Threshold properties of attractive and repulsive 1 Õ r

Threshold properties of attractive and repulsive 1 Õ r

²

potentials

Michael J. Moritz, Christopher Eltschka, and Harald Friedrich Physik-Department, Technische Universita¨t Mu¨nchen, 85747 Garching, Germany

共Received 22 September 2000; published 12 March 2001兲

We study the near-threshold (E→0) behavior of quantum systems described by an attractive or repulsive 1/r²potential in conjunction with a shorter-ranged 1/r^m(m⬎2) term in the potential tail. For an attractive 1/r² potential supporting an infinite dipole series of bound states, we derive an explicit expression for the threshold value of the pre-exponential factor determining the absolute positions of the bound-state energies. For poten- tials consisting entirely of the attractive 1/r² term and a repulsive 1/r^m term, the exact expression for this prefactor is given analytically. For a potential barrier formed by a repulsive 1/r² term共e.g., the centrifugal potential兲 and an attractive 1/r^m term, we derive the leading near-threshold behavior of the transmission probability through the barrier analytically. The conventional treatment based on the WKB formula for the tunneling probability and the Langer modification of the potential yields the right energy dependence, but the absolute values of the near-threshold transmission probabilities are overestimated by a factor which depends on the strength of the 1/r²term共i.e., on the angular momentum quantum number l) and on the power m of the shorter ranged 1/r^mterm. We derive a lower bound for this factor. It approaches unity for large l, but it can become arbitrarily large for fixed l and large values of m. For the realistic example l⫽1 and m⫽6, the conventional WKB treatment overestimates the exact near-threshold transmission probabilities by at least 38%.

DOI: 10.1103/PhysRevA.63.042102 PACS number共s兲: 03.65.Ge

I. INTRODUCTION

Encouraged by the interest in cold atoms and their inter- actions, there has recently been strong activity in the study of atomic and molecular systems near the threshold which sepa- rates the bound-state regime from the continuum 关1兴. The Schro¨dinger equation for a particle of massMin a potential V(r) is

␺

⬙

共r兲⫹2M

ប² 关E⫺V共r兲兴␺共r兲⫽0, r⬎0, 共1兲 and the behavior of its solutions near threshold depends cru- cially on the asymptotic共large-r) behavior of the potential.

For long ranged potentials falling off slower than 1/r² 共e.g., the Coulomb potential兲, the threshold E⫽0 represents the semiclassical limit 关2兴, and there are infinitely many bound states if the potential tail is attractive. The behavior of the quasicontinuum of bound states just below threshold and the real continuum above threshold is well understood, at least for the case of one Coulombic coordinate, on the basis of quantum defect theory.

For potentials tails falling off faster than 1/r², the thresh- old E⫽0 represents the anticlassical limit of the Schro¨dinger equation, and the potential supports at most a finite number of bound states. Near-threshold properties of bound and con- tinuum states for deep potentials with attractive tails falling off faster than 1/r² have been the subject of several recent publications关3–11兴.

This paper deals with potentials asymptotically propor- tional to 1/r² which represent the borderline separating the long-range tails from shorter-range tails. For a potential pro- portional to 1/r², the energy dependence scales out of the Schro¨dinger equation 共1兲; the semiclassical limit is reached neither for E→0 nor for兩E兩→⬁, but for large absolute val-

ues of the potential strength 关2兴. Repulsive 1/r² potentials appear commonly as centrifugal potential in the radial Schro¨- dinger equation. An attractive 1/r² potential can occur through the interaction of a charged particle with a perma- nent electric dipole, as in the scattering of electrons by polar molecules 关12兴 or by excited hydrogen atoms 关13–15兴. If such an attractive 1/r² potential is sufficiently strong, it sup- ports an infinite ‘‘dipole series’’ of bound states 关13–17兴. Moderately strong attractive 1/r² potentials have been seen as a probable mechanism for the generation of ‘‘quantum halo states’’关18兴. If the strength of the共attractive兲1/r²term is too weak, then the potential supports at most a finite num- ber of bound states 共see, e.g., Ref关19兴兲.

We study potentials behaving for large r as

V共r兲⫽ ប²

2M

冋

^{r␥2⫾␤mrm⫺2}

册

^{, m⬎2. 共2兲}

The dimensionless strength parameter␥ ^{of the 1/r2term can} be positive or negative, and for the centrifugal potential in the radial part of the three-dimensional Schro¨dinger equation with angular momentum quantum number l we have

␥⫽l(l⫹1). The strength of the 1/r^m term is expressed in terms of the 共non-negative兲 parameter ␤, which has the physical dimension of a length.

For m⫽4 the Schro¨dinger equation with potential 共2兲 posesses analytical solutions based on Mathieu functions 关20–22兴. More general potentials like Eq.共2兲have been stud- ied extensively over the years关23–28兴, mainly with the aim of understanding scattering properties. In the present paper we focus on two particular features of 1/r²potentials. In Sec.

II we study potentials with an attractive 1/r² term strong enough to support an infinite dipole series of bound states, and we calculate the threshold value of the factor determin- ing the absolute values of the energies in the series. In Sec.

III we study potential barriers consisting of a repulsive 1/r² term and an attractive 1/r^m term, and we derive the exact expression for the near-threshold behavior of the transmis- sion probability through the barrier. This enables us to give a founded judgement on the accuracy of the conventional pro- cedure for deriving transmission probabilities which is based on the WKB formula and the Langer modification of the potential.

