n-expansion for multidimensional problems

26 September 1994 PHYSICS LETTERS A

Physics Letters A 193 (1994) 165-172

ELSEVIER

Large orders of 1/n-expansion for multidimensional problems V.S. Popov a,b, A.V. Sergeev c t Institute of Theoretical and Experimental Physics, 117259, Moscow, Russian Federation b International Institute of Physics, 125040 Moscow, Russian Federation c S.L VavilovState Optical Institute, 199034 Saint Petersburg, Russian Federation

Received 14 March 1994; revised manuscript received 18 July 1994; accepted for publication 20 July 1994 Communicated by B. Fricke

Abstract

The asymptotics of large orders of the 1/n-expansion is investigated for multidimensional problems of quantum mechanics and atomic physics, including those with separable variables (the hydrogen molecular ion H + ), and those where separation of variables is impossible (a hydrogen atom in electric and magnetic fields). It is shown that the parameters of the asymptotics can be found by means of calculating sub-barrier trajectories with the help of the "imaginary time" method, as well as by solution of the eikonal equation.

1. The 1/n-expansion is widely used in quantum mechanics and atomic physics, see e.g. Refs. [ 1-12 ]. The energy eigenvalues (which are complex for quasistationary states, E = E r - ½iF) are represented in the form of an expansion [2,4,5 ] e=Et°)+

E(I)

n

+...+

E(k)

~

+ ....

'

(1)

where n is the principal quantum number, e = n2Enl is the "reduced" energy of the nl state and k is the order of the 1/n-expansion. This method was successfully applied to the problems of atoms in strong external fields [3,5-7 ], the hydrogen molecular ion n ~ [6,8,9], etc. [2]. At present the behaviour of large orders of the 1/ n-expansion, i.e. of the coefficients ~(k) with k>> 1, has been investigated numerically [ 9,11 ] and analytically [ 12,13 ]. Here we continue our previous investigations [ 11-13 ] and present some results for multidimensional problems of q u a n t u m mechanics. Elsevier Science B.V. SSDI0375-9601 (94)00615-6

In the problems considered earlier the asymptotics has the form [ 11,13 ] ~Ck)~,k!akk#(co + c t / k + ...),

k--,oo

(2)

or e ( k ) ~ k ! k # R e ( c o a k) [1 + O ( k - l )

] .

(2')

Here a, p, Co, etc. are calculable constants, the numerical values of which depend on the problem considered. For the one-dimensional case (e.g. for a potential with spherical s y m m e t r y ) the asymptotical parameter a can be calculated from the equation [ 12 ] r2

a - 1 = 2 3 / 2 1 [ U ( r ) - e ¢ ° ) ] 1/2 d r ,

(3)

where U(r) = n 2 V(n 2r) + (2r2) - ~ is the effective potential including the centrifugal energy, V(r) is the initial potential in the SchrSdinger equation and

V.S. Popov, A. V. Sergeev / Physics Letters A 193 (1994) 165-172

166

ro < r < r2 is the sub-barrier region 1. Since the energy ~to) corresponds to the m i n i m u m o f the potential U, two turning points coincide, rl = ro. The applications of Eq. (3) to the bound and quasistationary states in the Yukawa and Hulthen potentials can be found in Refs. [12,13]. 2. Here we consider the problem of two Coulomb centres, V ( r ) "= - -

(Z1/r

I

q~

+ ZE/r2) ,

t*

(4)

rl,2 = [ p 2 + ( 2 3- 1 R ) 2 ] 1/2

Proceeding to the elliptic coordinates ~= (rl + rE)/R, rl=(rl-r2)/R and performing the scaling transformation 2 E=2E,

which coincides with Eq. (28) in Ref. [ 12], see also Ref. [8]. The value R = R . corresponds to the collision o f two classical electron orbits in the field o f two Coulomb centres and is determined from the condition that the frequency o f small oscillations near the equilibrium orbit turns to zero (compare with the analogous calculations in Ref. [ 6 ] ). If R ' < R . , then t/o=0 and the turning points are complex, q2 = 1 + ( 2 e ) - 1 < 0 . Therefore,

