Tunnelling through the Morse barrier

Volume 157, number 1

PHYSICS LETTERS A

15 July 1991

Tunnelling through the Morse barrier Zafar Ahmed Nuclear Physics Division, BhabhaAtomic Research Centre. Bombay 400085, India Received 16 July 1990; revised manuscript received 16 January 1991; accepted for publication 17 May 1991

Communicated by J.P. Vigier

The exact and the WKB forms ofthe transmission coefficient of the Morse barrier have been obtained. Comparing the results obtained from the exact and the WKB analysis on different types of potentials, we present a possible criterion to ascertain the performance of the WKB method. Extraction of bound states of a potential from the transmission coefficient of the inverted potential is also suggested.

The tunnelling through one-dimensional potential barriers has, since the infancy of quantum mechanics, been playing an important role in many branches of physics [1,2]. However, there are not many potentials which admit a simple form for transmission amplitude t(k) and transmission coefficient T(k). Moreover, the fact that the simple poles in the k-plane oft(k) yield the bound states, metastable states, resonances and virtual states [3] of a potential adds to the motivation of obtaining t(k) for a potential. In this Letter, the transmission amplitude of the Morse barrier (inverted Morse oscillator potential) is obtamed. The Morse oscillator potential has long been used to investigate the anharmonicities of the vibrational spectra in molecular and nuclear physics. More recently, there has been a revival of interest in the quantum mechanics of the Morse oscillator [4,5], due mainly to an extensive study ofthe grouptheoretical properties of the corresponding Schrödinger equation. At this juncture when the bound states, the scattering states, the propagator and the supersymmetric properties have been well discussed, it seems natural to discuss the Morse barrier for which the transmission coefficient has remained elusive in The exact and the WKB expressions for the transthe past. mission coefficient of the Morse bamer are given. The Morse barrier and the Eckart barrier are used to infer a sharper condition on the validity of the semiclassical WKB approximation [6] and its insensi0375-9601/9 1/S 03.50 © 1991

—

tivity towards the asymptotic behaviour of the potentials. Thebound states ofthe Morse oscillator have been retrieved from the transmission coefficient. The Morse barrier potential (fig. 1) V(x) = V0[2 exp(x/a) —exp(2x/a)]

(1)

offers repulsion to an incoming particle (x< 0) and after a finite distance, it provides to the outgoing particle an asymptotically divergent attraction. Because of this divergent nature, finding T(k) by numerical integration is rendered infeasible. We insert the Morse barrier potential (1) in the Schrodinger equation 2 d2 ~x) h —

+V 0 [2

exp(x/a)

—

exp (2x/a)] 9’(x)

~

=EW(x)

forE> 0.

(2)

v

~

Elsevier Science Publishers B.V. (North-Holland)

_______

x -.

Fig. I. The Morse barrier potential.

Volume 157, number 1

PHYSICS LET1TERSA

15 July 1991

Substituting u =p exp (x/a) in (2) and eliminating the first derivative of W( u), we get

~P(x,t)= (2au)~2exp(xa/2)Mjp,ja(u)

d2w(u)+(fl22fl2+ ~+a2~ du2 ~2 pu u2 )w(u)=O~

or

Xexp(—iEtfll),

(3)

where~P(u)=(1/,.J~)w(u).Inordertobring(3)to the standard Whittaker equation [8] form, we set 2=—i2a2 to write p=2ifl and a d2~v(u)+(!+~+ ~—i2a2\ du2 4 u u2 ~ (4) 2/h2)”2and where we have introduced /J=(2~tV0a a= (2~Ea2/h2)”2or a=ka. There are eight forms for the solutions of ~v(u) (w( u) = Z~i = 1, 8) and their properties have widely been worked out [8]. Using Z 7, one time-dependent solution of the Schrodinger equation (4) can be written as 2Z x,t)=exp(xfl/2)u~ 7exp(—iEt/h) (5a) ...~