II. DIPOLE SERIES OF BOUND STATES

When the 1/r² term in the potential tail关Eq.共2兲兴is attrac- tive, we have g ⫽⫺^def ␥⬎0. If the strength parameter g is suf- ficiently large, viz. g⬎1/4, then the potential supports an infinite ‘‘dipole series’’ of bound states whose energies ap- proach an exponential behavior near threshold 关13–17兴:

E_n ⫽

n→⬁

⫺F exp

冉

^⫺

^冑

^g2⫺␲n1/4

冊

^{. 共3兲}

The limiting value of the ratio of successive energies in a dipole series is fixed by the strength of the 1/r² term in the potential tail, and is simply lim_n→⬁E_n/E_n⫹1

⫽exp(2␲^/

冑

^g⫺1/4). However, the constant of proportional- ity F in Eq.共3兲depends very sensitively on the potential at shorter distances, where it necessarily deviates from the 1/r² behavior.

We first look at the case where the shorter ranged term proportional to 1/r^m is attractive. At threshold, E⫽0, the Schro¨dinger equation共1兲with potential共2兲is

冋

^{drd22⫹rg2⫹␤rmm⫺2}

册

^{M共r兲⫽0. 共4兲}

We introduce the abbreviations

␶⫽

def

冑

^g⫺14, ␰⫽

def 2␶ m⫺2⫽ 2

m⫺2

冑

^g⫺14. 共5兲 Two linearly independent analytical solutions of Eq. 共4兲are

M_1;2共r兲⫽

冑

^rJ⫾i␰共␳兲, ␳⫽

def 2

m⫺2

冉

^␤r

冊

^{(m⫺2)/2. 共6兲}

Here J_⫾i␰ stands for the ordinary Bessel function 关29兴 of order⫾i␰. The asymptotic (r→⬁) behavior of solutions共6兲 is

M_1;2共r兲⬃共m⫺2兲^⫿i␰

⌫共1⫾i␰兲

冑

^r

冉

^␤r

冊

^⫾i␶

冉

^{1⫺共m⫺共␤2/r兲2兲共m1⫺⫾2i␰兲}

冊

^.

共7兲 For sufficiently large r, the shorter-ranged term in the potential tail can be neglected, so the Schro¨dinger equation for finite negative energy, E⫽⫺ប²␬^2/(2M), is

冋

^{drd22⫹rg2⫺␬2}

册

^{R⫽0. 共8兲}

The solutions of Eq. 共8兲 are functions of ␬r only, and the physically relevant solution is

R共␬^r兲⫽i exp

冉

^⫺␲2␶

冊 ^冑

^{␬rHi(1)␶ 共i␬r兲, 共9兲}

which behaves asymptocially (␬^r→⬁) as

R共␬^r兲⬃

冑

␲^{2 exp共⫺}␬^r兲. 共10兲 The function H_i␶⁽¹⁾ in共10兲is the Hankel function of order i␶ as defined in Ref. 关29兴,

H_i␶⁽¹⁾共z兲⫽exp共␲␶兲J_i␶共z兲⫺J_⫺i␶共z兲

sinh共␲␶兲 . 共11兲 For sufficiently small values of ␬, we may use the small argument expansion of the Bessel functions in Eq. 共11兲 to obtain

R共␬^r兲 ⬃^␬

r→0

冑

␲␶^sinh␬r共␲␶兲

冋

^e⫺i␪

冉

^␬2r

冊

^{i␶⫹e⫹i␪}

冉

^␬2r

冊

^⫺i␶

册

⫻关1⫹O„共␬^r兲²…兴, 共12兲 where␪ ⫽^defarg⌫(i␶). The r dependence of the leading terms in Eq.共12兲is the same as the large-r behavior关Eq.共7兲兴of the zero-energy wave functions 关Eqs. 共6兲兴 in the full potential tail 关Eq. 共2兲兴, so we can determine the near-threshold共real兲 solution of the Schro¨dinger equation to order below ␬^{2 by} taking the appropriate superposition of the solutions 共6兲,

R共␬^r兲⫽L M₁共r兲⫹L*M₂共r兲⫽

def

M_␬共r兲. 共13兲 The coefficient L is determined by the condition that the leading asymptotic (r→⬁) terms derived for Eq.共13兲 from Eq. 共7兲 agree with the corresponding leading terms in Eq.

共12兲:

L⫽

冑

␲␶^sinh␬共␲␶兲e^i␪⌫共1⫹i␰兲共m⫺2兲^i␰

冉

^␬␤2

冊

^⫺i␶.

共14兲 Solutions 共6兲 of the Schro¨dinger equation 共4兲 at energy zero are accurate as long as the energy term␬² is negligible in comparison with the smaller of the two potential terms, which is g/r² when the shorter-ranged term is dominant.

This implies

rⰆ

冑

^g

␬ ^{. 共15兲}

The Schro¨dinger equation with the full potential tail can be approximated by its asymptotic form关Eq.共8兲兴, when r is so

large that the shorter ranged term ␤^m⫺2/rm is negligible compared with the longer-ranged term g/r², implying

rⰇ ␤

g^1/(m⫺2). 共16兲 We can match the superposition 关Eq. 共13兲兴 of zero-energy solutions to solutions共9兲,共12兲of Eq.共8兲if there is a region of r values, where conditions共16兲and共15兲are satisfied simul- taneously. This is the case when the right-hand side of Eq.

共15兲is much larger than the right-hand side of Eq.共16兲, i.e., when

␬␤Ⰶg^{(1/2)⫹[1/(m⫺2)]}. 共17兲 In other words, matching is justified in the limit␬→0, which is sufficient to determine the leading near-threshold behavior of the energy eigenvalues. An estimate for the numerical accuracy of the near-threshold formulas derived below can- not, however, be given on the basis of the leading terms alone. For this we would require a knowledge of the next-to- leading terms, for which we would have to include correc- tions of order E in the wave functions.

