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A new treatment of singular potentials
Volume 101A, number 2
PHYSICS LETTERS
12 March 1984
A NEW TREATMENT OF SINGULAR POTENTIALS M. ZNOJIL Institute of Nuclear Physics, Czechoslovak Academy of Sciences, 250 68 Re~, Czechoslovakia
Received 7 December 1983
For a broad class of strongly singular potentials, the effective (finite-dimensional) forms of the hamiltonian H axe constructed by its "smooth" algebraic truncation in a large model space. The method is based on an asymptotic factorization of H into a product of matrices. Its efficiency (acceleration of convergence of the energies) is illustrated on the singularly anharmonic potential V(r) = r 2 + hr -4 .
In the various applications of quantum mechanics we often encounter phenomenological potentials which are strongly singular at the origin. For example, the Lennard-Jones forces between the molecules, nucleon-nucleon potentials with a core, singular models of fields in zero dimensions, etc. [ 1]. In the radial Schr6dinger equation [ - d Z / d r 2 + I(l + 1)/r 2 + V(r)] •(r) = E ~ ( r ) , /=0,1 .....
(1)
they may be approximated by their truncated powerlaw expansion
1)] ,
r~0,
g_p > O ,
r - p ~k(r) remain regular at the origin - they may be
p>12,
(2)
and/or by its various r ~ r const modifications [2]. In a purely mathematical context, a lot of difficulties related to the r -+ 0 singularity o f (2) has been described in the literature [ 1]. Our present intention is to demonstrate that at least some of them are not so serious in the bound-state problem (1), (2) with q /> 1 andgq > 0 : (i) We shall show that one of the most common eigenvalue methods, namely, an approximate solution of the Schr6dinger equation in the truncated harmonicoscillator basis, becomes easily applicable to (1), (2) after a slight modification. In this setting, the singular66
t~(r)~exp[-gl/2prl-p/(p-
of the physical solutions [2]. Obviously, the functions
q V(r) = Vpq(r) = ~ gk r2k , k=-p
ity of (2) does not violate the applicability of some semi-numerical (e.g., matrix continued fractional [3] ) methods designed originally for the regular p = 0 anharmonic oscillators only. (ii) An introduction of a perturbative expansion of the exact, finite-dimensional effective hamiltonian [4] is one of the most current improvements of (i). Here, in essence, we shall derive a closed formula representing its asymptotic summ to all orders. In more detail, let us start from the threshold behaviour
expanded into the complete set I n ) o f the ordinary harmic-oscillator eigenstates [3], Ho In) = en ln) ,
e n = 4n + 21+ 3 ,
H 0 = - d 2 / d r 2 + l(l + 1)/r 2 + r 2 , oo
Ir-Pqj)= ~
In)z n .
n=0
As a consequence, we may rewrite our Schr6dinger equation (1), (2) in the algebraic form e~
C~mnZ n = O,
m = O, 1 .....
(3)
n=O
where 0.375-9601/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 101A, number 2
PHYSICS LETTERS
q gkTP +k - ETP - TP +1 ,
~( = TSKo T s +
(4)
k=-p
with K0 =H 0 ,
p = 2s,
12 March 1984
As expected, the rate o f convergence o f the ordinary truncation m e t h o d is rather slow - this follows from the marked difference between ~(r) and at r ~ 0, as well as from the observation that [Zn/Zn+ll 1 for n >> 1 and p = 0 at least [5]. Hence, the exact effective (M × M)-dimensional form of (3) [4] M-1
_1
- I(HoT
+ THo) + I ,
s=0,
p = 2s + l ,
n=0
1.... ,
~¢(eff), ~ rnn ~n = 0 ,
m = 0 , 1 ..... M - 1
(5)
should be considered, with the definition
T/j =
~((eff)= p ~ p
= ai = e i / 2 ,
_ p~ Q(Q~ Q)- 1Q~p,
i =j = O, 1....
