–16]. Duru derived the wave functions and the energy spectrum of the Wood-Saxonpotential for s-wavesvia the radial path integral. Inomata obtained the energy spectrum and the normalized s-state eigenfunctions for theHulthen Potential using the Green function [11]. The kernel of the Hulthen potential can be exactly solved given thepath integral for the particlemotion on theSU(2) manifold S3 [10–12]. In Sec. II and III, we derive the energy dependent Green’s function of the PT- /Non-PT-Symmetric and non-Hermitian q-deformed Hulthen Potential. We obtained the energy eigenval- ues and the corresponding wave functions. 2 PT-Symmetric and Non-Hermitian Hulthen Potential The kernel of a point particle moving in the V (x) potential in one dimension is represented by the fol- lowing path integral K(xb, tb;xa, ta) = ∫ DxDp 2π · (1) exp{i ∫ dt[pẋ − p2 2m − V (x)]} where h̄ = 1. The kernel expresses the probability amplitude of a particlemoving to position xb at time tb fromposition xa at time ta. The time interval can be divided into n equal parts tj − tj−1 = tb − ta = T j =1,2,3, . . . , N (2) and taking initial position is xa and final position xb, the kernel [11] can be performed as K(xb, T ;xa,0)= ∫ ∞ −∞ n∏ i=1 dxi n+1∏ i=1 dpi 2π · (3) exp{i n+1∑ i=1 [pi(xi − xi−1)− p2i 2m − V (xi)]}. The PT- symmetric and non-Hermitian potential is V (x)= − Voe −ix/a 1− qe−ix/a (4) which is determined by taking 1 a −→ i a in the q-deformed Hulthen potential [5]. We will start by applying point transformation to get a solvable path integral form for the Hulthen potential 1 1− qe−ix/a =sin2 θ p = i 2a sinθcosθpθ (5) 38 Acta Polytechnica Vol. 51 No. 1/2011 Because of this transformation, there is a contribu- tion to the Jacobi performed kernel K(xb, T ;xa,0)= q 2a sinθb cosθb ∫ DθDpθ × exp[i ∫ dt(pθθ̇ + sin2 θcos2 θ 4α2 p2θ 2μ − V0 cos2 θ)]. (6) Here, the kinetic energy term becomes positive. We define a new time parameter s [12] to eliminate the sin2 θ cos2 θ 4α2 part in the kinetic energy term dt ds = − 4a2 sin2 θcos2 θ or t = −4a2 ∫ ds′ sin2 θcos2 θ . (7) If we use the Fourier transform of the δ− function, we can write 1 = ∫ dS ∫ dE 2π 4a2 sin2 θb cos2 θb · exp [ −i ( ET − ∫ ds 4a2E sin2 θcos2 θ )] (8) where S = sb − sa. The factor in front of the path integral reached from the Jacobian can be a symmetrization accord- ing to points a and b, as follows 1 sinθb cosθb = 2 √ sin2θa sin2θb · exp ( i ∫ S 0 ds(−i) cos2θ sin2θ θ̇ ) (9) Thus Eq. (6) happens K(xb, xa, T) = ∫ ∞ 0 dSeiS/2μ ∫ ∞ −∞ dE 2π eiET · 4iaq √ sin2θa cos2θb K (θb, θa;S) (10) where K (θb, θa;S)= ∫ DθDpθ · exp { i ∫ S 0 ds [ pθθ̇ − p2θ 2μ − (11) 1 2μ ( K (K −1) sin2 θ + λ(λ −1) cos2 θ ) − ipθ cos2θ 2μsin2θ ]} and K and λ are K = 1 2 [ 1+ √ 32μa2 (V0 + E) ] λ = 1 2 [ 1+ √ 32μa2E ] (12) if the factor contribution to the Jacobian is sym- metrized as [11] the contributions to the kernel be- come θ̇j −→ θ̇j ± icosθj 2μsinθj (13) So the problem is transformed into the path in- tegral for Pöschl-Teller potential, for which an exact solution is known [11]. K (θb, θa;S) can be obtained as K (θb, θa;S)= ∫ DθDpθ · exp { i ∫ S 0 ds [ pθθ̇ − p2θ 2μ − (14) 1 2μ ( K (K −1) sin2 θ + λ(λ −1) cos2 θ )]} The kernel can be obtained in the form K (θb, θa;S)= (15) ∞∑ n=0 exp [ −i(S/2μ)(K + λ +2n)2 ] ψn (θa)ψ ∗ n (θb) where ψn (θ) = √ 2(K + λ +2n) ·√ Γ(n +1)Γ(K + λ + n) Γ ( λ + n + 12 ) Γ ( K + n + 12 ) × (16) (cosθ) λ (sinθ) K P(K−1/2,λ−1/2)n ·( 1−2sin2 θ ) With integrating over dS, the Green’s function for the Hulthen potential can be obtained as G(xb, xa;E)= 8μaq √ sin2θa cos2θb · (17) ∞∑ n=0 ∞∫ −∞ dE 2π eiET (K + λ +2n) 2 −1 ψn (θa)ψ ∗ n (θb) Therefore, the kernel of the physical system is rewrit- ten as K(xb, xa;E) = ∞∑ n=0 e−iEnT ϕn(xa)ϕ ∗ n(xb)= ∞∑ n=0 exp {[ − 1 8μaq(n +1) 2 ·[ (n +1) 2 +2μa2V0 ]2] T } · φn(ub)φ ∗ n(ua). (18) Integrating over dE, we can get the energy eigenval- ues En = 1 8μa2 (n +1) 2 [ 2μa2 V0 q − (n +1)2 ]2 (19) 39 Acta Polytechnica Vol. 51 No. 1/2011 and the normalizedwave functions in terms of Jacobi polynomials are φ(x)= 1 2 √ 2 √ n +1 √ 4(n +1)2 − (λn − Kn)2 ·√ Γ(n +1)Γ(Kn + λn + n) Γ ( λn + n + 1 2 ) Γ ( Kn + n + 1 2 ) × exp[(Kn −1/2)x/2a]( 1+ e−x/a )(Kn+λn−1/2) P(Kn−1/2,λ−1/2)n ·( − 1+ e−ix/a 1− e−ix/a ) (20) where we got Kn = 1 8μa2 (n +1) 2 [ (n +1) 2 −2μa2 V0 q ] ; λn = 1 2 + 1 n +1 [ (n +1) 2 +2μa2 V0 q ] (21) Here we see that the PT Symmetric and Non- HermitianHulthen potential has real energy spectra. 3 Non- PT-symmetric and non-Hermitian Hulthen Potential The non PT-symmetric and non Hermitian Hulthen potential is determined by taking 1 a → i a , V0 → A + iB and q → iq as V (x)= − iVoe −ix/a 1− iqe−ix/a (22) We will follow the same steps for getting the wave function and the energy spectrum. A suitable coor- dinate transformation kernel is obtained as K(xb, T ;xa,0)= q 2a sinθb cosθb ∫ DθDpθ · (23) exp [ i ∫ dt(pθθ̇ − sin2 θ cos2 θ 4α2 p2θ 2μ − V0cos2 θ) ] . If we follow the steps in sec. (2), we will obtain energy eigenvalues En = 1 8μa2 (n +1)2 · [ (n +1) 2 +2μa2 (iA − B) q ]2 (24) and the normalizedwave functions in terms of Jacobi polynomials are φ(x) = 1 2 √ 2 √ n +1 √ 4(n +1) 2 − (λn − Kn) 2 · √ Γ(n +1)Γ(Kn + λn + n) Γ ( λn + n + 1 2 ) Γ ( Kn + n + 1 2 ) × exp[(Kn −1/2)x/2a]( 1+ ex/a )(Kn+λn−1/2) · P(Kn−1/2,λ−1/2)n ( 1− e−x/a 1+ e−x/a ) (25) Kn and λn are the same in Eq. (21). It is clear that the energy spectra are real only if Re(V0)= 0. 4 Conclusion We have calculated the energy eigenvalues and the corresponding wave functions for the PT-/non-PT Symmetric and non-Hermitian Deformed Hulthen Potential. We obtained that PT-/non-PT Symmet- ric and non-Hermitian forms of potentials have real energy spectra by restricting the potential parame- ters. References [1] Bender, C. M.: Reports on Progress In Physics 70(6), 947–1018 (2007); Bender, C. M., Darg, D. W.: Journal of Mathematical Physics 48(4), 042703 (2007); Bender, C. M., Boet- cher, S.: Pys. Lett. 80, 5243 (1998); Ben- der, C. M., Boetcher, S., Meisenger, P. N.: J. Math. Phys. 40, 2201 (1999); Bender, C. M., Dunne, G. V., Meisenger, P. N.: Phys. Lett. A, 252, 272 (1999). [2] Yesiltas, O., Simsek, M., Sever, R., Tezcan, C.: Phys. Scripta, T67, 472 (2003). [3] Berkdemir, C., Berkdemir, A., Sever, R.: Phys. Rev. C72, 027001 (2005). [4] Faridfathi, G., Sever, R., Aktas, M.: J. Math. Chem. 38, 533 (2005). [5] Egrifes, H., Sever, R.: Int. J. Theo. Phys. 46, 935 (2007). [6] Ikhdair, S.M., Sever,R.: J. of Math. Chem. 42, 461 (2007). [7] Ikhdair, S. M., Sever, R.: J. Mod. Phys. E 17, 1107 (2008). [8] Kandirmaz, N., Sever, R.: Chinese J. Phys. 47, 47 (2009). [9] Kandirmaz, N., Sever, R.: Physica Scripta, 87, 3, (2010). [10] Feynmann, R., Hibbs, A.: Quantum Mechan- ics and Path Integrals.McGrawHill, New York, (1965). 40 Acta Polytechnica Vol. 51 No. 1/2011 [11] Duru, I. H.: Phys. Rev. D, 28, 2689 (1983); Duru, I. H., Kleinert, H.: Phys. Lett. B84, 185 (1979); Fortschr. Phys. 30, 401 (1982); Duru, I. H.: Phys.Rev. D, 30, 2121 (1984); Duru, I. H.: Phys. Lett A, 119, 163 (1986). [12] Peak, D., Inomata, A.: J. Math. Phys. 10, 1422 (1969). [13] Erkoc, S., Sever, R.: Phys. Rev D, 30, 2117 (1984). [14] Erkoc, S., Sever, R.: Phys. Rev A, 37, 2687 (1988). [15] Pak, N. K., Sokmen, I.: Phys. Lett. 100A, 327 (1984). [16] Gradshteyn, I. S., Ryzhic, I. M.: Table of Inte- grals, Series, and Products, 2nd ed., Academic Press, New York, (1981). Nalan Kandirmaz E-mail: nkandirmaz@mersin.edu.tr Mersin University Department of Physics Mersin, Turkey Ramazan Sever Middle East Technical University Department of Physics Ankara, Turkey 41