As r decreases, the argument␳ in solutions共6兲becomes large, and we approximate the near-threshold wave function 共13兲via the large argument expansion of the Bessel functions 关29兴, J_⫾i␰(␳⁾⬃

冑

^2/(␲␳^)cos关␳⫿i(␲^/2)␰⫺¹4␲兴,

M_␬共r兲 ⬀

r→0

r^m/4cos

冉

^{␳⫺␲4⫹␦}

冊

^{, 共18兲}

where␦ is an angle defined by

tan␦⫽tanh

冉

^␲2␰

冊

^tan

冉

^{␪⫹␹⫹␲2 ⫺␶ln q}

冊

^,

q⫽ ␬␤

2共m⫺2兲^2/(m⫺2); 共19兲 in analogy to ␪⫽arg⌫(i␶⁾ 关see Eq. 共12兲兴 we have intro- duced the abbreviation␹ ⫽^defarg⌫(i␰^).

The regular solution␺reg(r) of the Schro¨dinger equation also depends on the potential at small r values, and vanishes at r⫽0. Bound states exist for energies at which the regular solution matches to the wave function 关Eq. 共13兲兴in the re- gion where the potential is already dominated by the two power-law terms of the tail 关Eq. 共2兲兴, so the condition of quantization, quite generally, is

␺reg

⬘

␺reg

⫽M_␬

⬘

M_␬. 共20兲

Semiclassical wave functions are defined with the help of the local classical momentum,

p共r兲⫽

冑

²M关E⫺V共r兲兴, 共21兲

and WKB wave functions,␺WKB⬀p^⫺1/2exp关⫾(i/ប)兰pdr兴, are accurate solutions of the Schro¨dinger equation when the fol- lowing condition is fulfilled 关2兴:

1

16␲²

冏 冉

^ddr␭

冊

^{2⫺2␭ddr2␭2}

冏

^{Ⰶ1, 共22兲}

where␭(r)⫽2␲ប/ p(r) is the共local兲de Broglie wavelength.

For potentials behaving as 1/r^m, m⬎2, condition共22兲is ful- filled increasingly well as r decreases. We assume that there is a range of r values in the potential well where condition 共22兲is well fulfilled so that the WKB wave function,

␺WKB共r兲⬀ 1

冑

^p共r兲cos

冉

^1ប

^冕

^rrinp共r

^⬘

^兲dr

^⬘

^⫺␾2in

冊

^共23兲

is an accurate solution of the Schro¨dinger equation, see Fig. 1. The angle␾inin Eq.共23兲is the reflection phase at the inner classical turning point r_in, which is defined so that the WKB wave function共23兲agrees with the exact quantum me- chanical wave function␺regin this ‘‘WKB region’’;␾incan be taken to be␲/2 if the conditions of the semiclassical limit are fulfilled near the inner classical turning point 关30兴.

The r dependence of both the amplitude and the phase of the wave function 共18兲 is that of the WKB wave function 共23兲 at E⫽0, when the potential near r is given by the shorter-ranged term in the tail alone,

2M

ប² V共r兲⫽⫺␤^m⫺2

r^m , p共r兲⫽ប␤^(m⫺2)/2 r^m/2 ,

FIG. 1. Schematic illustration of a potential with a tail关Eq.共2兲兴 consisting of an attractive 1/r² term and an attractive 1/r^m term.

Near-threshold solutions of the Schro¨dinger equation are well ap- proximated by WKB wave functions in the ‘‘WKB region.’’ We assume that this WKB region overlaps with a region of moderate r values where the potential is dominantly described by the 1/r^m term.

1 ប

冕

r_in

r

p共r

⬘

兲dr

⬘

^⫽const⫺␳^. 共24兲

If the WKB region overlaps with a range of r values where the potential is dominated by the 1/r^mterm共see Fig. 1兲, then the quantization condition can be formulated by matching wave functions 共23兲 and 共18兲 in this range of overlap. We expect the WKB wave function here to be a smooth 共ana- lytic兲 function of energy, so, to order less than␬^{2, we can} assume E⫽0 in Eq.共23兲. Equating the cosines in Eqs.共23兲 and共18兲leads to the quantization condition

1 ប

冕

^rin

r

p共r

⬘

兲dr

⬘

⫺␾in

2 ⫽n␲⫺␳⫹␲

4⫺␦^, 共25兲 where the action integral on the left-hand side is to be taken at threshold, E⫽0. In the region of overlap, where the WKB approximation is accurate and the potential is dominated by the 1/r^m term, the r dependence of the action integral is compensated for by the term ⫺␳ on the right-hand side of Eq. 共25兲 关cf. Eq.共24兲兴, so the expression

I₀⫽

def1 ប

冕

r_in

r

p共r

⬘

兲dr

⬘

⫹ 2

m⫺2

冉

^␤r

冊

^{(m⫺2)/2⫺␾2in⫺␲4 共26兲}

is independent of r. With the help of Eq.共19兲the quantiza- tion condition共25兲thus reduces to

tan

冉

^{␪⫹␹⫹␲2⫺␶ln q}

冊

^{⫽tanhtan共␰␲␦/2兲⫽⫺tanhtan I共␰␲0/2兲,}

共27兲 which is equivalent to

␬²⫽4共m⫺2兲^4/(m⫺2)

␤^{2 exp}

再

^2␶

冋

^{␪⫹␹⫹␲2}

⫹arctan

冉

^{tanhtan I共}␰␲^0/2兲

冊册冎

^{. 共28兲}

The multivalued nature of the arcus tangent in the exponent on the right-hand side of Eq. 共28兲 allows the subtraction of n␲ (n is an integer兲, and this leads to the known asymptotic (E→0) behavior of the energies of the dipole series,

