The WKBJ Wavefunctions in the Classically Forbbiden Region: the Connection Formulae

Transcription

1 Apeiron, Vol. 1, No., April The WKBJ Wavefunctions in the Classically Forbbiden Region: the Connection Formulae K. J.Oyewumi*, C. O. Akoshile, T. T. Ibrahim Department of Physics, University of Ilorin, P.M.B. 1515, Ilorin, Nigeria. D. A. Ajadi Department of Pure and Applied Physics Lautech, P.M. B. 000, Ogbomoso, Nigeria. * As well known, the WKBJ approimation method provides an approimate solution to the Schrödinger equation of a quantum mechanical system; but the method fails at the classical turning points (classically forbidden region) of the system s potential energy function. Solutions about these inaccessible regions are derived by the transformation of Schrödinger equation to either the Airy or modified Airy differential equation. The asymptotic epansion of these solutions are appropriately connected (connection formulae) with the WKBJ solutions to provide full range solutions and these are used to derive the standard energy level formula (Semiclassical quantization rule), which is applied to obtain the modified semiclassical quantization rule.

2 Apeiron, Vol. 1, No., April KEY WORDS: - Schrödinger Equation, Asymptotic epansion, Connection formulae, 1. Introduction An eact solution of the Schrödinger equation is an impractical proposition ecept for the simplest of potentials. In most cases of practical interest, one has to settle for an approimate solution. Thus, several methods of approimation have been devised for tackling various types of problems in Quantum mechanics. The asymptotic or WKBJ approimation provides analytical epressions that pave the way for an approimate description of the mechanisms underlying some physical phenomena [1-7]. With it one obtains provision of simple and good approimate solution to the Schrödinger equation that has been widely used for many approimate calculations of quantum mechanical quantities [8-1]. In particular, this approimate method finds useful applications in theoretical nuclear physics [- ]. The condition for the application of WKBJ approimation is not satisfied at the regions surrounding the classical turning points of the system s potential energy function. However, transformation of the equation of either the Airy or modified Airy differential equation provides more accurate solutions at such inaccessible regions. These solutions are asymptotically etendable to the regions where the WKBJ method is valid, leading to connection formulae. The purpose of this paper is to present a method for the construction of the WKBJ wavefunctions based on the eplicit consideration of matching between the results obtained from the classically allowed and classically forbidden region which is a pure asymptotic approimation technique via the asymptoticity of the Airy and modified Airy functions. These connection formulae are later used to derive the semiclassical quantization condition and also provide its etension to the modified semiclassical quantization

3 Apeiron, Vol. 1, No., April condition. The paper is organized as follows. A brief review of the WKBJ concepts is presented in section ; section 3 contains the derivation of the connection formulae. Section contains the semiclassical quantization condition, while section 5 contains the conclusion.. Review of the WKBJ approimation methods The principle underlying the method is elucidated in the following way; the stationary wave equation which involves a one-dimensional potential V() is d Ψ ( ) + K ( ) Ψ ( ) = 0, (1) d with the local wavenumber 1 µ K( ) = E V ( ), if E > V() () 1 µ = [ V( ) E] = ir( ), if E<V(). (3) If V() = V 0 (a constant), then equation (1) has solution where ( ) Ψ ( ) = ep ± ik, () 0 1 µ K0 = [ E V 0]. (5)

4 Apeiron, Vol. 1, No., April Equation () suggests that if V() is no longer a constant but varies slowly with, slow variation of V() with means, that it varies appreciably only over a small distances and it does approimate a constant potential, i.e., dv ( ) d is small, and provided is not near classical turning point for which we would have [E V()] = 0. We may try a solution of equation (1) in the form Ψ ( ) = ep iu ( ), (6) this converts the linear, time-independent Schrödinger equation for Ψ( ) into the nonlinear Riccati s equation for the function u(), i.e., ( ) ( ) = ( ) iu u K (7) The nonlinearity of equation (7) can be used to develop an iteration procedure to obtain the zero-order approimation solution. 0( ) = ± () u K t dt C, (8) where C 0 is arbitrary constant of integration. Suppose U n () is the n th order iteration, then on re-writing equation(7). Wehave, ( ) () () 1 n+ 1 =± + n + n+ 1 0 u K t iu t dt C. (9) For n = 0, equation (9) gives ( ) () () 1 u =± K t ± ik t dt+ C. (10) Consequence of this procedure to the correct functions, u(), demands that u 1 () should be close to u 0 (). This is possible only if K << K. (11) ( ) ( )