R'=2R,

r=2r,

1

2=m2_~n-2

,

a-1=23/2

_1 [ V(q) - V(O) ]1/2 dq o

=-2(Arth(-O

<0

(7)

(0 < ( < 1 ). The dependence o f the variable ( on the reduced internuclear distance R ' = n - Z R is determined parametrically, (=(2e+1)1/2= (l_3z)l/Z(l_r)

we arrive at the Schrrdinger equations with the effective energy e and effective potentials U(~) and V(~/),

1

0
I

,SR' (5)

a(R)=A+_[hl-3/2[l+O(h)], (here Z1 = Z2 = 1, which corresponds to the hydrogen molecular ion H f , m = 0, + 1, + 2 ... is the magnetic quantum number and R is the internuclear distance). The potential V(rl) is shown in Fig. 1. If R ' > R . = 3 ~/2 × 2-2, then qo /"

a-l=2

J

IP, I d~/

-- q0

=4(

(8)

( z = ] corresponds to ( = 0 and R ' = R , ) . So, the asymptotical parameter a ( R ) in Eq. (2) is obtained in analytical form for any R. From Eqs. ( 6 ) - ( 8 ) it follows that at R'--,R,,

ER 2 e= ~ER'2 = 4n 2 ,

V ( q ) = 2 ( 1 - r 1 2 ) z + ~1-rl

R ' = z l / 2 ( 1 - - ' c ) -2,

-l ,

,6,

An analogous, though more complicated, formula can be obtained for the pre-exponential factor Co. It can be also shown that r = - ~ for nodeless states in a spherically symmetrical potential

[12]. 2 Further we consider the states with a maximal possible value of m = n - 1 and choose 2 = l / (n 2_ 2n). If the value 2 = n - 2 is chosen, the parameters a and fl in Eq. (2) remain the same, unlike the pre-exponential coefficientsCo,cl, etc. Evidently, for the highly excited states, n >> 1, both values of the scaling factor 2 are equivalent.

h= R ' - R * R,

(9) '

where A+ = 3-l/Z and A_ = - ( 2 / 3 ) 1/2. Calculation o f the parameter a by Eqs. ( 6 ) - (8) gives curve 1 in Fig. 2. If Zl #Z2, the potential V(~/) is not symmetrical any more, see Fig. 1c. The calculations are performed in a similar way as in the case Z1 = Z2, but the analytical formulae become more complicated (the calculation details will be published elsewhere). Just as for the discrete states in the Yukawa potential [4], the turning point r 2 moves into the complex plane, the parameter a becomes complex and Eq. (2') holds. Here [ a ( R ) l
V..S. Popov, A. E Sergeev /Physics Letters A 193 (1994) 165-172 (a)

-

(c)

(b)

-70

o

167

T

?0 ¢

Fig. 1. The effective potential V(~/) : (a) at R ' > R , = 1.299, (b) at 0 < R ' < R , , (c) for Z~ ~Z:. Here R'=n-2R, R is the internuclear distance.

lal

rx/

4.0

0.3-

0 ~

,

-/L

~ ~

~

~~

~,

•

Fig. 2. The parameter of the asymptotics (2) for the problem of two Coulomb centres. Curves 1-4 correspond to ZI/Z2-~ 1, ½, 2 and 3.

Borel transform closest to the origin, ~0= (2a) - ' , was calculated numerically with the help of the quadratic Pad6 approximants. The agreement between Ref. [9 ] and our formula (7) is excellent, see Table 1. Note that only the case of R' < R , was considered in Ref. [9], where a(R') <0, so the series (2) is alternating in sign and the singularity of the Borel transform does not lie on the path of integration. 3. We consider a hydrogen atom in external fields, the case of uniform and parallel fields 8 and 9ff and (0, 0, n - 1 ) states with magnetic quantum number m = n - 1, which correspond to circular electron orbits (at n>> 1). The 1/n-expansion is constructed

around the classical orbit with radius ro=ro(F, B) determined from the equation [ 7 ] r( 1 - F 2 r 4) 2( 1 + ~B2r 3) = 1

(10)