E_n⫽⫺ប²␬n 2

2M ⫽

n→⬁

⫺F exp

冉

^⫺2␲␶n

冊

^{. 共29兲}

The theory above now allows us to give an explicit expres- sion for the prefactor F in Eq.共29兲, namely,

F⫽ 2ប²

M␤^2共m⫺2兲

4/(m⫺2)exp

再

^2␶

冋

^{␪⫹␹⫹␲2}

⫹arctan

冉

^{tanhtan I共␰␲0/2兲}

冊册冎

^{. 共30兲}

Result 共30兲 holds under the condition that there is a WKB region where condition 共22兲 is well fulfilled, and that this region overlaps with a region of r values where the potential is dominated by the 1/r^m term; see Fig. 1. A definite choice of the branch of arctan关tan I₀/tanh(␰␲^/2)兴 in Eq. 共30兲fixes the quantum numbers n assigned to the individual levels. If, in a given potential, we fix the numbering of levels, e.g., by starting with n⫽0 for the ground state, then this determines which branch of the arcus tangent is to be taken. The choice of branch remains undetermined in our present theory based on the near-threshold wave functions.

We now consider the case that the shorter ranged term proportional to 1/r^mis repulsive, and that the two terms关Eq.

共2兲兴constitute the whole potential共see Fig. 2兲. This potential again supports an infinite dipole series if the strength of the 共attractive兲1/r² term is large enough (g⬎1/4).

The Schro¨dinger equation is given by

冋

^{drd22⫹rg2⫺␤rmm⫺2⫺␬2}

册

^{␺共r兲⫽0. 共31兲}

In order to obtain near-threshold wave functions for small and moderate values of r, we neglect the energy term in Eq.

共31兲, and the resulting equation

冋

^{drd22⫹rg2⫺␤rmm⫺2}

册

^{Z共r兲⫽0 共32兲}

can be solved with the help of Bessel functions. We keep the abbreviations ␶ ^and ␰ as defined in Eq. 共5兲 with ␪

⫽arg⌫(i␶^{) and} ␹⫽arg⌫(i␰). The real solution of Eq. 共32兲 which obeys the physical condition of vanishing at the origin is

FIG. 2. Potential关Eq.共2兲兴consisting entirely of a repulsive 1/r^m term and an attractive 1/r²term. Here m⫽4 and g⫽⫺␥⫽200, so the potential corresponds to the case␩⫽5 studied by Varshni关32兴. The potential is given in units of its depth D_eat its minimum r_e; see Eq.共39兲.

Z共r兲⫽i exp关⫺␲␰^/2兴

冑

^rHi(1)␰ 共i␳兲, ␳⫽ 2

m⫺2

冉

^␤r

冊

^(m⫺2)/2,

共33兲 which behaves as

Z共r兲⬃

冑

^m⫺␲ ␤²

冉

^␤r

冊

^{m/4exp共⫺␳兲 共34兲}

for small r.

For large values of r we use the small argument expansion 关29兴of the Hankel function H_i⁽¹⁾_␰ in Eq.共33兲, and obtain the leading terms

Z共r兲⬃

冑

␲␰^sinhr共␲␰兲

冋

^{共m⫺2兲i␰ei␹}

冉

^␤r

冊

^i␶

⫹共m⫺2兲^⫺i␰e^⫺i␹

冉

^␤r

冊

^⫺i␶

册

^{. 共35兲}

These leading terms must, except for a common constant of proportionality, agree with the near-threshold limit of wave function共9兲as given in Eq.共12兲, i.e., the ratios of the coef- ficients of r^⫺i␶and r^i␶must be the same in Eq.共35兲as in Eq.

共12兲. This leads to the condition,

冉

^␬␤2

冊

^{2i␶⫽共m⫺2兲2i␰e2i(␪⫹␹), 共36兲}

which is equivalent to exp关2i(␪⫹␹⁾兴⫽exp(2i␶ln q), with q

⫽¹2␬␤^(m⫺2)^⫺2/(m⫺2) as in Eq. 共19兲. This corresponds to the quantization condition

␬n

2⫽4共m⫺2兲^4/(m⫺2)

␤^{2 exp}

冋

^{2␶共␪⫹␹⫺n␲兲}

册

^{, 共37兲}

where the multivalued contribution ⫺n␲ on the right-hand side originates from the multivalued nature of the exponen- tials in Eq.共36兲. For the energies E_n we again obtain expres- sion 共29兲, but for the prefactor we now have the analytical formula

F共m,g兲⫽ 2ប²

M␤^2共m⫺2兲

4/(m⫺2)e^{2(␪⫹␹)/␶}. 共38兲

The Schro¨dinger equation 共31兲was studied by Papp关31兴 and Varshni 关32兴for the case m⫽4 with the aim of testing approximation schemes such as the 1/N expansion and the WKB approximation. As is customary in molecular physics, the parameters defining the potential are taken as the position r_e of the potential minimum and the depth D_e⫽⫺V(r_e) of the well. The energies are normalized to the depth, ␧n

⫽E_n/D_e, and these normalized energies now depend only on the strength g of the attractive 1/r² term, which is related to a parameter called␩^{2 in Ref.}关32兴. In terms of our poten- tial parameters g and␤, the parameters of Ref. 关32兴are

r_e²⫽2␤²

g , D_e⫽ ប²g²

8M␤^2, ␩²⫽g

8. 共39兲 For m⫽4 the parameters ␶ ^and␰ defined by Eq. 共5兲are the same, and, according to Eqs.共37兲and共38兲, the threshold behavior of the normalized energies is