5 Apeiron, Vol. 1, No., April If condition (11) holds, then Binomial epansion of the integrand in equation (10) gives, 0 1 u1( ) ± K( t) dt+ Ln[ K( )], (1) where the constant of integration, C 1, can be absorbed into the normalization of Ψ( ). This first order iteration equation (1) constitutes the WKBJ approimation; and leads to the general approimate wavefunction. By equation (6), ( ) [ 1 ( )] { ep[ ( ) ] ep[ ( ) ]} 0 0 Ψ = K A i K t dt + B K t dt. (13) Whereas in the classically forbidden region (or classically inaccessible region) E < I min (), where the wave number K() = ir(). Then, the general approimate wave function r() > 0 is obtained as, 1 ( ) [ ( )] { ep[ ( ) ] ep[ ( ) ]} 0 0 Ψ = r A rtdt+ B rtdt (1) 3. Derivation of the connection formulae The validity of equations (13) and (1) depends on the condition (11) being satisfied and by virtue of equations () and (3), we have 3 V ( ) µ [ E V ( )] << 1, (15) according to E > V() or vice-versa. Thus, condition(11) breaks either at the classical turning points, where V() = E, or anywhere V() has a very steep gradient, since our iteration procedure already assumed that V() varies slowly with, which implies that V ( ) < 1, it follows that equations (13) and (1) become invalid around the

6 Apeiron, Vol. 1, No., April classical turning points. Hence, the necessity for another approimate solution of the Schrödinger equation that is valid near a classical turning point. Connection formulae give the correct transfer of the approimate solution valid in the classical region, i.e., E > V() or vice-versa. The formulae also depend on whether V ( ) is positive or negative at such turning point. The connection formulae which relate oscillatory and eponential behaviour of the wave forms on the opposite sides of the classical turning points can be matched. Figure 1: Graph of V() showing the intervals (a, b) and (c, d) around the turning points 1 and, where WKBJ approimation is valid. F and H are the classically forbidden regions, while G is the classically allowed region. In Fig. 1, V( 1 ) = V( ) = E, so that the WKBJ approimation is invalid in the interval a< < b and c< < d, since V() is assumed to vary slowly with, we take V() to be a linear function of in ( ab, ) and ( cd, ) ; which are respectively neighbourhoods of the turning points 1 and. For a< < 1, we epand V() in power series about 1 and retain only the first two terms to give V( ) V( ) ( ) V '( ) = E+ ( ) F (16) 1 1 1

7 Apeiron, Vol. 1, No., April Where F = V ( 1), equation (16) is the required linear approimation of V() in ( a, 1) and its use in equation (1) gives µ F Ψ ( ) ( 1 ) Ψ ( ) = 0. (17) The substitution 1 3 µ F ξ = ( 1 ), (18) transforms equation (17) into Airy differential equation Ψ ( ξ) ξψ ( ξ) = 0. (19) In a similar fashion, for 1 <<b V( ) = E ( ) F. (0) Then, equation (1) becomes Ψ ( η) + ηψ ( η) = 0, (1) with µ η F = ( 1). () Equation (1) is the Modified Airy differential equation and has general solution [5] Ψ ( η) = Aη J1 ( η ) + Bη J 1 ( η ) (3)

