Download SEMICLASSICAL AND LARGE QUANTUM NUMBER LIMITS

PROCEEDINGS OF THE FOURTH INTERNATIONAL CONFERENCE ON
DYNAMICAL SYSTEMS AND DIFFERENTIAL EQUATIONS
May 24 – 27, 2002, Wilmington, NC, USA
pp. 288–294
SEMICLASSICAL AND LARGE QUANTUM NUMBER LIMITS
OF THE SCHRÖDINGER EQUATION
Harald Friedrich
Physik-Department
Technische Universität München
85747 Garching, Germany
Abstract. For bound one-dimensional systems, the semiclassical limit ~ → 0 of the
Schrödinger equation generally corresponds to the limit of infinite quantum numbers,
and conventional WKB quantization becomes increasingly accurate in this limit. A
potential well with a sufficiently strong attractive inverse-square tail supports an
infinite dipole series of bound states, but the limit of infinite quantum numbers is not
the semiclassical limit in this case. Semiclassical eigenvalues derived via conventional
WKB quantization tend to a constant relative error in the large-quantum-number
limit when the Langer modification is used. Without the Langer modification the
relative error grows exponentially in the limit of large quantum numbers.
1. Introduction. Semiclassical approximations of the Schrödinger equation are
usually expected to work well in the limit of high energies and/or large quantum
numbers, and this is indeed the case for bound states in one-dimensional potential
wells which go to infinity as some positive power of the coordinate r at large distances. Potential wells converging to a constant, e.g. zero, on at least one side can
support an infinite number of bound states, if the potential approaches its limiting
threshold value slower than 1/r2 , as is the case for Coulombic potentials.
Inverse-square potentials define the borderline between long-ranged potentials
supporting an infinite number of bound states, and shorter ranged potentials supporting at most a finite number of bound states. A potential well with a sufficiently
attractive inverse-square tail still does support an infinite number of bound states,
but the limit of infinite quantum numbers does not correspond to the semiclassical
limit, defined as the limit of vanishing Planck’s constant ~ [1]. The aim of the
present contribution is to illustrate this statement with a concrete example demonstrating how conventional semiclassical WKB quantization leads to a constant or
exponentially increasing relative error in the energy eigenvalues as the quantum
number goes to infinity.
2. Semiclassical and anticlassical limits of the Schrödinger equation. The
semiclassical limit [1] of the Schrödinger equation
~2 d2 Ψ
+ F V (r)Ψ(r) = E Ψ(r) ,
(1)
2M dr2
is reached in the formal (but unrealistic) limit ~ → 0, which, for a homogeneous
potential V (r) of degree d,
V (αr) = αd V (r) ,
(2)
−
1991 Mathematics Subject Classification. 81Q20.
Key words and phrases. semiclassical limit, large quantum numbers, inverse-square potentials,
WKB quantization, quantum-classical correspondence.
288
SEMICLASSICAL AND LARGE QUANTUM NUMBER LIMITS
289
is equivalent to realistic limits, which can be reached by varying the energy E and/or
the strength parameter F . Whether the semiclassical limit is reached for large or
small energies or for large or small potential strengths depends on the degree d of
homogeneity of the potential, as summarized below (see also Ref. [2], p. 322 of
Ref. [3]),
0<d :
F →0
−2 < d < 0 : |F | → ∞
d = −2 : |F | → ∞
d < −2 : |F | → ∞
or
or
and
or
|E| → ∞
E→0
E arbitrary
|E| → ∞ .
(3)
Conversely, the anticlassical or extreme quantum limit is reached for the opposite
conditions to those listed in (3), e.g. |F | → ∞ or E → 0 for d > 0.
For positive degrees d, e.g. all sorts of homogeneous oscillators, the first line of
(3) expresses the widely appreciated fact, that the semiclassical limit is reached for
high energies, whereas E → 0 defines the anticlassical limit. Billard systems, where
freely moving particles are reflected at infinitely hard walls, can be integrated into
the scheme (3) by putting d = +∞. For bound systems the limit of high energy
implies the limit of large quantum numbers.