␧n⫽E_n

D_e⫽⫺f e^⫺2␲n/␶, f⫽F共m⫽4,g兲 D_e ⫽e^4␪/␶

␩^{4 . 共40兲} For potentials with ␩⫽5, 15, and 25 Varshni 关32兴 listed normalized eigenvalues for quantum numbers from n⫽0 for the ground state to n⫽9, 30, and 55, respectively. In order to demonstrate how these dipole series approach the limiting behavior关Eq.共40兲兴, we plot the logarithms ln f_nof the effec- tive strength parameters

f_n⫽⫺␧n⫻e^2␲n/␶ 共41兲 against the quantum number n. In the limit n→⬁, these ef- fective strengths converge to the strength f in Eq.共40兲. Here f is defined only to within a factor consisting of an arbitrary integer power of exp(2␲^/␶), so ln f is only defined modulo 2␲^/␶. A definite choice of f fixes the quantum numbers n assigned to the individual states关see the discussion after Eq.

共30兲兴.

The results are shown in Fig. 3. The values of ln f follow- ing from Eq.共40兲for the three values of␩are listed in Table I, and shown as dashed horizontal lines in Fig. 3. The con- vergence of the effective strengths to the respective threshold limits is obvious. This convergence implies that the energies of the near-threshold states are, for growing n, given with increasing共absolute and relative兲accuracy by formula共29兲, with the appropriate prefactor 关Eq. 共38兲兴, just as the near- threshold energies of a Rydberg series in a Coulomb poten- tial are given with increasing共absolute and relative兲accuracy by the Rydberg formula with the appropriate threshold value of the quantum defect关2兴.

The well known fact that conventional WKB quantization breaks down near threshold, for potentials falling off faster than 1/r²asymptotically, was recently interpreted as a break- down of Bohr’s correspondence principle关33兴. The threshold is, however, an unusual place to expect quantum classical correspondence for such potentials 关8,10兴, because it does not correspond to the semiclassical limit. A potential falling off faster than 1/r²supports at most a finite number of bound states, so the limit of infinite quantum numbers, which is fundamental to the usual formulation of Bohr’s correspon- dence principle, cannot be taken. Dipole series form an in- teresting special case for this discussion. For potentials pro- portional to 1/r², the accuracy of semiclassical approximations does not depend on energy—the semiclassi- cal limit is reached for large absolute values of the strength parameter关2兴. However, a sufficiently attractive 1/r² poten- tial tail with a fixed strength parameter g⬎1/4 does support an infinite number of bound states. As is obvious from the tables in Ref.关32兴, the relative errors of the energy eigenval- ues obtained via conventional WKB quantization become

larger with increasing quantum numbers for all potentials studied. 关Note, however, that higher-order WKB results are very accurate for all quantum numbers.兴 Thus dipole series in potentials with an attractive 1/r² tail are a genuine ex- ample where the naive expectation that semiclassical ap- proximations necessarily improve in the limit n→⬁ is not fulfilled. This naive interpretation of Bohr’s correspondence principle fails in the present case. Here, as elsewhere, the correspondence principle refers to the semiclassical limit.

For the attractive 1/r² potential tail, the semiclassical limit can be realized by taking the limit of large strength param- eters, independent of energy. For a discussion of potential tails falling off faster than 1/r²; see Refs.关8,10兴.

III. TUNNELING

When the 1/r² term is repulsive and the shorter-ranged 1/r^mterm is attractive, then potential共2兲represents a barrier typical for the radial Schro¨dinger equation with nonvanish- ing angular momentum; see Fig. 4. The Schro¨dinger equa- tion for this barrier is

冋

^{drd22⫺r␥2⫹␤rmm⫺2⫹k2}

册

^{␺共r兲⫽0, ␥⬎0. 共42兲}

FIG. 3. Logarithmic plot of the effective strength parameters f_n 关Eq.共41兲兴for the共normalized兲energy eigenvalues␧ncalculated by Varshni关32兴for a potential关Eq.共2兲兴consisting entirely of a repul- sive 1/r⁴term and an attractive 1/r²term for␩⫽5, 15, and 25. The dashed horizontal line in each panel shows the threshold limit ln f of the ln fnas the behavior of the energies approaches the dipole series form关Eq. 共40兲兴; also see Table I. This limit is defined only modulo 2␲/␶, and the magnitudes of 2␲/␶for the various values of

␩ are shown as vertical bars in the respective panels.

TABLE I. Values of ln f for the threshold limits f which deter- mine, via Eq.共40兲, the explicit values of the normalized energies in the dipole series generated by the Schro¨dinger equation 共31兲 for m⫽4.

␩ ^{g 2}␲/␶ ln f

5 200 0.444566 ⫺0.0675707

15 1800 0.148106 ⫹0.0843669

25 5000 0.0888599 ⫹0.1142865

FIG. 4. Potential barrier consisting of a repulsive 1/r²term and an attractive 1/r^mterm. Here m⫽6 as for a van der Waals interac- tion, and the strength of the 1/r²term corresponds to a centrifugal potential with angular momentum quantum number l⫽1.

For an angular momentum quantum number l, the strength of the 1/r² term is ␥⫽l(l⫹1).

In the following we shall calculate the threshold behavior of the probability for transmission from a region of low-r values to the left of the barrier to large-r values beyond the barrier. Reflection by and transmission through the potential tail 关Eq. 共2兲兴 can also be discussed in the absence of the centrifugal term, ␥⫽0 关9,34兴, as long as there is a range of small r values where Eq.共22兲is well fulfilled so that incom- ing and reflected WKB waves are accurate solutions of the Schro¨dinger equation. The following theory can be applied for ␥⫽0, and even for a weakly attractive 1/r² term, mean- ing that␥ can be negative but must be larger than⫺1/4.