(6b)

1(x,t)=(2au)_2exp(7a/2)exp(_u/2)u2+I~~

x

1 F1 (~+ ia — ifl, 1 + 2ia; u) exp ( — iEt/h). (6c)

Using 1F1(a, ~)=exp(~)1F1(b—a, —~) and 1F1(a, b; O)=1b; [7], we observe that asb;x—’—~x ~P(x,t)’-.~exp(x/2a) xexp{i[flexp(x/a)+kx—Et/h]} (6d) indicating that eqs. (6) represent a wave incident on ,

the barrier from the left. Another choice of the function çv( u) is Z2 which leads to a wave function 2exp(— xa/2)Z W(x, t)= (2au)~ 2

,

Xexp(—iEt/fz),

(7a)

where u = 2ifièxp (x/a) or in Whittaker’s notation (MK,m, Wrm)

2 ~P(x,t)=exp(xfl/2)[2iflexp(x/a)]” X W_jp,ja(2iflexp(x/a)) exp(—iEt/Fz). (5b) In order to find the asymptotic behaviour of the wavefunction (Sb), we use its confluent hypergeometric function representation [81and write W(x,t)=exp(xfl/2)u~”2exp(u/2)(—u)”2~1’~ xU(1+icr+i/3,1+2ia;—u)exp(—iEt/h).

or W(x,t)=(2au)~’2exp(—xa/2) XMjp,_j~(—u)exp(—iEtfll),

(7b)

which in terms of the confluent hypergeometric functions can be expressed as !P(x,

t) =

(2au)

—

1/2

exp ( — xa/2) exp (

—

u/2)

Xuh/2~~xiFi(~_ia_ifl,1_2ia;u) Xexp(—iEl/h).

(7c)

(5c)

which denotes a wave moving from left to right (transmitted) through the barrier. Another relevant form of the solution çv( u) ofour interest is Z1, which gives 2exp(xa/2)Z ~P(x,t)=(2au)~’ 1 exp(—iEt/h), (6a)

Since (7b) can be obtained from (6b) by changing a to —a, one can easily check that Y’(x, 1) (7) denotes a wave travelling to the left after being reflected from the barrier. Henceforth, for convenience, the wave function W(x, t) in eqs. (5), (6) and (7) will be denoted by !P, ~t~and!P 1.respectively. We note that ZT=Mip,_ja(—u)=Z4, Z~=M_~,~ ( — u) = Z3 and Z~= Wjp, —ia (u) = Z6 or Z5. Next, we use [8] Z4=exp[—i~(~—ia)]Z2, Z3 = exp [— i~t( ~+ ia) ] Z~and the Wronskian relations [Z~, Z2] = —2ia, [Z5, Z7] =exp( —,~fl)to obtam the following Wronskian relations for the wavefunctions,

or

[~Pr,Y~]=i,

Using the limit ~ x—~x,we get

U(a, b,

~

~P(x,t)—~exp(—x/2a) xexp{i[flexp(x/a)—flx/a—Et/1l]},

2

[8]as

(Sd)

[!1rl~~,lFr]=_i,

Volume 157, number 1

[~P~’, !P]

PHYSICS LETTERS A

= i.

(8)

By virtue of a standard connection formula among ZI=MKm(U), Z

2=Mic,_m(U)

[8] viz

and Z7

15 July 1991

smallness of which makes the transmission through the barrier behave semi-classically, thereby resulting in a good performance of the WKB method. In the

W_pc,m(U)

(9)

limit (4-+0) h—~0thetransmission coefficients (l2a) and (13) degenerate into the classical limit, i.e. T(E)—~O(E—V0). Further, the curvature of the top of the barrier, 2V/dx2 I — [1+(dV/d\213/21 c0—~ d I J Ix=O

one can write the desired linear connection among ~r and !P as

for the Morse barrier becomes 2 V 2, suggesting that the WKB will work well for wider0/a barriers with lesser