(we use atomic units, h=e=me=l, and reduced variables F = n4g and B = n39f, convenient in the case of Rydberg, n >> l, states). To determine the asymptotical parameter a the "imaginary time" method was used, previously developed [ 14 ] for calculation of the tunneling probability through barriers, varying in time (e.g. in the theory of multiphoton ionization of atoms and ions by a strong light wave [ 15 ], as well as for calculation of e+e - pair production from vacuum by a varying

V.S. Popov, A. V. Sergeev / Physics Letters A 193 (1994) 165-172

168

Table 1 Values of the Borel parameter ~o for the hydrogen molecular ion 1.5R 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.5 3.0 5.0 10.0

Jo

Method of calculation

- 1.720950 - 1.062376 -0.705527 -0.474800620 -0.474795 -0.313841191 -0.31384121 -0.19751662152 -0.19751661876 -0.112797 -0.052563 --0.013569 0.003681 0.038463 0.119351 0.294164 1.202939 3.943259

a a a a c a c a c a a a b b b b b b

l1 - I m S / S o l 1

¢2)1 + ( 0 2

electric field [16]). The classical trajectory r(t), which joins the point ro of maximum effective potential U, 1 1 U ( r ) = - U = - - - - + F z - ~BZp 2 r 2,0 z (

11 )

with the turning point r2~o", was found by numerical integration (here tr is an isoenergetical surface, e.g. U(r2) = U ( r o ) ) . The condition ~o=~2=0 selects the sub-barrier trajectory, which minimizes Im S [ 14 ]. Then the parameter a = a ( F , B ) in Eq. (2) is equal to a = ( 2 I m S ) -1,

S=

pdr= ~

/'2dt,

~ ( A / 9 / L ) 2,

z~ - - I n ( L / l A p [

Notations: ( a ) t h e calculation of~o= (2a) -I by Eq. (7), ( b ) b y Eq. (6), (c) ~o from Ref. [9]. Note that the definition of the scaled internuclear distance in Ref. [ 9 ] differs from ours by the factor -~.

19= ( X 2 + y 2) 1/2,

above trajectory is infinite. The lines of constant potential 0 (dashed curves) and a few classical trajectories (solid curves) are given in Fig. 3. The integral S in Eq. (12) was calculated from the initial point r2~ a to the point rm of closest approach of the trajectory to the point of unstable equilibrium r0. Since the potential U(r) is approximated at r ~ ro by the potential of the "upturned oscillator", any trajectory is unstable. The trajectories in Fig. 3 correspond to initial deviations Ap, shown in Table 2. Although it is impossible to get from r2~ a precisely to ro in numerical calculations, the value of Im S is readily determined. It can be shown that at small Ap

(12)

0

where i'= d r / d t and t = iz is "imaginary time". Note that 0 differs in sign from the potential U entering the SchrSdinger equation, which exactly corresponds to the description of the sub-barrier motion of a particle in terms of the imaginary time method. Since ~=VU=0 at r=ro, the time of motion z along the

),

(13)

where L = l r2-roJ, mt are the frequencies of normal vibrations at r ~ to, and the action So corresponds to the above trajectory, which connects the point ro with the isoenergetic surface a. The results of the calculations (at F < F , ) are shown in Fig. 4. Here F, is the classical ionization threshold, F , ( B ) = 0 . 2 0 8 1 , 0.2207, 0.2532 and 0.3449 at B=0, 0.25, 0.5 and 1.0. At F - ~ F , the points r2 and ro converge, and the potential ( 11 ) takes the form V = Vo + o t q - ~flq 3, where a o c F , - F - - , O and fl~ 1. Taking Eq. (3) into account, we find that q2 t~

a - l = 2 J (__p2)l/2dqocot5/4 ' q0

(14)

aoc ( F . - F ) -5/4 .

So the parameter a has a power singularity at F = F . of the same type as in the one-dimensional case [ 12 ]. 4. Let us also give a brief description of the alternative calculation method for the action S along the sub-barrier trajectory, based on solution of the Hamilton-Jacobi (H J), or eikonal equation, (VS)2=2(U-Uo),

Uo=U(ro).

(15)

Choosing the origin at the point ro and axes x, y (normal coordinates), we get at r ~ ro U ( x , y ) = Uo + ]VJl~ l ,2 ~ 2 n±~ 2l Y,2, 2 ~ ...,

V.S. Popov,A. V. Sergeev/ PhysicsLettersA 193 (1994) 165-172 I

!

