Green's Function of a General PT-Symmetric Non-Hermitian Non-central Potential
aa r X i v : . [ qu a n t - ph ] J un Green’s function of a general PT-symmetricnon-Hermitian non-central potential
Brijesh Kumar Mourya and Bhabani Prasad Mandal
Abstract
We study the path integral solution of a system of particle moving incertain class of PT symmetric non-Hermitian and non-central potential. The Hamil-tonian of the system is converted to a separable Hamiltonian of Liouville type inparabolic coordinates and is further mapped into a Hamiltonian corresponding totwo 2- dimensional simple harmonic oscillators (SHOs). Thus the explicit Green’sfunctions for a general non-central PT symmetric non hermitian potential are cal-culated in terms of that of 2d SHOs. The entire spectrum for this three dimensionalsystem is shown to be always real leading to the fact that the system remains inunbroken PT phase all the time.
Feynman’s path integral (PI) approach to quantum mechanical systems is an ele-gant formalism and powerful in semi-classical calculations [1]. PI formalism whichis generally tied to the Lagrangian formalism of mechanics is an extremely power-ful technique in quantum mechanics. In a class of problems it provides the Green’sfunction with tremendous ease and also provides valuable insight into the relationbetween classical and quantum mechanics. Green’s functions which in mathematicsare to solve non-homogeneous boundary value problems are the backbone of anycalculations of physical quantities in quantum field theory [2]. Thus it is extremelyimportant for any physical theory that the Green’s functions are well defined. The
Brijesh Kumar MouryaDepartment of Physics, Banaras Hindu University, Varanasi-221005,India.e-mail: [email protected]
Bhabani Prasad MandalDepartment of Physics, Banaras Hindu University, Varanasi-221005,India. e-mail: [email protected] purpose of the present work is to discuss the PI formulation of a general non-central,combined parity (P) and time reversal (T) symmetric non-Hermitian system in 3dby calculating the explicit Green’s functions for such a system. Consistent quan-tum theory with real energy eigenvalues, unitary time evolution and probabilisticinterpretation for PT symmetric non-Hermitian theories in a different Hilbert spaceequipped with positive definite inner product has been the subject of intrinsic re-search in frontier physics over the last one and half decades [3]. Such non-HermitianPT symmetric systems generally exhibit PT phase transition or more specifically aPT breaking transition [4] which has been realised experimentally [5]. In-spite ofhuge success and wide applicability [6, 7, 8, 9, 10, 11] of this field the study of non-Hermitian quantum mechanics is mostly restricted to one dimensional or centralpotentials in higher dimension potential problems. In this article we consider a gen-eral and very important non-central PT symmetric non-Hermitian system in the PIformulation. First we show that how the Hamiltonian corresponding to this potentialis reduced to a separable Hamiltonian of Liouville type [12] in a different coordinatesystem. This further enables us to map the system into two non-interacting 2d har-monic oscillators with the appropriate choice of coordinates. We then calculate theGreen’s functions of the system in terms of the Green’s functions of 2d harmonicoscillators. Further we write the Hamiltonian in terms of appropriate creation andannihilation operators to calculate the energy eigenvalues of this non-central non-Hermitian system. We find that the energy eigenvalues are always real as long as theparameters in the potential are real. This indicates that system is always in unbrokenPT phase.Now we present the plan of the paper. In Sec. 2 we calculate the Green’s functionfor the system in terms of SHO Green’s functions. The reality of the spectrum isshown in Sec. 3. Sec. 4 is kept for concluding remarks.