For negative degrees d, the potentials behave as inverse powers of the coordinate
and vanish asymptotically. The spectrum of the Hamiltonian is continuous for
positive energies, and bound states with discrete energy eigenvalues occur for E ≤ 0.
For d < −2, the semiclassical limit is reached for large energies and the threshold
E = 0 corresponds to the anticlassical or extreme quantum limit, as in the case of
positive degrees d. A one-dimensional potential well with a tail proportional to rd
with d < −2 can support at most a finite number of bound states and the limit
of (infinitely) large quantum numbers cannot be taken in the bound state regime.
The semiclassical limit can be reached, however, e.g., by taking the limit of large
potential strengths, see (3).
When −2 < d < 0, large energies correspond to the anticlassical or extreme
quantum limit. This is seen, e.g. in the differential cross section for scattering by
a 1/r potential in two spatial dimensions, where the exact quantum mechanical
result approaches that of the Born approximation but differs substantially from the
classical result at high energies [4, 5, 6]. A one-dimensional potential well with
a tail proportional to rd with −2 < d < 0 supports an infinite number of bound
states, and the semiclassical limit is reached for E → 0 corresponding to the limit
of large quantum numbers. For 1/r potential in more than one spatial dimension,
the continuum limit can occur well below the “break-up threshold” E = 0. E.g., for
a two-electron atom, the continuum threshold is the ionization threshold, and the
physical states between the ionization threshold and the break-up threshold are not
strictly bound states but resonances of finite width. The structure of this spectrum
immediately below the break-up threshold is not at all understood yet and is a
subject of intense ongoing research, see, e.g. Refs [7, 8].
3. WKB quantization. One-dimensional systems in which the classical motion is
bound by two turning points, rl and rr , can be quantized via the WKB quantization
rule,
Z rr
³
ν´
,
(4)
I=
p(r) dr = π~ n +
4
rl
290
HARALD FRIEDRICH
p
where p(r) = 2M[E − U (r)] is the local classical momentum and ν is the Maslov
index accounting for a phase loss of the WKB wave at each classical turning point.
In the semiclassical limit, this phase loss is generally equal to π/2 [1] and the
Maslov index just counts the number of turning points, ν = 2. Assuming an energy
independent Maslov index and differentiating Eq. (4) with respect to energy yields,
1 dI
1
1
dn
=
=
Tcl =
dE
π~ dE
2π~
hνcl
,
(5)
where we have used the fact that dI/dE = 12 Tcl ; Tcl is the classical period of the
whole orbit, so 21 Tcl is the time of passage from rl to rr .
For a unit difference of quantum number, Eq. (5) reduces to the familiar formulation of Bohr’s correspondence principle which relates the energy difference
∆E = En+1 − En between initial and final states in a photoemmission process to
the energy hνcl of the photon, whose frequency corresponds to the frequency νcl of
the corresponding classical orbit,
En+1 − En = hνcl
.
(6)
Accurate fulfillment of Bohr’s correspondence principle thus implies a good approximation of the energy eigenvalues via the WKB quantization rule (4) (with
energy-independent Maslov index) and vice versa.
For a one-dimensional potential well which grows to infinity as rd with d > 0 or
d < −2 on both sides, e.g. homogeneous oscillators U (r) ∝ |r|d , d > 0, or “spiked
oscillators” [9] such as U (r) = A r2 + B/r4 , the limit of large quantum numbers,
n → ∞, is the high-energy limit E → ∞. The semiclassical energies obtained via
WKB quantization (4) become increasingly accurate in this limit.
For a one-dimensional potential well consisting of a short-ranged part Usr and a
Coulombic tail,
F
U (r) = Usr (r) −
,
(7)
r
the bound-state energy spectrum is a Rydberg series,
R
En = −
,
(8)
(n − µn )2
with a Rydberg constant R depending on the potential strength F and smoothly
energy-dependent quantum defects µn , which behave as analytical functions of E
near threshold [10],
µ ¶
1
n→∞
µn ∼ µ0 + O(E) = µ0 + O
;
(9)
n2
µ0 is a constant. The energies EnWKB obtained via WKB quantization (4) also
behave as given in Eq. (8) with the correct Rydberg constant (see, e.g., Sect. 3.1.1
of Ref. [3]), but the quantum defects may converge to a slightly different constant
µ00 at threshold. Near threshold, the WKB energies are,
¶
µ ¶
µ
1
µ00 − µ0
WKB n→∞
+O
.