(10)

topcurvature. The fully repulsive Eckart potential, V(x) = V 2(x/a), which converges on both sides (x—~ ±0sech cc), is another idealization of potential barriers which admits an analytic form for the transmission coefficient [1] expressible as

sin(2itm) it

~(

exp[—iit(~+m)]Z1 f’(~—m±K)F(1+2m)

=

~

+F(i+m+K)T(l2m))~ exp[—ix(~—m)]Z2 \

~,

with the transmission amplitude t(k) and the reflection amplitude r(k) defined by t(k) = (2a)”2 exp[

—

7t(a+fl)/2]

1(~—ia+ifl) X

(ha)

I’(1—2ia)

T(E)=

and

cosh(2xa)—l cosh(2xa)+cos(2,tw)’ (14)

r(k)=exp(—ita)

For very small values of A, or E> 4 and also V

F(~—ia+ifl)r(l+2ia) X F(~+ia+ifl)F(l—2ia)

(llb)

respectively. The current densities are calculated using (8), subsequently we obtain —

T(k)=t*(k)t(k) =

exp( 4ica)

(1 2a)

—

1+ exp [2~t(fl—a)]

and

exp ( 2ica) +exp (2itfl) R(k)=r~(k)r(k)= exp(2ita)+exp(2irfl) —

transmission coefficient of the Eckart barrier. The coincidence of the WKB transmission coefficient for the Morse and the Eckart barrier (which behave differently at x—’ no) suggests an insensitivity of the WKB approximation towards the asymptotic behaviour of the potentials. 2], purely incidentally, the WKB and the transFor the parabolic barrier, V(x)= V0[l exact (x/a) mission are the same i.e., T(E)={l+ exp[2x(V 2]}’ [2]. On the other hand, the 0—E)/(4V04)” Morse and the Eckart barriers which are two different and generic instances, do render a sharper condition for the validity of the WKB approximation. Herein the smallness of 4 ensures the validity of the WKB approximation, or else when —

—

‘

a=(E/4)’1’2, fl=(V

2. (12b) 0/4)” In an interesting approximation when exp ( — 4ita) is ignored in (1 2a), we get T(E) = (1 +exp{2it [ ( V~/4)1/2

0> 4 (unlike the Morse banier), T(E) degenerates into (13) which once again turns out to be the WKB

(E/A)”2 1 } ) —

(13)

V

0 >4 the

an expression that can otherwise be obtained by using the WKB method which is known to work well for the slowly varying potential profiles at asymptotically large energies. Therefore, the WKB approximation is2/2~ta2). expected4to very well for energies is work an interesting parameter, the E>A (= h

WKB method will work very well at ener-

gies E> 4. In the one-dimensional barrier penetration model [2] of the heavy-ion fusion the calculation of the fusion cross section is done by obtaining the transmission coefficient effectivebarrier formed by short-range nuclearofthe attraction and Coulomb plus 3

Volume 157, number 1

PHYSICS LETTERS A

centrifugal repulsion. To this end, one either uses the WKB approximation for T(E) or parametrizes the fusion interaction by a potential barrier (V0 and C0 or A are the parameters) which admits an analytic form for T(E). The typical values of the barrier heights are given by V 0(MeV) Z1Z2/ (A ~ + A ~/3), where Z and A refer to the atomic number and atomic mass number of the colliding nuclei, respectively. The typical values of A are given by A(MeV)~4/V0. In the light of the present discussion, we would like to emphasize that the smallness of A and the large values of V0 underlie the success of the WKB method in heavy-ion fusion. A fusion barrier should have a Coulombic tail; however, owing to the insensitivity of the WKB method towards the tail ofa potential, its parametrization by the parabolic [2] shape works satisfactorily. The physical interpretation of the singularities of t ( k) in the complex k-plane facilitates the nature of the energy eigenspectrum of the potential [3]. For instance, the simple poles of 1(k) lying on the upper half of the imaginary line in the complex k-plane (physical sheet) are known to represent the possible bound states of the potential [3]. By changing fi to iô (o=~JIV0/AI) and k to ik~(note that a=ka) in eq. (11 a) we get the transmission amplitude for the Morse oscillator potential t0 ( k) as I 2

t0(k)=(2ia~) / exp[—jt(ia~+iô)] x1(~+a~—ô)/F(l+2a~)