We consider a system described by a general non-central non-Hermitian potentialin 3-dimension in spherical polar coordinates as H = P r m + P q mr + P f mr sin q + V ( r , q ) (1)where the non-Hermitian potential is V ( r , q ) = − a r + B ¯ h mr sin q + iC ¯ h cos q mr sin q . (2) a , B and C are real constants. It is straight forward to check that this non-Hermitiansystem is PT symmetric, where in 3-d in spherical polar coordinates the parity trans-formation is defined as, r → r ; q → p − q , f → f + p .This particular potentialis very important as the Coulomb and the ring- shaped potentials are particular cases reen’s function of a general PT-symmetric non-Hermitian non-central potential 3 of this potential. For C = H = V ( q ) + V ( q ) (cid:26) mW ( q ) p + mW ( q ) p + U ( q ) + U ( q ) (cid:27) (3)This is reduced to a simpler form by using some canonical transformation and re-defining U ’s and V ’s as [12] H = V ( q ) + V ( q ) (cid:26) m p + m p + U ( q ) + U ( q ) (cid:27) (4)The time independent Schrodinger equation corresponding to this Hamiltonian isthen written as ˆ H T y = H T = m p + m p + U ( q ) + U ( q ) − E ( V ( q ) + V ( q )) (5)Now, to obtain the path integral for this Hamiltonian ˆ H T , let us consider theevaluation of this operator for a arbitrary parameter t : (cid:28) q b , q b (cid:12)(cid:12)(cid:12)(cid:12) exp (cid:26) − i ˆ H T t ¯ h (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) q a , q a (cid:29) = (cid:28) q b (cid:12)(cid:12)(cid:12)(cid:12) exp (cid:26) − ( i ¯ h ) (cid:20) m ˆ p + U ( q ) − EV ( q ) (cid:21) t (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) q a (cid:29) × (cid:28) q b (cid:12)(cid:12)(cid:12)(cid:12) exp (cid:26) − ( i ¯ h ) (cid:20) m ˆ p + U ( q ) − EV ( q ) (cid:21) t (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) q a (cid:29) (6)The RHS of Eq. 6 is written in terms of path integral [12]. (cid:28) q b , q b (cid:12)(cid:12)(cid:12)(cid:12) exp (cid:26) − i ˆ H T t ¯ h (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) q a , q a (cid:29) = Z D q D p exp (cid:26) ( i ¯ h Z t (cid:20) p ˙ q − (cid:18) ˆ p m + U ( q ) − EV ( q ) (cid:19)(cid:21) d t (cid:27) × Z D q D p exp (cid:26) ( i ¯ h Z t (cid:20) p ˙ q − (cid:18) ˆ p m + U ( q ) − EV ( q ) (cid:19)(cid:21) d t (cid:27) (7)The parameter t is arbitrary and one can obtain physically meaningful quantity outof Eq. 6 by integrating over t from 0 to ¥ . (cid:28) q b , q b (cid:12)(cid:12)(cid:12)(cid:12) ¯ h ˆ H T (cid:12)(cid:12)(cid:12)(cid:12) q a , q a (cid:29) Semi-classical
Brijesh Kumar Mourya and Bhabani Prasad Mandal = Z ¥ d t (cid:28) q b (cid:12)(cid:12)(cid:12)(cid:12) exp (cid:26) − ( i ¯ h ) (cid:20) m ˆ p + U ( q ) − EV ( q ) (cid:21) t (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) q a (cid:29) × (cid:28) q b (cid:12)(cid:12)(cid:12)(cid:12) exp (cid:26) − ( i ¯ h ) (cid:20) m ˆ p + U ( q ) − EV ( q ) (cid:21) t (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) q a (cid:29) (8)The meaning of RHS of Eq. 8 has been explained in [12]. And the LHS of it iswritten as, (cid:28) q b , q b (cid:12)(cid:12)(cid:12)(cid:12) ¯ h ˆ H T (cid:12)(cid:12)(cid:12)(cid:12) q a , q a (cid:29) = (cid:28) q b , q b (cid:12)(cid:12)(cid:12)(cid:12) ¯ h ˆ H − E (cid:12)(cid:12)(cid:12)(cid:12) q a , q a (cid:29) V ( q a ) + V ( q a ) (9)with the completeness relation Z | q q i dVV ( q ) + V ( q ) h q q | = H = P r m + P q mr + P f mr sin q − a r + b r sin q + i g cos q r sin q (11)where b = B ¯ h m ; g = C ¯ h m . This system will be reduced to separable system of Liou-ville type in