X

3

\\ \\\ \\\ '\

i

~

,

,

\,

169

\\j

,

\

i

I

-3

I

II

-2

I

I

-~

i

i

0

z

Fig. 3. The sub-barrier trajectories for the potential ( 11 ). The values of parameters for trajectories Nos. 1-8 are given in Table 2. Here p and Z are the cylindric coordinates of the electron, the nucleus (proton) is located at p = z = 0. Table 2 Convergence of the Im S values ( F = 0.15, B = 0.2 and F , = 0.2163) Imaginary time method

Solution o f t h e HJ equ~ion

No.

Ap

Ar

r

Im S

N

Im S

1 2 3 4 5 6 7 8

0.1 -0.1 0.01 -0.01 10 -4 - l0 -4 l0 -6 - 10 -6 10 - s - 10 -8 ~0

1.396 1.401 0.692 0.675 0.1347 0.1335 0.0247 0.0246 0.0045 0.0045 ~ 10 -3

6.16 4.92 8.35 7.80 11.35 11.24 13.93 13.91 16.45 16.45 ~20

0.345 0.424 0.393 0.412 0.40106 0.40126 0.4011264 0.4011278 0.40112688 0.40112689 0.401126879

15 18 28 35 38

0.39924 0.40072 0.4011278 0.4011270 0.401126879

-

Nos. 1-8 in the table correspond to the numbers of the curves in Fig. 3. Here A t = Ir m - r o l is the distance of closest approach of the classical trajectory to the point of (unstable) equilibrium to, r is the time of motion from the initial point r2 to rm.

S(x,y)=

5cdx+ 0

~dy

U= ~, UjkXJyk, S= E SikXJyk j,k=O

o

= ½(to, x 2 + t o 2 y 2) + .... Substituting the expansions

(16)

into the HJ equation, relations

(17)

j,k

we arrive at the recurrence

V.S. Popov,A. V. SergeevI PhysicsLettersA 193 (1994) 165-172

170

o 0.5 i.O

I

o.~,

o.~

0.5

I

O. 8

Fig. 4. The parameter la(F, B) I versus the ratio F/F, (B). The values of the reduced magnetic field B are shown besides the curves.

n--2

Sjk=(j0), +k0)2) -1 ujk--½ ~

p

~, [ ( q + l )

p=2 q=0

X (j--q+ 1 )Sq+l,lSj__q+l,k__ l + (l+ 1 ) ( k - l + 1 )Sq,l+~Sj-q,k-l+l ] )

( 18)

(n = j + k~> 3, l = p - q ) , from which the coefficients Sjk can be calculated subsequently. For example, S03 = (30)2)-1Uo3, S21 =

(20)1

$12 = (0)1 +20)2)-1U12,

-{-O)2)--1/,/21 ,

S13 = ( 0 ) 1 + 3 0 ) 2 )

-1

× [ul3 - 2s12 (s21 + 3So3) ] .....

(17')

It is easy to obtain Sjk up to n ~ 50 with a computer, thus allowing one to use summation methods for divergent (or slowly convergent) series. So we proceed to a power series in one variable 2,

(0 < 2 < 1 ). Then the value f ( 1 ) is calculated by Pad6 (PA) or Hermite-Pad6 ( H P A ) approximants. The sub-barrier trajectory ends at the turning point r2, where the line S = So contacts the isoenergetic surface U=Uo, and 0 ~ S = 0 y S = 0 . The derivatives OiS have a square root singularity at r = r2, therefore the calculation of rE with high precision requires some effort 2. Table 2 illustrates the rapid convergence o f the S values with growing order of approximation ( N is the number o f the coefficients f , included into calculation). It is evident from Table 2, that both methods used are in full agreement, thus providing the accuracy ~ 10 -s in the So values. Note that the series (16) for S(x, y) should be summed at the isoenergetic surface a, where the anharmonicity of the potential U(x, y) is not small any more. The precise values o f So can be obtained only due to the availability of high-order coefficients Sjk with n=j+k>> 1 and the application o f the effective methods for the summation of series 3.

f(2)=_S(~x,~y)= nZ L2", =O L= ~ ~.-jx~ "-j j=0

(19)

3 For this purpose, we have used a special modification of the HPA, where the solution of the equations for r2 was obtained by a minimization procedure (calculation details will be published elsewhere).

