摘要:

A sufficient condition for the angle parameter of the complex dilatation transformation (3) where S� is the generator (4) Application of Û( q) to the wave function y(r) and to the Hamiltonian operator H �yields (5) And where p�is the conjugate momentum operator. It has been shown7, 8 that the complex scaled Hamiltonian operator H �(rei q, p�e–i q) for a system with resonances has the following series expansion: (8) (9) is the unscaled Hamiltonian operator and V0(r)=V(r) is the potential function. Meishan Zhao and Stuart A. Rice Department of Chemistry and James Fanck Institute, The University of Chicago, Chicago, Illinois 60637, USA Received 18th January 1999, Accepted 4th February 1999, Published 12th February 1999 We discuss a sufficient condition for the determination of the angle parameter of the complex dilatation transformation that is often used for the location of dynamical resonances.We show, as one example, that the proposed condition yields the exact value of this angle parameter for the case of a harmonic potential. The formulation of the sufficient condition permits it to be used for general potentials. We illustrate the utility of the proposed method by calculating the location of the resonance supported by a potential introduced in a model study by Moiseyev, Certain and Weinhold. The results obtained are very close to, but more accurate than, those previously reported, and they are obtained with less computational effort. Resonances1,2 are ubiquitous in the spectra of decaying systems.Since the analysis of nuclear a-decay by Gamow, it has been popular to represent the energy of a resonance in the form Eres=E – i G/ 2 , where G is the full width at half-maximum of the resonance. However, the Hilbert space structure of quantum mechanics requires, for a given state, that the physical observables are Hermitian and that their expectation values are real. Thus, the assumption that the energy of a resonance is complex implies that the Hamiltonian operator is not Hermitian or that the resonance wave function is not in the Hermitian domain of the Hamiltonian in the basic Hilbert space. Put another way, the Hilbert space that is commonly used to develop the basic theory of quantum mechanics does not describe resonance states naturally.A better representation of the relationships between resonances and the other states of a system was introduced in the pioneering work of Balslev and Combes3,4 in the 1970s. The key element of their analysis is the definition of an extended Hilbert space which includes the non- Hermitian domain. This extended Hilbert space, called a rigged Hilbert space, is generated by complex scaling (dilatation transformation) of the coordinates of the particles in the system. In this communication, we analyze a sufficient condition for the determination of the scaling parameter that appears in the complex dilatation transformation used to study dynamical resonances. The dilatation transformation is defined by the replacement of the ordinary particle coordinates by complex coordinatesvia the substitution3–6 (1) where h and q are scaling parameters.The parameter h does not have direct physical significance; it is used only to enhance the rate of convergence in numerical calculations. Therefore, we take h=1 and write (2) The dilatation transformation is executed by introducing the scaling operator PhysChemComm, 1999, 1 (6) (7) where (10) andConventionally, q is determined via the so called complex virial theorem, which requires that (11) where the complex scaled "energy" E( q) is the expectation value of the complex scaled Hamiltonian operator H �(rei q, p�e–i q), (12) In practical computations, q is determined by an iterative process, since for each value of q an eigenvalue problem must be solved first.We note the real and imaginary parts of complex valued "energy" E( q) can be obtained, using the Hamiltonian displayed in eqn. (7), in the form (13) (14) The non-Hermitian operator H �(rei q, p�e–i q) has admissible eigenfunctions, y(r), only when its real and imaginary parts commute. Therefore, we shall calculate q by enforcing the commutation relation9 (15) In order to simplify the analysis, we write ÊR( q) and ÊI( q) in a different form. One can show that ÊR( q) and ÊI( q) can be written (16) (17) Then we have (18) or (19) where we have used the identity (20) The commutator can be further simplified by the use of standard operator algebra.Specifically, we find that (21) Thus, a sufficient condition for eqn. (15) to be valid is (22) eqn. (22) is a linear operator non-polynomial differential equation; it belongs to the class of ordinary homogeneous differential equations of the typel0 (23) Obviously, the solution to eqn. (22) is different for