(6)
M-I = b k = [(~ + 1)(k + l + 3 / 2 ) ] 1 / 2 ,
k =i=j-l(ork=j
=i-l)
P= I - Q =
Eq. (3) is solvable numerically by a standard numerical method, namely, by the diagonalization o f the (M × M)-dimensional submatrices of Jf in the limit M - + ,~. A sample of the results obtained for the simplest nontrivial potential V21(r ) = g_2 r - 4 + g - 1 r - 2 + go + gl r2 is given in the first column of table 1 here.
~
In>(nl,
n=0
=0,1 .....
employed at least in an approximate way. In the present context, it has the following two merits: (m 1) In the light of (4), 9~ is a band, 2t + 1 - a diagonal in general - we get (i/if) = ~ i/,
L/= O, 1..... M - 1 ,
(i,j)@(M-rn, M-n),
Table 1 Ground state energies in V(r) = r 2 + hr -4 and the acceleration of their convergence. h
M
Ordinary
Smooth
truncation
truncation (9)
0.02
10 30 50 100 300 500
3.466 3.327 3.300 3.285 3.283 3.282
3.358 3.290 3.281 3.281 3.283 3.282
0.04
10 30 50 100 300 500
3.509 3.403 3.390 3.386 3.386 3.385
3.419 3.383 3.384 3.386 3.385 3.385
10 30 50
4.03906 4.03579 4.03312
4.03841 4.03366 4.03204
100 300 500
4.03202 4.03198 4.03197
4.03195 4.03197 4.03197
0.4
m , n = 1 , 2 ..... t ,
(7)
directly from the definition (6). Thus, the exact equation (5) does not differ " t o o much" from the truncated forms o f (3). (m2) The matrix elements TM+i M+/ = M X (2+i_i) + O(1) increase quickly with growing M. Hence, we may assume that the finite, P-projected model space is large, M >> 1, and consider the truncated asymptotic expansions of Q ~ Q only. Let us restrict our attention here to the leading-order asymptotic contributions only, Q ~ Q ,~ QTtQ, t = p + q - ~ql [for q = 1, the scaling r ~ const × r gives gl -+ 1 in (4)]. Then, introducing the notation Qi-1 T = a i - 1 T Q i = Hi,
ai =
~ In>
i = 1, 2 ..... M, the factorizability of QTtQ gives, in a highly formal manner, the cancellations p ~ Q ( Q ~ Q ) - 1a ~ e
,~ p H T H T_ 1 "" H1
X (H1H 2 ... H m K H £ H T _ 1 "'" H ? ) - 1H1H 2 ...HtP ~ P H T H T _ I •. . H mT+ I K - 1 H m + l H m + 2 . . . H t P , (8)
67
Volume 101A, number 2
PHYSICS LETTERS
where t = 2m and K = I, or t = 2m + 1 and K = T. With the particular matrix elements as given by (4), the leading-order formula (8) may be further reduced. Indeed, after simple binomial multiplications with
H1H2... Hm ~ H~ Mm
( 0 ) ( 1 2m) .." (2m 2 m ) 0 0 .... f2m 1 ~0 (2m) •"" (2ra_ 2m 1 ) ( 22m m ) 0 ..
\--.
,
t
X tt+i-]
)-1
G (ti-k)k]-k ,rt ) ' k=l
i , j = 1 , 2 .... t , complementing (7) and completing our asymptotic
68
algebraic definition of the exactly solvable eigenvalue condition (5), det ~ ( e f f ) = 0. In the same way as above, the M>> 1 numerical algorithm (5), (7), (9) has been tested (second column in table 1). The approximate energies converge, with increasing M and possible oscillations, to the exact variational values. Obviously, the " s m o o t h cut-off" (9) causes an extremely efficient acceleration of convergence. A systematic inclusion of the higher-order corrections is under current investigation at present.
References
etc., we get finally the compact M >~ 1 prescription
[2t
12 March 1984
(9)
[1] R. Graeber and H.P. Diirr, Nuovo Cimento 40A (1977) 11; R.G. Newton, Scattering theory of waves and particles, (MeCraw-HiU, New York, 1965) Ch. 12.4, and references therein. [2] M. Znojil, J. Phys. A15 (1982) 2111. [3] S. Graffi and V. Grecchi, Lett. Nuovo Cimento 12 (1975) 425. [4] H. Feshbach, Ann. Phys. (NY) 5 (1958) 357. [5] M. Znojil, J. Phys. A16 (1983) 3313.