parabolic coordinate system. To express the Hamiltonian in paraboliccoordinates ( x , h , f ) , it is useful to first express this potential V ( r , q ) in cylindricalcoordinates ( r , f , z ) . In cylindrical coordinate the potential looks like, V ( r , z ) = − a p r + z + br + i g z r p r + z ; r = x + y (12)The parabolic coordinates are expressed in terms of cylindrical coordinates as x = (cid:18)q r + z − z (cid:19) ; h = (cid:18)q r + z + z (cid:19) ; f = f . (13)Now the potential in Eq. 12 in terms of these parabolic coordinates, is V ( x , h ) = − ax + h + b xh + i g ( h − x ) hx ( h + x ) (14)and the Hamiltonian in parabolic coordinate is written as reen’s function of a general PT-symmetric non-Hermitian non-central potential 5 H ( x , h , f ) = m ( x + h ) h x ˆ p x + h ˆ p h i + m hx ˆ p f + V ( x , h ) (15)We define variables u , v x = u ; 0 ≤ u < ¥h = v ; 0 ≤ v < ¥ (16)and perform a canonical transformation p x ˆ p x = ˆ p u ; √ h ˆ p h = ˆ p v (17)to simplify the kinetic term in H in Eq. 15 as H ( u , v , f ) = u + v (cid:26) m (cid:20) ˆ p u + ˆ p v + ( u + v ) ˆ p f (cid:21) − a + b + i g u + b − i g v (cid:27) (18)This is further written compactly as H ( u , v , f ) = u + v (cid:26) m (cid:20) ˆ p u + ˆ p v + u ˆ p f + v ˆ p f (cid:21) − a (cid:27) (19)where ˆ p f = ˆ p f + m ( b + i g ) ; ˆ p f = ˆ p f + m ( b − i g ) (20)Note ˆ p f and ˆ p f are not Hermitian but complex conjugate to each other. ThisHamiltonian is still not a separable one of Liouville type. We further consider thetotal Hamiltonian H T (= H − E ) with E = − m w for the bound state case ( E < H T = m (cid:20) ˆ p u + ˆ p v + u ˆ p f + v ˆ p f (cid:21) − a + m w ( u + v ) (21)Now we introduce the components of 2-dimensional vectors u and v as u = ( u , u ) = ( u cos f , u sin f ) v = ( v , v ) = ( v cos f , v sin f ) (22)to have p u = ˆ p u + u ˆ p f ; p v = ˆ p v + v ˆ p f ; (23)Putting all these in Eq. 21 we obtainˆ H T = m p u + m w u + m p v + m w v − a (24) Brijesh Kumar Mourya and Bhabani Prasad Mandal
This is the Hamiltonian which is separable of Liouville type. Thus, the Hamilto-nian for the non-central potential has been reduced to that of two 2-dimensionaloscillators (apart from some constant shift in ground state energy ).Now by using Eq. 6 for this separable Hamiltonian, we find the path integral forthe non-Hermitian non-central potential exactly as (cid:28) u b , v b (cid:12)(cid:12)(cid:12)(cid:12) exp (cid:20) − i ˆ H T t ¯ h (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) u a , v a (cid:29) = e i at ¯ h (cid:28) u b (cid:12)(cid:12)(cid:12)(cid:12) exp (cid:20) − ( i ¯ h ) (cid:18) p u m + m w u (cid:19) t (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) u a (cid:29) × (cid:28) v b (cid:12)(cid:12)(cid:12)(cid:12) exp (cid:20) − ( i ¯ h ) (cid:18) p v m + m w v (cid:19) t (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) u a (cid:29) = e ( i at ¯ h ) (cid:18) im w p i ¯ h sin wt (cid:19) exp (cid:26) im w h sin wt (cid:2)(cid:0) u b + v b + u a + v a (cid:1) cos wt − u b · u a − v b · v a (cid:3)(cid:27) (25)where the exact result for one dimensional simple harmonic oscillator has been used[1]. (cid:28) q b (cid:12)(cid:12)(cid:12)(cid:12) exp (cid:20) − i ¯ h (cid:18) ˆ p m + m w q (cid:19) t (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) q a (cid:29) = (cid:16) m w p i ¯ h sin wt (cid:17) exp (cid:26) im w h sin wt (cid:2) ( q a + q b ) cos wt − q a q b (cid:3)(cid:27) (26)The Eq. 25 contains the arbitrary parameter t and has to be eliminated to obtainphysically meaningful quantity. This can be done