each potential V(r). To illustrate the use of eqn. (22) we consider two cases. (i). Suppose (24) If the eigenvalues and eigenfunctions of this linear operator L �are l and y, respectively, (25) then (26) and the sufficient condition becomes (27) The necessary conditions are then (28) or, alternatively, (29) The eigenvalue equation Ly= ly can thus be writtenand solved formally to yield where A is an arbitrary constant.It is now clear that the special case defined by eqn. (24) corresponds to power law potentials. For the harmonic potential V(r) ® r2, we have11 (32) or (33) which is the known exact solution. (ii). Now we extend the condition in eqn. (22) in a somewhat different representation by writing sin (2 q + qrd/dr) as so that Recalling the definition of the complex scaling operator Û( q), and using the standard commutation relation (36) we obtain and then we have We now define the complex coordinate (40) (30) (31) (34) (35) (37) (38) (39) to thereby obtain (41) Eqn. (41) is another expression of the sufficient condition for the determination of the parameter q. Again considering the harmonic oscillator V(z) ® z2, we have (42) Thus (43) or (44) as before.The use of the proposed sufficient condition for other potentials looks promising, since the needed numerical calculations should be less tedious than the iterative calculations required when the complex virial theorem condition for the determination of q is employed. We illustrate this point by examining one example of the calculation of the location of a resonance supported by a nontrivial potential using the method outlined above. Consider the potential function (45) where a and b are parameters. This potential (expressed in a.u.) has the following attributes: (i) It supports predissociation resonances analogous to those found in diatomic molecules.(ii) For the particular case that a=0.8 a.u. and b=0.l a.u. it supports a single bound state with energy E=0.502 a.u. (iii) For the same values of a and b it supports a broad resonance just below the top of the barrier, near E=2.1 a.u. The potential displayed in eqn. (45) was used by Moiseyev et al.12 in studies of the locations of dynamical resonances calculated using the complex scaling analysis. In particular, these investigators used the so called q-trajectory method to calculate the resonance locations. They found that the resonance is located at (46) corresponding to q » 17.8º. We now present a calculation of the location of the resonance supported by the potential displayed in eqn. (45) using the method proposed in this paper.We have(47) For the same parameter values as used by Moiseyev et al., namely a=0.8 a.u. and b=0.l a.u., the potential barrier is located at the solution of (48) or x2 =11.6. We then obtain q from the solution to (49) or (50) This equation can be solved easily with very high accuracy. We find q=18.14º, to be compared with q » 17.8º determined from the q-trajectory analysis reported by Moiseyev et al. In closing we note that the condition 0=Im[ei3 qdV(z)/dz]z=rei q may support an infinite number of solutions for q. However, the dynamical resonances are determined on the second Riemann sheet of the complex energy plane, and the rotational parameter q is restricted to lie in the range (0, p/2). Then the only physically meaningful valuethe domain (0, p/2).All other values of q are excluded from consideration. This research has been supported by a grant from the National Science Foundation. References 1 E. P. Wigner, Phys. Chem. (Munich), 1932, B19, 203. 2 See, for example, Resonances, ed. E. Brandas and N. Elander, Springer-Verlag, Sweden, 1987, Resonances- Models and Phenomena, ed. S. Albeverio, L. S. Ferreira and L. Streit, Springer-Verlag, New York, 1984. 3 Ch. Obcemea and E. Brandas, Ann. Phys. (NY), 1983, 151, 383. 4 E. Balslev and J. M. Combes, Commun. Math. Phys., 1971, 22, 280. 5 W. P. Reinhardt, in Mathematical Frontiers in Computational Chemical Physics, ed. D. G. Truhlar, Springer-Verlag, New York, 1988, p. 41. 6 C. A. Chatyidinmitriou-Dreismann, Adv. Chem. Phys., 1991, 80, 201. 7 M. Zhao and S. A. Rice, J. Phys. Chem., 1994, 98, 3444. 8 S. A. Rice, S. Jang and M. Zhao, J. Phys. Chem., 1996, 100, 11893. 9 See, for example, E. Merzbacher, Quantum Mechanics, 2nd edn., John Wiley & Sons, New York, 1970. 10 See P. M. Morse and H. Feshbach, Methods of Theoretical Physics, McGraw-Hill, New York, 1953, ch. 5. 11 O. Atabek, R. Lefebvre, M. Garcia Sucre, J. Gomez- Llorente and H. S. Taylor, Int. J. Quantum Chem., 1991, 40, 211. 12 N. Moiseyev, P. R. Certain and F. Weinhold, Mol. Phys., 1978, 36, 1613. Paper 9/00479C PhysChemComm © The Royal Society of Chemistry 1999

ISSN:1460-2733    

DOI:10.1039/a900479c

出版商:RSC    

年代:1999

数据来源: RSC