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Remark on the relation between the generator-coordinate method and the random-phase approximation
Nuelear Physics A157 (1970) 358- 362; @ ~o~rh-~oIIa~d P~bZi~hin~Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
REMARK ON THE RELATION BETWEEN THE GENERATOR-COORDINATE METHOD
AND THE RANDOM-PHASE
APPROXIMATION
1. da PROVIDENCIA Laboratdrio de F&a, ~nive~si~de de Coimbra, Portugal Received 9 June 1970 (Revised 17 July 1970) ABxdract:The equivalence between the generator-coordinate method and the random-phase approximation has been established by several authors. In the present note it is shown that a trial wave function 4(x; a), dependent on the time by the parameter a = u(t), oscillates in time with a frequency which is also given by the equations provided by both the generator-coordinate method and the RPA for the excitation energy.
1.Summary of previous results The relation between the generator-coordinate method of Grifk, Hill and Wheeler, in the Gaussian overlap approximation, and the random-phase approximation has been extensively investigated by several authors lw3). We will consider the related question of the connexion between those methods and the problem of the approximate time-evoIution of variational parameters a appearing in trial many-body wave functions 4(a) = 4(x; a). In the generator-coordinate method one considers linear combinations
Y(x) =
/4(x; M4da,
(1)
where the weight functionf(u) is determined by minimizing the expectation value of the energy for the state Y(x). In this way one arrives at the eigenvafue equation
J
f #(a,
a’)-El@,
u’))~(u’)du’
= 0,
(2)
where
(3)
K(% a’> = (4(4lHl4(0, I(& 4
= (4644(4).
In the Gaussian overlap approximation notation,
~
f
= &=-(a,a’) =
one may write, closely following Wong’s “)
(f ~)(~,) .
J%+s(u’u) 358
(4)
GENERATOR-COORDINATE
METHOD
359
The quantities Ea”)s, A and B are expressed in terms of the many-body wave functions +(xta). By expanding K(a, a’) and I@, a’), as given by the defining eqs. (3), we obtain &” = (MW),
(5a)
s = 3(%?MU $)9
(5b)
A = (2s)-‘((a,~I~la,~)-~*,(~,~l~,~>>,
B = (~s)-‘((~,~~IHI~)-E.,(~.Z~I~)),
(5c) (54
where 4 = 4(u) and the derivatives a,@~= &#J/&, etc., are taken at the point u = 0, for which
For simplicity, we have assumed that the wave functions 4, and therefore the parameters A and B, are real quantities. This assumption may obviously be removed without difficulty. It has been shown that the eigenvalue problem expressed by eq. (4) may be cast in the usual RPA form {E”-[E,,+AC+C+~B(C+Z+C2)]}ln)
= 0,
(6)
where C, C+ are boson operators for the Hilbert subspace spanned by the functions 4(x; a), [C, C’] = 1. (7) One can then solve eq. (6) by standard techniques. Let Q = xc-yc+,
@a)
with
(8b)
X2- y2 = 1. One may write eq. (6) in the form {E,-CE,,+~(~-A)+WQ+QI}I~)
= 0,
(9)
provided that
so that E,, = E,,++(w--A)+nw.
(12)
2. The time evolution of generator coordinates It is the purpose of the present note to show that, if we allow the parameter a, occurring in 4(x; a), to become time dependent, so that 4(x; u(t)) describes a nonstationary state, then the time evolution (small oscillations) of the function u(t) is
360
J. DA PROVIDENCIA
also determined by eq. (10). In order to obtain the equation governing the time evolution of the function a(t) we consider the time dependent variational principle, which, for the present purpose, may be written (13) On physical grounds one expects the equation determining the time evolution of the parameter a should be of second order. In order to obtain an equivalent system of two first-order equations it is necessary to extend the set of functions 4(x; u) by replacing the parameter a by a complex parameter “) c = a+ i b, where a and b are real. By expanding eq. (13) up to second order in c we may cast it in the form
-i(h*h)
(!i*) +td(c*c>6 1)(:) =0,
(14
where the quantities A and B are still given by eqs. (5~) and (5d). From eq. (14) we obtain -i(_$)
+ (t
z)(cc*(
=O.