by integrating over t from 0 to ¥ in both side of the Eq. 25. When we integrate over t in the LHS of the Eq. 25, itis nothing but the Green’s functions of the operator H − E as discussed at beginningof this section. And the integration in the RHS can be done in a straightforwardmanner [12]. Thus we obtain the explicit Green’s functions for the system of noncentral non-Hermitian potential. Since this system with non-Hermitian, non-central potential is equivalent to two 2dSHOs, conjugate to each other we define the creation and annihilation operators forthis theory as a k = √ (cid:20)r m w ¯ h u k + i √ m w ¯ h ˆ p u k (cid:21) ˜ a k = √ (cid:20)r m w ¯ h v k + i √ m w ¯ h ˆ p v k (cid:21) (27)where k = , reen’s function of a general PT-symmetric non-Hermitian non-central potential 7 creation and annihilation operators as follows:ˆ H T = ¯ h w " (cid:229) k = (cid:16) a † k a k + ˜ a † k ˜ a k (cid:17) + − a (28)and the conjugate momentum variables are written asˆ p f = i ¯ h h a †1 a − a †2 a i ˆ p f = i ¯ h h ˜ a †1 ˜ a − ˜ a †2 ˜ a i (29)We further perform an unitary transformation of the following type, a = √ ( b − ib ) a = √ ( − ib + b ) (30)and similar transformations for ˜ a , ˜ a also in Eqs. 28 and 29 to get,ˆ H T = ¯ h w " (cid:229) k = (cid:16) b † k b k + ˜ b † k ˜ b k (cid:17) + − a (31)and ˆ p f = ¯ h h b †1 b − b †2 b i ˆ p f = ¯ h h ˜ b †1 ˜ b − ˜ b †2 ˜ b i (32)The number operators, n k = b † k b k , ˜ n k = ˜ b † k ˜ b k are defined for this system. In termsof number operators the total Hamiltonian in Eq. 31 is now written asˆ H T = ¯ h w [ n + n + ˜ n + ˜ n + ] − a = h w (cid:20) n + ˜ n + + ˆ p f + ˆ p f ¯ h (cid:21) − a (33)Now considering (cid:2) ˆ p f + ˆ p f (cid:3) f phy ≡ lf phy , the physical state condition is [12], [ ( n + ˜ n + ) ¯ h w + wl − a ] f phy = l as, E n , ˜ n , l ≡ − m w = − m a [ ( n + ˜ n + ) ¯ h + l ] . (35) l can be calculated easily using (cid:2) ˆ p f + ˆ p f (cid:3) f phy ≡ lf phy and Eq. 20 as Brijesh Kumar Mourya and Bhabani Prasad Mandal l = ¯ h (cid:20)(cid:0) n + B + iC (cid:1) + (cid:0) n + B − iC (cid:1) (cid:21) = (cid:20)(cid:0) n ¯ h + m ( b + i g ) (cid:1) + (cid:0) n ¯ h + m ( b − i g ) (cid:1) (cid:21) (36)where n is non-negative integer and l is real as l = l ∗ . Therefore, the completereal bound state spectrum for the problem is E n , ˜ n , n = − m a h (cid:20) n + ˜ n + + √ n + B + iC + √ n + B − iC (cid:21) = − m a (cid:20) ( n + ˜ n + ) ¯ h + √ n ¯ h + m ( b + i g )+ √ n ¯ h + m ( b − i g ) (cid:21) (37)The corresponding result for the real potential agrees with that of in Refs. [15, 16]where energy spectrum has been calculated by solving Schroedinger equation usingcomplicated KS transformation [17, 18]. The Hamiltonian corresponding to the PT symmetric non-Hermitian non-central po-tential in Eq. 2 has been mapped into a Hamiltonian of two 2d harmonic oscillatorsby choosing appropriate coordinate system and using a suitable canonical transfor-mation. Next we have calculated the Green’s functions for the system using path in-tegral method for this separable Hamiltonian of Liouville type. The exact spectrumare calculated by writing this Hamiltonian in terms of creation and annihilation op-erators of 2d SHO. The entire spectrum is real for any real values of the parameters a , b and g indicating that system is always in unbroken phase. Acknowledgments : BPM acknowledge the financial support from the Depart-ment of Science and Technology (DST), Govt. of India under SERC project sanctiongrant No. SR/S2/HEP-0009/2012.
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