PHYSICS LETTERS
12 March 1984
A NEW TREATMENT OF SINGULAR POTENTIALS M. ZNOJIL Institute of Nuclear Physics, Czechoslovak Academy of Sciences, 250 68 Re~, Czechoslovakia
Received 7 December 1983
For a broad class of strongly singular potentials, the effective (finite-dimensional) forms of the hamiltonian H axe constructed by its "smooth" algebraic truncation in a large model space. The method is based on an asymptotic factorization of H into a product of matrices. Its efficiency (acceleration of convergence of the energies) is illustrated on the singularly anharmonic potential V(r) = r 2 + hr -4 .
In the various applications of quantum mechanics we often encounter phenomenological potentials which are strongly singular at the origin. For example, the Lennard-Jones forces between the molecules, nucleon-nucleon potentials with a core, singular models of fields in zero dimensions, etc. [ 1]. In the radial Schr6dinger equation [ - d Z / d r 2 + I(l + 1)/r 2 + V(r)] •(r) = E ~ ( r ) , /=0,1 .....
(1)
they may be approximated by their truncated powerlaw expansion
1)] ,
r~0,
g_p > O ,
r - p ~k(r) remain regular at the origin - they may be
p>12,
(2)
and/or by its various r ~ r const modifications [2]. In a purely mathematical context, a lot of difficulties related to the r -+ 0 singularity o f (2) has been described in the literature [ 1]. Our present intention is to demonstrate that at least some of them are not so serious in the bound-state problem (1), (2) with q /> 1 andgq > 0 : (i) We shall show that one of the most common eigenvalue methods, namely, an approximate solution of the Schr6dinger equation in the truncated harmonicoscillator basis, becomes easily applicable to (1), (2) after a slight modification. In this setting, the singular66
t~(r)~exp[-gl/2prl-p/(p-
of the physical solutions [2]. Obviously, the functions
q V(r) = Vpq(r) = ~ gk r2k , k=-p
ity of (2) does not violate the applicability of some semi-numerical (e.g., matrix continued fractional [3] ) methods designed originally for the regular p = 0 anharmonic oscillators only. (ii) An introduction of a perturbative expansion of the exact, finite-dimensional effective hamiltonian [4] is one of the most current improvements of (i). Here, in essence, we shall derive a closed formula representing its asymptotic summ to all orders. In more detail, let us start from the threshold behaviour
expanded into the complete set I n ) o f the ordinary harmic-oscillator eigenstates [3], Ho In) = en ln) ,
e n = 4n + 21+ 3 ,
H 0 = - d 2 / d r 2 + l(l + 1)/r 2 + r 2 , oo
Ir-Pqj)= ~
In)z n .
n=0
As a consequence, we may rewrite our Schr6dinger equation (1), (2) in the algebraic form e~
C~mnZ n = O,
m = O, 1 .....
(3)
n=O
where 0.375-9601/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 101A, number 2
PHYSICS LETTERS
q gkTP +k - ETP - TP +1 ,
~( = TSKo T s +
(4)
k=-p
with K0 =H 0 ,
p = 2s,
12 March 1984
As expected, the rate o f convergence o f the ordinary truncation m e t h o d is rather slow - this follows from the marked difference between ~(r) and
_1
- I(HoT
+ THo) + I ,
s=0,
p = 2s + l ,
n=0
1.... ,
~¢(eff), ~ rnn ~n = 0 ,
m = 0 , 1 ..... M - 1
(5)
should be considered, with the definition
T/j =
~((eff)= p ~ p
= ai = e i / 2 ,
_ p~ Q(Q~ Q)- 1Q~p,
i =j = O, 1....
(6)
M-I = b k = [(~ + 1)(k + l + 3 / 2 ) ] 1 / 2 ,
k =i=j-l(ork=j
=i-l)
P= I - Q =
Eq. (3) is solvable numerically by a standard numerical method, namely, by the diagonalization o f the (M × M)-dimensional submatrices of Jf in the limit M - + ,~. A sample of the results obtained for the simplest nontrivial potential V21(r ) = g_2 r - 4 + g - 1 r - 2 + go + gl r2 is given in the first column of table 1 here.