At threshold, E⫽0, the Schro¨dinger equation共42兲is

冋

^{drd22⫺r␥2⫹␤rmm⫺2}

册

^{W共r兲⫽0. 共43兲}

In analogy to Eq.共5兲, we introduce the abbreviations

␮⫽

def

冑

^␥⫹14, ␯⫽

def 2␮ m⫺2⫽ 2

m⫺2

冑

^␥⫹14. 共44兲 The condition ␥⬎⫺1/4 mentioned above implies that both

␮ ^and␯ are positive real numbers. Two linearly independent solutions of Eq.共43兲are

W_1;2共r兲⫽

冑

^rJ_⫾␯共␳兲, ␳⫽ 2

m⫺2

冉

^␤r

冊

^{(m⫺2)/2. 共45兲}

The asymptotic (r→⬁) behavior of solutions共45兲is

W_1;2共r兲⬃共m⫺2兲^⫿␯

⌫共1⫾␯兲

冑

^r

冉

^␤r

冊

^⫿␮

冉

^{1⫺共m共⫺␤2/r兲2兲共m1⫺⫾2␯兲}

冊

^.

共46兲 For large values of r the 1/r² term dominates the poten- tial, and the Schro¨dinger equation 共42兲corresponds to

冋

^{drd22⫺r␥2⫹k2}

册

^{U⫽0. 共47兲}

The solutions of Eq. 共47兲are functions of kr only, and the solution which describes an outward traveling wave is

U共kr兲⫽exp

冉

^i␲2␮

冊 ^冑

^{krH␮(1)共kr兲, 共48兲}

which behaves asymptotically (kr→⬁) as

U共kr兲⬃

冑

_␲^2exp

冋

ⁱ

冉

^kr⫺␲4

冊册

^{. 共49兲}

Near threshold, kr→0, the leading contribution to Eq.共48兲is 关29兴

U共kr兲 ⬃

kr→0

⫺i

冑

²

␲^{ei␲␮/2⌫共}␮兲

冉

^kr2

冊

^{(1/2)⫺␮. 共50兲}

The r dependence of Eq. 共50兲 agrees with the r depen- dence of the leading asymptotic (r→⬁) behavior of the so- lution W₁(r) 关see Eq. 共46兲兴, so in the near-threshold limit k→0 these solutions can be matched according to

U共kr兲 ⫽

kr→0

LW₁共r兲, 共51兲 and the k-dependent coefficient is given by

L⫽

冑

^{k ei}_i^␲␮_␲^/2⌫共1⫹␯兲⌫共␮兲共m⫺2兲^␯

冉

^k2␤

冊

^{⫺␮. 共52兲}

To the left of the barrier, r→0, the large argument expan- sion of the Bessel function J_␯(␳^{) yields}

W₁共r兲⬃

冑

^m⫺_{␲ ␤}²

冉

^␤r

冊

^m/4cos

冉

^{␳⫺␲2␯⫺␲4}

冊

^{, 共53兲}

so the wave function LW₁(r) has the form¹

LW₁共r兲 ⬃

r→0

冑

^m4⫺␲ ␤²

冉

^␤r

冊

^{m/4L•共e⫺i␲␯/2ei(␳⫺␲/4)}

⫹e^⫹i␲␯/2e^{⫺i(␳⫺␲/4)}兲. 共54兲 The amplitude and phase of the two terms in Eq.共54兲corre- spond to the amplitude and phase of leftward traveling 共re- flected兲 and rightward traveling 共incoming兲 waves in the WKB approximation, which becomes increasingly accurate for small-r values where the 1/r^m term dominates the potential. The associated current densities, J⫽Im关(ប/M)␺*d␺^/dr兴, are

J_in/refl⫽ ប

4␲M共m⫺2兲兩L兩², 共55兲

1In a recent paper, Gao关35兴studied potential tails consisting of a centrifugal term and an attractive term proportional to 1/r^m, m⫽6, and he found the following rule: If the potential well supports a zero-energy bound state for an angular momentum quantum number l_b, it will also do so for l⫽l_b⫾4, l_b⫾8, . . . 共as long as l⭓0).

This rule and its generalization to any m⬎2 follow immediately from the properties of the wave function W₁ 关Eq. 共45兲兴, which solves the Schro¨dinger equation 共43兲 with the correct asymptotic 共large r) boundary conditions and is to be matched to the regular solution coming from the origin. To the left of the barrier, W1(r) becomes proportional to Eq. 共53兲, and depends on l only via the contribution ⫺␯␲/2⫽⫺(l⫹¹2)␲/(m⫺2) to the argument of the cosine, so the wave function is invariant up to a sign when l changes by an integral multiple of m⫺2. If matching to the regular solution yields a bound state at threshold for one angular momen- tum quantum number l_b, then it will do so also for l⫽l_b⫾(m

⫺2), lb⫾2(m⫺2), . . . (l⭓0). An important condition for this rule to hold is, of course, that the wave function to the left of the matching point be essentially unaffected by the centrifugal poten- tial; for a potential well of finite depth this cannot be fulfilled for an arbitrarily large l.

and they are equal to leading order, because the transmitted current density is of higher order in k. The transmitted wave traveling rightward at large r values is given by Eq.共49兲, and the associated current density is

J_trans⫽2បk

␲M. 共56兲

In order to obtain the current density of the reflected wave to an accuracy sufficient to fulfill the continuity condition, J_in