(10)
En
∼ En 1 + 2
n
n4
Not only the absolute error, but also the relative error in the WKB energies tends
to zero as we approach the semiclassical limit E → 0. These considerations can
be generalized to potential tails proportional to rd with other (non-integer) values
of d between −2 and 0; the generalized form of the Rydberg series (8) is En ∝
−(n − µn )2d/(2+d) [11].
SEMICLASSICAL AND LARGE QUANTUM NUMBER LIMITS
291
A potential well of fixed strength with a tail falling off faster than 1/r2 supports
at most a finite number of bound states, so we cannot take the limit of infinite
quantum numbers in this case. For a deep well with a large but finite number of
bound states, the accuracy of the WKB quantization rule (4) actually improves
towards smaller quantum numbers corresponding to larger binding energies and
deteriorates towards threshold [12, 13], because this represents the anticlassical
limit according to (3) [12, 14, 15]. The appropriate modification of the conventional
WKB quantization rule, which gives exact energies in the anticlassical limit, involves
energy dependent non-integral Maslov indices and has been discussed in detail in
Refs. [12, 16].
4. Potential wells with inverse-square tails. On the borderline between potential wells with long-ranged attractive tails, which fall off slower than 1/r2 and
support an infinite number of bound states, and those with shorter ranged tails
which fall off faster than 1/r2 and support at most a finite number of bound states,
there are the potentials with inverse-square tails,
~2 g
.
(11)
2M r2
Such potentials occur as centrifugal terms in the radial Schrödinger equation (−g ≡
l(l+1) for quantum number l of three-dimensional angular momentum) and through
monopole-dipole interactions [17]. Inverse-square potentials which are more attractive than the s-wave centrifugal potential in two dimensions [18, 19],
1
g>
,
(12)
4
support an infinite series (“dipole series”) of bound states whose energies behave as


2πn
n→∞

(13)
En ∼ −E0 exp − q
1
g− 4
r→∞
U (r) ∼ −
towards threshold [17, 20]. The limiting value of the ratio of successive energies
depends only on the strength parameter g of the tail, but the prefactor E0 defining
the energy scale is determined by the potential at small distances, where it must
necessarily deviate from the asymptotic form (11).
In the Schrödinger equation with an inverse-square potential (11), the energy can
be scaled out, there is no natural energy scale and the semiclassical limit cannot
be approached by varying the energy. The strength g alone determines how close
we are to the semiclassical or the anticlassical limit. As listed in (3), |g| → ∞
leads to the semiclassical limit whereas g → 0 defines the anticlassical limit. For a
fixed potential strength g > 1/4 the potential supports an infinite number of bound
states, the energies of which converge to zero according to Eq. (13), but this limit
is not the semiclassical limit. In the following, this is illustrated using the concrete
example
µ 2
¶
~2
β
g
1
U (r) =
− 2 , g>
,
(14)
4
2M r
r
4
which has been studied in detail by Varshni [21] and others.
The potential (14) has a minimum value −De at r = re ,
r
2
~2 g 2
, re =
β ,
(15)
De =
2
8Mβ
g
292
HARALD FRIEDRICH
and the energy eigenvalues relative to the potential depth,
def
εn =
En
De
,
(16)
depend only on g and not on β. The Schrödinger equation for the potential (14)
is not solvable analytically [21], but Moritz et al. [11] have derived an analytical
expression for the pre-exponential factor defining the actual energies in the dipole
series (13),



Ãr
!
64
4
1
2πn
n→∞
, f=
 .