(15)
By making the ansatz c = x e-‘“‘+y eiW we recover eq. (10). At this point one may enquire about the physical significance of the parameter b, the imaginary part of c, and about the meaning of the operators C+, C. In order to provide answers for these questions we note that eq. (15) is equivalent to ci = (A-B)b,
Wa)
b = -(A+B)u,
which may be written d=IL--
Ia2 4s ab ’
b=
(16b)
-Lax 4s
au
where.%@is the expectation value of H s!P =
qc*, c) = (4(4mb(4) (4(aNc))
= &“+s(c*c)(‘j ;)
(;) .
(174
The analogy between eq. (16b) and Hamilton equations shows that pa = 4sb is the variable canonically conjugate to a. In terms of a and pa the energy becomes X = E,,+
$4-B)p.:+2s(A+B)a2.
Wb)
GENERATOR-COORDINATE
METHOD
361
The wave function 4(x; a+ ib) appears, therefore, to represent a state possessing definite expectation values a and p. of some collective conjugate operators 21and 8,. Indeed, a wave packet lc), belonging to the Hilbert subspace spanned by the wave functions 4(x; c) and characterized by appropriate average values a and pa = 4 sb of the conjugate operators a = t(2s)-+(~+ +c), (18) hll = i(2s)*(C+ -C), is conveniently defined as the eigenvector lc) of C corresponding eigenvalue (2s)+ c, Clc) = (2s)*clc).
to a complex (19)
’ p recisely such a wave packet lc), which is It has been shown 1, “) that 4( x,. c ) is usually referred as a coherent state. It is clear that the expectation value of the Hamiltonian 2 = E,,+AC+C+3B(C+2+C2), (20)
appearing in eq. (6), is given by (clJ?lc) = &? = E,,+2s[Ac*c+3B(c*2+cZ)],
(21)
i.e.,%? is the same as given by eq. (17a). The expectation value of2 for the wave packet lc) coincides with the expectation value of H for the wave function 4(x; c). Since 2 is a representation of H restricted to the subspace spanned by the wave functions 4(x; c), it is obvious that 4(x; c) coincides with lc), which, by construction, possesses definite average values a and pa of the operators d and h,. These observations partially justify the usual quantization procedure for a semiclassical Hamiltonian X, which consists in the replacement of classical conjugate variables a and pa by operators il and 8,. However, such a simple-minded quantization of* would replace in eq. (20) the term AC+ C by 3 A(C+C+ CC+) and would lead to an incorrect correlation energy. The complex method described may also be useful for obtaining anharmonic collective Hamiltonians, which are still quadratic inp, but may contain an u-dependent mass parameter. We find indeed that (Oia210) = w/(8s(A+B)), <016210) =
362
J.
DA PROVIDENCIA
References I) D. M. Brink and A. Weiguny, Phys. L&t. 26B (1968) 494; Nucl. Phys. A12@:(1968) 59 2) H. Ui and L. C. Biedenham, Phys. Lett. 26B (1968) 608 3) Chun Wa Wong, Nucl. Phys. A147 (1970) 545 4) J. da Providencia, Portugaliae Physica 4 (1965) 171; 5 (1968) 71
REMARK ON THE RELATION BETWEEN THE GENERATOR-COORDINATE METHOD
AND THE RANDOM-PHASE
APPROXIMATION
1. da PROVIDENCIA Laboratdrio de F&a, ~nive~si~de de Coimbra, Portugal Received 9 June 1970 (Revised 17 July 1970) ABxdract:The equivalence between the generator-coordinate method and the random-phase approximation has been established by several authors. In the present note it is shown that a trial wave function 4(x; a), dependent on the time by the parameter a = u(t), oscillates in time with a frequency which is also given by the equations provided by both the generator-coordinate method and the RPA for the excitation energy.