8 Apeiron, Vol. 1, No., April where J ( η) are the independent solutions of Bessel s differential ±n equation of order 1/3; with A and B representing the arbitrary constants of integration. From equations (18) and (), it is easily seen that: η = ξ. () Therefore, the solution of equation (19) is: Ψ ( ξ ) = ξ 1 ( ξ ) + 1 ( ξ ) AI 3 3 BI 3 3, (5) where I ( ) satisfies the Modified Bessel s differential equation ±n of order n. With the use of the asymptotic epansions of J n () and I n () [5, 6] and the facts that Ψ(ξ) must be normalized (for Ψ(ξ) to be normalized, A = B), then, the above equation becomes 1 3 Ψ ( ξ ) = B 0ξ ep( ξ ), (6) 3 where 3B B 0 =, (7) π is an arbitrary constant. Similarly, putting A = B and using the asymptotic epansion of J ( ) ±n [5, 6] in equation (3) gives

9 Apeiron, Vol. 1, No., April ( ) π Ψ η = B0 η Cos η 3. (8) Equations (6) and (8) give the connected formula from region F(< 1 ) (i. e. classically inaccessible region) to region G( 1 <) with B 0 = 1 as π ξ ep[ ξ ] η Cos η < < (9) Now, using equations (16), (18) and (3) in equation (9), we have the connection formula as, π [ r ( )] ep[ rtdt ( ) ] [ K ( )] Cos Ktdt ( ), 1 (30) so also, the second connection formula at 1 can be obtained as: π [ r ( )] ep[ rtdt ( ) ] [ K ( )] Cos Ktdt ( ). 1 (31) In the similar manner, the two connection formulae at are obtained as:

10 Apeiron, Vol. 1, No., April π 1 [ K ( )] Cos Ktdt ( ) [ r ( )] ep[ ( ) ] rtdt (3) and 1 π 1 [ K ( )] Cos Ktdt () [ r ( )] ep[ () ] rtdt. (33). SEMICLASSICAL QUANTIZATION CONDITION: BOHR-SOMMERFELD QUANTIZATION CONDITION. A simple eample of the application of the WKBJ approimation is presented here that served as a derivation of the lowest-order Bohr-Sommerfeld quantization rule [7, 13, 16, 7, 8]. The aim is to find the energy levels of a particle moving in the 0nedimensional potential well as shown in Fig. 1. For any assumed energy level E, there are just two turning points of the classical motion such that V( 1 ) = V( ) = E. Consider equation (30) which is the connection formula at the classical turning point 1 (i.e. the formula that connects the solution at the classically allowed region with the solution at the classically forbidden region), if

11 Apeiron, Vol. 1, No., April ( ) α1[ ( )] ep[ ( ) ], 1 Ψ = r rtdt < (3) this implies that 1 π Ψ ( ) = α[ ( )] K Cos K( t) dt, 1 < (35) 1 and from equation (3) which is the connection formula at the classical turning point, if then 1 3( ) α3[ ( )] ep[ ( ) ], Ψ = r rtdt <, (36) 1 π ( ) α[ ( )] [ ( ) ], 1 Ψ = K Cos K t dt <. (37) Where α i( i = 1,,3, ) are non-zero arbitrary constants. Hence, in the interval 1 << we have Ψ () = Ψ (). 1 Since[ K( )] 0, Ktdt () = Ktdt () + Ktdt (). (38) 1 1 We can write that:

12 provided Apeiron, Vol. 1, No., April π π π Cos [ K () t dt ] = Cos{[ () ] [ () ]} K t dt K t dt n π = ( 1) Cos[ K( t) dt ], (39) π [ Ktdt ( ) ] = nπ ; n = 0,1,,3..., (0) 1 which implies that i.e. π n α π α Cos [ K( t) dt ] = ( 1) Cos[ ( ) ] K t dt (1) α = ( 1) n α. () Equations (0) and () are the conditions for which Ψ ( ) =Ψ ( ) in 1 << ; in particular equation (0) is re-written as : 1 or classically, 1 1 { µ [ E V( t)]} dt = ( n+ ) π ; n = 0,1,,3.. (3) 1