εn ∼ −f exp − q
exp  q
arg Γ i g −
2
g
4
1
1
g−
g−
4
4
(17)
On the other hand, the WKB integral entering the semiclassical quantization
rule (4) is known analytically for the potential (14) [21],
r
µ
¶
I
η
3 5
g
= (1 + ε) 2 F1
, ; 2; 1 + ε , η =
,
(18)
π~
2
4 4
8
where ε is the normalized energy, ε = E/De < 0. Expanding the hypergeometric
function 2 F1 for small absolute values of the normalized energy ε gives,
∞
X
I
η
= (1 + ε)
Aj |ε|j (aj − ln |ε|) ,
π~
2
j=0
with
¡3¢ ¡5¢
(19)
µ
¶
µ
¶
3
5
¡ ¢ ¡ ¢ , aj = 2ψ(j + 1) − ψ j +
Aj =
−ψ j+
,
(20)
4
4
(j!)2 Γ 34 Γ 54
√
£ ¡ ¢ ¡ ¢¤−1
in particular, A0 = Γ 34 Γ 54
= 2 2/π and a0 = 6 ln 2 − 4. In (20), ψ stands
for the digamma function [22]. The semiclassical energies derived via conventional
WKB quantization thus behave as
µ
¶
µ
¶
2πn
π
WKB
WKB n→∞
WKB
∼ −f
exp − √
εn
, f
= 64 exp −4 − √
.
(21)
g
g
4 j
4 j
Comparing Eq. (21) with Eq. (17) shows that the WKB energies have the wrong
energy dependence all the way to threshold,
q because the exponent in the n-dependent
√
factor is −2πn/ g instead of −2πn/ g − 14 . This defect can be remedied by invoking the Langer modification [1, 3], whereby the term 14 ~2 /(2Mr2 ) is added to
the (radial) potential to be used in the WKB calculation. For a centrifugal potential in three dimensions with angular momentum quantum number l, this cor¡
¢2
responds to replacing l(l + 1) by l + 12 ; in the present situation it amounts to
replacing g by g − 14 in Eq.(21). The Langer modification repairs the defect in
the energy dependence of the WKB energies (21), but the absolute values of the
energy keep an error due a wrong pre-exponential factor. In the limit of large
strengths g both f and f WKB (with or without Langer modification) approach the
value 64/ exp (4) = 1.1722 . . .. Explicit values for the exact factor f and its WKB
approximation f WKB are given in Table 1 ³for a number
parameters g.
q of strength
´
Also listed are the values f WKBL = 64 exp −4 − π/
the Langer modification.
g−
1
4
obtained by invoking
SEMICLASSICAL AND LARGE QUANTUM NUMBER LIMITS
293
When the Langer modification is invoked, the ratio of the WKB energies to the
exact eigenvalues tends to a finite constant different from unity as the quantum
number goes to infinity. Its value is given by the ratio of the pre-exponential factors
f WKBL and f listed in Table 1. This corresponds to a constant relative error in the
semiclassical eigenvalues in the limit n → ∞.
If we choose not to invoke the Langer modification, then the ratio of the WKB
energies to the exact eigenvalues acquires an additional n-dependent factor,



WKB
WKB
εn
1
1
n→∞ f
∼
exp 2πn  q
− √  ,
(22)
εn
f
g
1
g−
4
so the relative error in the semiclassical eigenvalues actually grows exponentially
with n as n goes to infinity.
The smallest potential strength studied by Varshni in Ref. [21] was g = 200 (corresponding to η = 5), for which the errors in the WKB calculation are quite small.
The relative error of 0.4% which Varshni quotes for the lower WKB eigenvalues
agrees with the expectation from the pre-exponential factors in Table 1, and the
increase of the error with n is due to the exponential on the right-hand side of
Eq. (22).
5. Summary. The semiclassical limit of the Schrödinger equation for a one-dimensional potential well depends crucially on the asymptotic behaviour of the potential.
For potentials growing to infinity as a positive power of the coordinate on both sides
of the well, the high-energy limit is the semiclassical limit and corresponds to the
limit of infinite quantum numbers. For an attractive potential tail falling off to
a constant threshold more slowly than the inverse square of the coordinate, there
are infinitely many bound states with energies converging to the threshold; this
threshold represents the semiclassical limit and the absolute and relative plural of
semiclassical energies obtained via conventional WKB quantization (4) vanish in
the limit of infinite quantum numbers.