(15)

,

which has simple poles (ku) in the k-plane, k~= [ö— (n + ~)] Ia if ô> n+ ~ (physical sheet)

(16)

.

From these, the well-known bound states of the 2k~J2fL) as Morse oscillator are recovered (E~= —h n=0,h,2,..., [(V

2—fl (17) 0/4)” Alternatively, by changing k to ik~and fi to iô (V 0—* — V0) in the transmission coefficient T(k) of eq. (1 2a), we can locate the possible simple poles of T(k). They are k~=

[d±

(n+~)]/a.

.

(18)

In order to keep k~on the physical sheet (k~>0)and 4

15 July 1991

the number of bound states finite, the positive sign here has to be discarded and n has to be restricted 2k~/2j~ in as n<ô— = of h the Morse (16) we againFinally, retrieveby thesetting bound E~ states oscillator potential as given by (17). ~.

—

In fact, when V0 is changed to — V0, similar poles of the WKB transmission coefficient (13) should yield the semiclassical bound states consistent with the Bohr—Sommerfeld quantization law, which incidentally turn out to be the exact eigenvalues as given by (17). For a potential that has a minimum, if there exist two real turning points (roots of E= V(x)) at any energy (positive or negative), an infinite number of bound states will be realized (finite otherwise). Due to the very definition of T( k), i.e. T(k) = t” (k) 1(k), it should be clear that half the number of poles of T(k) are redundant and do not give correct bound states. Therefore the condition k~>0and the knowledge of the cardinality ofthe bound states enable one to rule out those redundant poles. It may be interesting to check that this constitutes a general method of obtaining the eigenvalues of the bound states of a potential from the transmission coefficient of the inverted potential. Lastly, it may be noted that the wave functions given by eqs. (5)— (7) suggest corrections to the wave functions of the Morse barrier as obtained by the complex coordinate method in ref. [4]. In the light of the present results, the Morse barrier comes in the class of the barrier models that admit a simple analytic form for the transmission coefficient. The other interesting potential barrier models which have lately been solved can be found in refs. [3,9,10]. I would like to thank Dr. Asish Kumar Dhara and Sudhir Ranjan Jam for discussions. References

[1] S. Bjørnholm and J.E. Lynn, Rev. Mod. Phys. 52 (1980) 725. [2] M. Beckerman, Rep. Prog. Phys. 51(1988)1047; C.Y. Wong, Phys. Rev. Lett. 31(1973) 766. [3]J.N. Ginocchio, Ann. Phys. (NY) 152 (1984) 203. [4] A.O. Barut, A. Inomata and R. Wilson, .1. Math. Phys. 28 (1987) 605.

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Gursey and F. Iachello, Ann. Phys. (NY) 148 (1987) 346; M. Berrondo and A. Palma, J. Phys. A 13 (1980) 773; P.C. Ojha, J. Phys. A 21(1988) 875. [6] P. Froman and P.O. Froman, JWKB approximations: contributions to the theory (North-Holland, Amsterdam, [5] Y. Alhassid, F.

1965).

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[7] M. Abrainowitzand L.A. Stegun, Handbook ofmathematical functions (Dover, New York, 1970). [8] U. Slater, Confluent hypergeometnc functions (Cambridge Univ. Press, Cambridge, 1960). [9) W.M. Zheng, J. Phys. A 16 (1983) 43. [10] A.O. Barut, A. Inomata and R. Wilson, J. Phys. A 20 (1987) 4083.

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