13 Where Apeiron, Vol. 1, No., April J = π ( n+ ) = ( n+ ) h, n = 0,1,,3,... () J= Ptdt () = Ptdt (). (5) Which is the required semiclassical quantization rule, which can be used to determine the allowed energy values E of a given potentials. As earlier stated WKBJ approimation is a semiclassical approimation, since it is epected to be most useful in the nearly classical limit of large quantum numbers. The method will not be good for, say, the first few lowest states, so in order to overcome this shortcomings there is a need for a modified semiclassical quantization condition. For a particle oscillating between the two classical turning points 1 and, we obtain the semiclassical quantization condition by requiring that the total phase during one period of oscillation to be an integral multiple of π; [13] such that J 1 φ φ = πn, (6) 1 where φ 1 is the phase loss due to reflection at the classical turning point 1 and φ is the phase loss due to reflection at. Equation (6) becomes 1 Ptdt () φ φ = πn. (7) 1

14 Apeiron, Vol. 1, No., April π Taking φ1 and φ to be equal to leads to the modified semiclassical quantization rule, i. e. τ ( φ1 φ) Ptdt () = ( n + ) π ; τ =, (8) π+ 1 where τ is the Maslov inde, which denotes the total phase loss during one period in units of π. It contains contributions from the phase losses φ 1 and φ due to reflections at points 1 and, π respectively. It is pertinent to note that taking φ 1 = φ = and an integer Maslov inde τ = in equation (8), we have the familiar semiclassical quantization rule i. e. equation (3).. 5. Conclusion We have presented a method for the construction of WKBJ wavefunctions based on the eplicit consideration of matching between the solutions obtained from classically allowed and classically inaccessible regions which involves a pure asymptotic approimation technique. The connection formulae derived are applied to obtain the Bohr-Sommerfeld quantization rule (Standard WKBJ quantization rule) and also, this quantization condition is etended to obtain the modified semiclassical quantization rule which involves phase losses at the classical turning points. It is obvious that the modified semiclassical quantization rule will produce more accurate results than familiar (Standard) WKBJ quantization rule. To demonstrate the efficiency of this method, one

15 Apeiron, Vol. 1, No., April can apply this method to find the quantized energy of the Pöschl- Teller potential, Cornell potential and Wood-Saon potential. 3. Acknowledgement We wish to thank Prof. P. Fröman and Prof.(Mrs.) N. Fröman for their interest in this work and for making available their research materials. KJO thanks Emeritus Professor M. E. Grypeos and the Department of Studies Aristotle University of Thessaloniki, Greece for the award of collaborative research to visit the Theoretical Physics Group of the institution where a part of the work has been done.. References [1] G. Z. Wentzel, Physik, 38 (196) 518. [] H. A. Kramers, Z. Physik, 39 (196) 88. [3] L. Brillouin, C. R. Acad. Sci. Paris, 183 (196). [] H. Jeffreys, Proceedings of London Math. Soc., 3 (193) 8. [5] D. L. Dunham, Phys. Rev. 1 (193) 713. [6] N. Fröman and P.O. Fröman, JWKB Approimations, Contribution to the theory, North-Holland, Amsterdam(1966); J. Math. Phys. 18 (1977) 96; J. Math. Phys. (1981) 1190; J. Math. Phys. 9 (1988) 91; J. Math. Phys. 39 (1998) 17. [7] N. Fröman and P. O. Fröman, Physical problems solved by the Phase-Integral method, Cambridge University Press. (00). [8] C. B. Duke, Solid State Phys. Suppl. 10. (1969). [9] J. P. Vigneron and Ph. Lambin, J.Phys. A: Math. Gen. 13 (1980) [10] A. A. Lucas, A. Moussiau, M. Schmeits and P. H. Cutler, Comm. Phys., (1977) 169. [11] J. E. G. Farina, J. Phys. A: Math. Gen. 1 (1988) 57. [1] S. Giler, J. Phys. A: Math. Gen. 1 (1988) 909. [13] V. P. Maslov and M. V.Fedoriuk, Semiclassical Approimation in Quantum Mechanics, Reidel, Dordrecht. (1981). [1] E. A. Bangudu and M. G. Tanko, Nig. J. Math. and Appl. 6 (1993) 39. [15] M. Robnik and L. Salasnich, J. Phys. A: Math. Gen., 30 (1997) 1711 & 1719.

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