Potential wells with inverse-square tails represent the boundary between longranged potentials supporting an infinite number of bound states, and shorter ranged
potentials supporting at most a finite number of bound states. A potential well with
a sufficiently attractive inverse-square tail (11), g > 14 , supports an infinite dipole
series of bound states, but the limit of infinite quantum numbers is not the semiclassical limit. This is demonstrated for a concrete example (14) by comparing the
semiclassical energies obtained via conventional WKB quantization (4) with the analytically available asymptotic form of the exact eigenvalues (17). The semiclassical
energies have the correct asymptotic energy dependence only if the Langer modification, g → g − 41 , is used in the WKB quantization, but the relative errors of the
semiclassical energies still tend to a finite constant in the limit of infinite quantum
numbers. Without the Langer modification, the relative errors of the semiclassical
energies grow exponentially with the quantum number.
REFERENCES
[1] M. V. Berry and K. E. Mount, Rep. Prog. Phys. 35 (1972) 315.
[2] H. Friedrich, in: Atoms and Molecules in Strong External Fields, ed. P. Schmelcher and W.
Schweizer, (Plenum Press, New York, London, 1998), p. 153.
[3] H. Friedrich, Theoretical Atomic Physics, 2nd. Edition, Springer-Verlag, Berlin, 1998.
[4] G. Barton, Am J. Phys. 51 (1983) 420.
294
HARALD FRIEDRICH
Table 1. Pre-exponential factors in the spectra (17), (21) for various strength parameters g of the inverse-square term in the potential (14); f WKBL is obtained by invoking the Langer modification,
i.e. replacing g by g − 41 in the expression for f WKB in Eq. (21).
g
0.5
1
2
5
10
20
50
100
200
500
1000
2000
∞
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
f
0.00013
0.01095
0.06870
0.23320
0.39370
0.55428
0.73841
0.84873
0.93466
1.01682
1.06045
1.09222
1.17220
f WKB
0.01379
0.05066
0.12713
0.28763
0.43406
0.58066
0.75171
0.85618
0.93870
1.01856
1.06135
1.09268
1.17220
f WKBL
0.00219
0.03116
0.10905
0.27732
0.42860
0.57809
0.75087
0.85584
0.93857
1.01852
1.06133
1.09268
1.17220
Q.-g. Lin, Am. J. Phys. 65 (1997) 1007.
M. J. Moritz and H. Friedrich, Am. J. Phys. 66 (1998) 274.
J.-M. Rost, Phys. Reports 297 (1998) 271.
R. Püttner, B. Grémaud, D. Delande, M. Domke, M. Martins, A. S. Schlachter and G. Kaindl,
Phys. Rev. Lett. 86 (2001) 3747.
J. Trost and H. Friedrich, Phys. Lett. A 228 (1997) 127.
M. J. Seaton, Rep. Prog. Phys. 46 (1983) 167.
M. J. Moritz, C. Eltschka and H. Friedrich, Phys. Rev. A 64 (2001) 022101.
J. Trost, C. Eltschka, and H. Friedrich, J. Phys. B 31, 361 (1998).
B. Gao, Phys. Rev. Lett. 83, 4225 (1999).
C. Eltschka, H. Friedrich and M. J. Moritz, Phys. Rev. Lett. 86 (2001) 2693.
C. Boisseau, E. Audouard and J. P. Vigué, Phys. Rev. Lett. 86 (2001) 2694.
H. Friedrich and J. Trost, Phys. Rev. Lett. 76 (1996) 4869; Phys. Rev. A 54 (1996) 1136.
T. Purr, H. Friedrich and A. T. Stelbovics, Phys. Rev. A 57 (1998) 308; T. Purr and H.
Friedrich, Phys. Rev. A 57 (1998) 4729.
M. A. Cirone, K. Rzazweski, W. P. Schleich, F. Straub and J. A. Wheeler, Phys. Rev. A 65
(2002) 022101.
K. Kowalski, K. Podlaski and J. Riembilinski, Phys. Rev. A 66 (2002)
M. J. Moritz, C. Eltschka and H. Friedrich, Phys. Rev. A 63 (2001) 042102.
Y. P. Varshni, Europhys. Lett. 20 (1992) 295.
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New
York, 1972).
Received August 2002; in revised March 2003.
E-mail address: [email protected]