KS. Popov,A. V. Sergeev/ PhysicsLettersA 193 (I 994) 165-172 5. If 8 = 0 or o f = 0 , the parameter a can be calculated analytically. Let us consider the Zeeman effect for the (0, 0, n - 1 ) states in a hydrogen atom. It is convenient to begin with the case of imaginary magnetic field, or the funnel-like potential V ( r ) = - r- 1_ ½gr2, where g = - ~of 2 > 0. Using Eq. ( 3 ) and calculating the penetrability of the barrier in this potential, after some algebra we get ( 2 a ) _ 1 _ 31/2(1-½ z2) 31/2Z -- (1__72)1/2 arctg2(l~-~--z2)l/2 3z -Arth z2+2,

(20)

z=( l +tr) 1/2 = 1 + ] B 2 - ~5--~8B4+....

B--.O,

=2-3X2-3/2B-l/2+O(B-l),

B~m,

where t r = 3 ( 1 - r o ) and ro(B) is the radius of the classical orbit determined by Eq. (10). Since z > 1, the value of a for the Zeeman effect is obtained from Eq. (20) with the help of analytic continuation, the parameter a(B) becomes complex and finite for all B. In particular, in the region of weak magnetic fields

a(B)=+iB+_~

2 B2 l n B + ..., ~-i

B~O

(21)

and ] a [ = zc- ~B + O ( B 2 ). The parameter Ia (B) I has a m a x i m u m at B = B m ~ l . 9 5 , where l a ( B m ) l = 0.2133. Note that la(oo)l = 1/2zc=0.1592, but the limit is achieved only in very strong magnetic fields,

a(B)= +_--~ (I + (2B) -I/2 + 23-~ B-l/21nB+ ...),

(22)

[a(B) I = ~ ( I + (2B) -'/2 1 ln2B + O ( B _ llnB)~) + 16~r~ ~ B--,oo

(22a)

Note that the two signs in Eqs. (21), (22) correspond to the complex conjugate singularities, 8o and 8~, of the Borel function, so the asymptotics of E~k) is of the type (2').

171

If of = 0 (the Stark effect in hydrogen), then a ( F ) is determined by Eqs. (16), (17) in Ref. [ 13 ]. 6. Thus, the coefficients of the 1/n-expansion e
References [ 1] A. Chatterjee, Phys. Rep. 186 (1990) 249. [2]D.R. Herschbach, J. Avery and O. Goscinsky, eds. Dimensional scaling in chemical physics (Kluwer, Dordrecht, 1993). [3 ] C.M. Bender, L.D. Mlodinow and N. Papanieolaou, Phys. Rev. A25 (1982) 1305. [4 ] V.S. Popov, V.M. Weinberg and V.D. Mur, Pis'ma Zh. Eksp. Teor. Fiz. 41 ( 1985 ) 439; Yad. Fiz. 44 (1986) 1103. [ 5 ] V.S. Popovet al., Phys. Lett. A 124 ( 1987) 77; A 149 (1990) 418, 425. [ 6 ] V.D. Mur, V.S. Popov and A.V. Sergeev,preprint ITEP 11489 (1989); Zh. Eksp. Teor. Fiz. 97 (1990) 32. [7] V.M. Weinberg, V.S, Popov and A.V. Sergeev, Zh. Eksp. Teor. Fiz. 98 (1990) 847. [ 8 ] S. Kais, J.D. Morgan and D.R. Herschbach, J. Chem. Phys. 95 (1991) 9028. [9] M. Lopez-Cabrera et al., Phys. Rev. Lett. 68 (1992) 1992. [10] S. Kais and D.R. Herschbach, J. Chem. Phys. 98 (1993) 3990. [ 11 ] V.S. Popov and A.V. Scheblykin, Yad. Fiz. 54 ( 1991 ) 1582.

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ES. Popov, A. E Sergeev / Physics Letters A 193 (1994) 165-172

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