~
In>(nl,
n=0
=0,1 .....
employed at least in an approximate way. In the present context, it has the following two merits: (m 1) In the light of (4), 9~ is a band, 2t + 1 - a diagonal in general - we get (i/if) = ~ i/,
L/= O, 1..... M - 1 ,
(i,j)@(M-rn, M-n),
Table 1 Ground state energies in V(r) = r 2 + hr -4 and the acceleration of their convergence. h
M
Ordinary
Smooth
truncation
truncation (9)
0.02
10 30 50 100 300 500
3.466 3.327 3.300 3.285 3.283 3.282
3.358 3.290 3.281 3.281 3.283 3.282
0.04
10 30 50 100 300 500
3.509 3.403 3.390 3.386 3.386 3.385
3.419 3.383 3.384 3.386 3.385 3.385
10 30 50
4.03906 4.03579 4.03312
4.03841 4.03366 4.03204
100 300 500
4.03202 4.03198 4.03197
4.03195 4.03197 4.03197
0.4
m , n = 1 , 2 ..... t ,
(7)
directly from the definition (6). Thus, the exact equation (5) does not differ " t o o much" from the truncated forms o f (3). (m2) The matrix elements TM+i M+/ = M X (2+i_i) + O(1) increase quickly with growing M. Hence, we may assume that the finite, P-projected model space is large, M >> 1, and consider the truncated asymptotic expansions of Q ~ Q only. Let us restrict our attention here to the leading-order asymptotic contributions only, Q ~ Q ,~ QTtQ, t = p + q - ~ql [for q = 1, the scaling r ~ const × r gives gl -+ 1 in (4)]. Then, introducing the notation Qi-1 T = a i - 1 T Q i = Hi,
ai =
~ In>
i = 1, 2 ..... M, the factorizability of QTtQ gives, in a highly formal manner, the cancellations p ~ Q ( Q ~ Q ) - 1a ~ e
,~ p H T H T_ 1 "" H1
X (H1H 2 ... H m K H £ H T _ 1 "'" H ? ) - 1H1H 2 ...HtP ~ P H T H T _ I •. . H mT+ I K - 1 H m + l H m + 2 . . . H t P , (8)
67
Volume 101A, number 2
PHYSICS LETTERS
where t = 2m and K = I, or t = 2m + 1 and K = T. With the particular matrix elements as given by (4), the leading-order formula (8) may be further reduced. Indeed, after simple binomial multiplications with
H1H2... Hm ~ H~ Mm
( 0 ) ( 1 2m) .." (2m 2 m ) 0 0 .... f2m 1 ~0 (2m) •"" (2ra_ 2m 1 ) ( 22m m ) 0 ..
\--.
,
t
X tt+i-]
)-1
G (ti-k)k]-k ,rt ) ' k=l
i , j = 1 , 2 .... t , complementing (7) and completing our asymptotic
68
algebraic definition of the exactly solvable eigenvalue condition (5), det ~ ( e f f ) = 0. In the same way as above, the M>> 1 numerical algorithm (5), (7), (9) has been tested (second column in table 1). The approximate energies converge, with increasing M and possible oscillations, to the exact variational values. Obviously, the " s m o o t h cut-off" (9) causes an extremely efficient acceleration of convergence. A systematic inclusion of the higher-order corrections is under current investigation at present.
References
etc., we get finally the compact M >~ 1 prescription
[2t
12 March 1984
(9)
[1] R. Graeber and H.P. Diirr, Nuovo Cimento 40A (1977) 11; R.G. Newton, Scattering theory of waves and particles, (MeCraw-HiU, New York, 1965) Ch. 12.4, and references therein. [2] M. Znojil, J. Phys. A15 (1982) 2111. [3] S. Graffi and V. Grecchi, Lett. Nuovo Cimento 12 (1975) 425. [4] H. Feshbach, Ann. Phys. (NY) 5 (1958) 357. [5] M. Znojil, J. Phys. A16 (1983) 3313.