⫽J_refl⫹J_trans, we would have to include higher-order terms in the solution of the Schro¨dinger equation, in particular the contribution proportional to (kr)^1/2⫹␮ in the near-threshold limit 关36兴. From Eqs.共56兲 and 共55兲 the transmission prob- ability T⫽J_trans/J_in is, to leading order,

T⫽ 4␲²

共m⫺2兲^2␯␮␯关⌫共␯兲⌫共␮兲兴²

冉

^k2␤

冊

^{2␮ ⫽defP共m,␥兲共k␤兲2␮.}

共57兲 The parameters␮ ^and␯ are defined in Eq.共44兲, and␮ ^is related to the angular momentum quantum number l of the centrifugal potential by ␮⫽

冑

␥⫹1/4⫽l⫹1/2. The propor- tionality of T to k^2␮, i.e., to E^l⫹1/2, is simply an expression of Wigner’s threshold law 关2兴. Since ␤ is the only length scale in the Schro¨dinger equation, the dimensionless trans- mission probability is 共to leading order兲 naturally propor- tional to (k␤^)2␮. The derivation above, however, also gives, for all potential barriers consisting of a repulsive共or weakly attractive兲1/r² term and an attractive 1/r^mterm (m⬎2), the exact analytical expression for the coefficient of (k␤^)2␮:

P共m,␥兲⫽ 4␲²

共m⫺2兲^2␯2^2␮␮␯关⌫共␯兲⌫共␮兲兴². 共58兲 The numerical values of P are listed in Table II for m⫽3, 4, 5, and 6, and for strength parameters ␥ corresponding to angular momentum quantum numbers l⫽0, . . . ,10.

When the strength of the 1/r² term vanishes, ␥⫽0, we have ␮⫽1/2 and ␯⫽1/(m⫺2); result 共57兲 for this case agrees with the result derived for the reflection probability 1⫺T of an attractive 1/r^mpotential tail关9,34兴. For arbitrary

␥ ⁽⬎⫺1/4) and the special case m⫽4, the Schro¨dinger equation 共42兲 can be solved analytically with the help of Mathieu functions 关20兴. The transmission probability near E⫽0 can be derived from the asymptotic (r→0 and r

→⬁) forms of the wave functions given in Ref. 关20兴, and this leads exactly to result 共57兲with m⫽4. Note, however, that the aim of Ref.关20兴was to derive scattering lengths and effective range parameters based on an expansion of the co- tangent of the scattering phase shifts as functions of the asymptotic wave number k. This expansion is not really valid for potentials behaving like Eq. 共2兲, and the number of us- able leading terms it contains depends on the angular mo- mentum quantum number l and on the power m of the attrac- tive 1/r^mterm. In contrast, formula共57兲derived above is not restricted in such a way. It is valid for any strength parameter

␥⬎⫺1/4 and for any, not necessarily integer, power m⬎2.

The derivation above requires the compatibility of ap- proximations共43兲and共47兲to the Schro¨dinger equation共42兲 for a common range of r values. In analogy to Eq.共17兲this leads to the condition

k␤Ⰶ兩␥兩^{(1/2)⫹[1/(m⫺2)]}, 共59兲 which is fulfilled in the limit k→0 for any finite value of兩␥兩. As mentioned above, result共57兲also gives the correct lead- ing behavior for the case␥⫽0 关9,34兴.

Tunneling probabilities are frequently approximated with the help of the WKB formula关37兴,

T_WKB⫽exp共⫺2I兲, I⫽

冕

r_in r_out1

ប兩p共r兲兩dr, 共60兲 where r_in and r_out are the two classical turning points, be- tween which the local classical momentum p(r) is purely imaginary. For potentials falling off faster than 1/r², formula 共60兲 fails near threshold, because it yields a finite value at TABLE II. Values of the coefficient 关Eq. 共58兲兴 of (k␤)^2␮ in the leading term describing the near-

threshold behavior of the transmission probabilities关Eqs.共57兲兴through a potential barrier consisting of an attractive 1/r^m potential (m⬎2) and a repulsive 1/r² 共centrifugal兲 potential with a strength parameter ␥ corresponding to angular momentum quantum number l,␮⫽冑␥⫹1/4⫽l⫹1/2.

P(m,␥) m⫽3 m⫽4 m⫽5 m⫽6

l⫽0 4␲ 4 2.52537 1.91196

l⫽1 ␲/9 4/9 0.465421 0.464911

l⫽2 ␲/32400 4/2025 0.00527976 0.00849758

l⫽3 0.219870⫻10^⫺8 0.161250⫻10^⫺5 0.0000143161 0.0000421688 l⫽4 0.865576⫻10^⫺14 0.406273⫻10^⫺9 0.144770⫻10^⫺7 0.856395⫻10^⫺7 l⫽5 0.883151⫻10^⫺20 0.414522⫻10^⫺13 0.687112⫻10^⫺11 0.878061⫻10^⫺10 l⫽6 0.299916⫻10^⫺26 0.202710⫻10^⫺17 0.176390⫻10^⫺14 0.517031⫻10^⫺13 l⫽7 0.402416⫻10^⫺33 0.533097⫻10^⫺22 0.269760⫻10^⫺18 0.190840⫻10^⫺16 l⫽8 0.241743⫻10^⫺40 0.819834⫻10^⫺27 0.263645⫻10^⫺22 0.470489⫻10^⫺20 l⫽9 0.715162⫻10^⫺48 0.785816⫻10^⫺32 0.173684⫻10^⫺26 0.812985⫻10^⫺24 l⫽10 0.112305⫻10^⫺55 0.493600⫻10^⫺37 0.804279⫻10^⫺31 0.102260⫻10^⫺27

E⫽0, whereas the exact tunneling probability vanishes in this limit, as pointed out in Ref.关38兴. For a potential barrier asymptotically equal to ␥^/r2 关times ប²/(2M)兴, the WKB tunneling probability关Eq. 共60兲兴is proportional to k^2冑␥ near threshold, so the correct energy dependence T⬀E^␮ can be recovered with the help of the Langer modification ␥→␥

⫹1/4, which amounts to replacing l(l⫹1) by (l⫹1/2)² for the centrifugal potential关2,37兴. Criticisms and improvements of the Langer modification were recently discussed in vari- ous contexts 关2,19,30,39兴. The present derivation of the as- ymptotically (E→0) exact formula for the transmission probability for potential barriers of the special form关Eq.共2兲兴 allows us to give a founded judgement on the accuracy of the usual procedure involving the WKB formula关Eq.共60兲兴with the Langer modified potential.

The integrand of the action integral in Eq.共60兲is 1

ប兩p共r兲兩⫽

冑

^␮r22⫺␤r^mm⫺2⫺k², 共61兲 where we have invoked the Langer modification and re- placed ␥ ^by␥⫹1/4⫽␮². Note that the condition␥⬎⫺1/4, for which the above theory is applicable, corresponds to the condition that the Langer modified potential is asymptoti- cally repulsive. We obtain an upper bound for the integral I if we neglect one of the subtracted terms in the square root.

For a given point r˜ with rin⬍˜r⬍r_out, we thus have

I⭐

冕

r_in

˜r

冑

^␮r22⫺␤r^mm⫺2dr⫹

冕

˜r r_out

冑

^{␮r22⫺k2dr. 共62兲}

Inequality共62兲remains valid if we replace the inner classical turning point r_in by its threshold value r_in0 and the outer classical turning point r_out by the value r_out0 obtained by neglecting the 1/r^mterm in the potential:

r_in0⫽ ␤

␮^{2/(m⫺2)⭐rin, rout0⫽}

␮

k⭓r_out. 共63兲 The right-hand side of inequality 共62兲 can then be easily evaluated analytically. We choose the value of r˜ such, that the two terms neglected in the respective integrals in Eq.共62兲 have equal magnitudes at r˜:

␤^m⫺2

˜r^m ⫽k², kr˜⫽共k␤兲^1⫺2/m⫽共k␤兲^1⫺2/m. 共64兲 The leading orders of the approximated action integral are then

I_approx⫽

def

冕

^rin0

˜r

冑

^{␮r22⫺␤rmm⫺2dr⫹}

^冕

^˜rrout0

^冑

^{␮r22⫺k2dr}

⫽ m␮

m⫺2

冋

^ln

冉

^{共k␤2兲␮1⫺2/m}

冊

^{⫺1⫹O„共k␤兲2⫺4/m…}

册

^.

共65兲

Since I_approxis an upper bound for the action integral enter- ing the WKB expression 关Eq. 共60兲兴, the corresponding ex- pression exp(⫺2I_approx) is a lower bound for the WKB ap- proximation to the tunneling probability:

T_WKB⭓exp共⫺2I_approx兲

⫽

冉

^2e␮

冊

^{2m␮/(m⫺2)共k␤兲2␮关1⫹O„共k␤兲2⫺4/m…兴.}

共66兲 The coefficient of (k␤^)2␮ on the right-hand side of Eq.

共66兲 is larger than the coefficient P of (k␤^)2␮ in the exact expression 关Eq. 共57兲兴for the near-threshold tunneling prob- ability. The usual WKB treatment overestimates the exact tunneling probability by at least the factor

G⫽

def

lim

k→0

exp共⫺2I_approx兲

T ⫽

冉

^{2e␲␮␮⫹␯␮⫺⌫共1/2␮兲⌫共␯␯⫺␯1/2兲}

冊

^{2. 共67兲}

For large values of the strength ␥ ^{of the 1/r2} term in the potential, ␮ ^and ␯ are also large and we can express the gamma functions in Eq.共67兲via Stirling’s formula关29兴. This yields

G^␥→⬁⬃ 1⫹ m

12␮^⫹O

冉

^␮12

冊

^{, 共68兲}

showing that the WKB treatment 关with the additional ap- proximation according to Eq. 共65兲兴becomes exact for large angular momentum quantum numbers.

For smaller strength parameters␥ corresponding to lower angular momentum quantum numbers, the error in the con- ventional WKB treatment of the tunneling probability can, however, be quite large. For m⫽3, 4, 5, and 6, the numerical values of G are displayed in Fig. 5 as functions of ␮ (⬅l⫹1/2). For a given strength of the 1/r² term in the po- tential, the relative error in the WKB tunneling probability increases with the power m of the shorter-ranged 1/r^mterm, and it becomes proportional to m for large m values. For the realistic and important case m⫽6 and ␮⫽3/2, correspond- ing to a van der Waals interaction with an l⫽1 centrifugal potential, the WKB tunneling probability is too large by at least 38%.

Because of the large errors in the WKB tunneling prob- abilities for low partial waves, such methods should be re- garded critically in the near-threshold regime. Consider, e.g., a reaction leading to a compound particle, where the forma- tion cross section is typically given by an expression of the form 关40,41兴

␴C⫽_l

兺

_⫽^⬁₀ ^␴l⫽_l

兺

_⫽^⬁₀ ^共2l⫹_k^{21兲␲Tl. 共69兲}

Here T_lis essentially the probability of transmission through the effective potential barrier in partial wave l. We mention in passing that the transmission probability through a poten- tial barrier does not depend on the direction of propagation 关42兴. Due to Wigner’s threshold law, the contributions from low partial waves dominate the cross section 共69兲 near threshold, so the large errors from these partial waves will