1.Summary of previous results The relation between the generator-coordinate method of Grifk, Hill and Wheeler, in the Gaussian overlap approximation, and the random-phase approximation has been extensively investigated by several authors lw3). We will consider the related question of the connexion between those methods and the problem of the approximate time-evoIution of variational parameters a appearing in trial many-body wave functions 4(a) = 4(x; a). In the generator-coordinate method one considers linear combinations
Y(x) =
/4(x; M4da,
(1)
where the weight functionf(u) is determined by minimizing the expectation value of the energy for the state Y(x). In this way one arrives at the eigenvafue equation
J
f #(a,
a’)-El@,
u’))~(u’)du’
= 0,
(2)
where
(3)
K(% a’> = (4(4lHl4(0, I(& 4
= (4644(4).
In the Gaussian overlap approximation notation,
~
f
= &=-(a,a’) =
one may write, closely following Wong’s “)
(f ~)(~,) .
J%+s(u’u) 358
(4)
GENERATOR-COORDINATE
METHOD
359
The quantities Ea”)s, A and B are expressed in terms of the many-body wave functions +(xta). By expanding K(a, a’) and I@, a’), as given by the defining eqs. (3), we obtain &” = (MW),
(5a)
s = 3(%?MU $)9
(5b)
A = (2s)-‘((a,~I~la,~)-~*,(~,~l~,~>>,
B = (~s)-‘((~,~~IHI~)-E.,(~.Z~I~)),
(5c) (54
where 4 = 4(u) and the derivatives a,@~= &#J/&, etc., are taken at the point u = 0, for which
For simplicity, we have assumed that the wave functions 4, and therefore the parameters A and B, are real quantities. This assumption may obviously be removed without difficulty. It has been shown that the eigenvalue problem expressed by eq. (4) may be cast in the usual RPA form {E”-[E,,+AC+C+~B(C+Z+C2)]}ln)
= 0,
(6)
where C, C+ are boson operators for the Hilbert subspace spanned by the functions 4(x; a), [C, C’] = 1. (7) One can then solve eq. (6) by standard techniques. Let Q = xc-yc+,
@a)
with
(8b)
X2- y2 = 1. One may write eq. (6) in the form {E,-CE,,+~(~-A)+WQ+QI}I~)
= 0,
(9)
provided that
so that E,, = E,,++(w--A)+nw.
(12)
2. The time evolution of generator coordinates It is the purpose of the present note to show that, if we allow the parameter a, occurring in 4(x; a), to become time dependent, so that 4(x; u(t)) describes a nonstationary state, then the time evolution (small oscillations) of the function u(t) is
360
J. DA PROVIDENCIA
also determined by eq. (10). In order to obtain the equation governing the time evolution of the function a(t) we consider the time dependent variational principle, which, for the present purpose, may be written (13) On physical grounds one expects the equation determining the time evolution of the parameter a should be of second order. In order to obtain an equivalent system of two first-order equations it is necessary to extend the set of functions 4(x; u) by replacing the parameter a by a complex parameter “) c = a+ i b, where a and b are real. By expanding eq. (13) up to second order in c we may cast it in the form
-i(h*h)
(!i*) +td(c*c>6 1)(:) =0,
(14
where the quantities A and B are still given by eqs. (5~) and (5d). From eq. (14) we obtain -i(_$)
+ (t
z)(cc*(
=O.
(15)
By making the ansatz c = x e-‘“‘+y eiW we recover eq. (10). At this point one may enquire about the physical significance of the parameter b, the imaginary part of c, and about the meaning of the operators C+, C. In order to provide answers for these questions we note that eq. (15) is equivalent to ci = (A-B)b,
Wa)
b = -(A+B)u,
which may be written d=IL--
Ia2 4s ab ’
b=
(16b)
-Lax 4s
au
where.%@is the expectation value of H s!P =
qc*, c) = (4(4mb(4) (4(aNc))
= &“+s(c*c)(‘j ;)
(;) .
(174
The analogy between eq. (16b) and Hamilton equations shows that pa = 4sb is the variable canonically conjugate to a. In terms of a and pa the energy becomes X = E,,+
$4-B)p.:+2s(A+B)a2.
Wb)
GENERATOR-COORDINATE
METHOD
361
The wave function 4(x; a+ ib) appears, therefore, to represent a state possessing definite expectation values a and p. of some collective conjugate operators 21and 8,. Indeed, a wave packet lc), belonging to the Hilbert subspace spanned by the wave functions 4(x; c) and characterized by appropriate average values a and pa = 4 sb of the conjugate operators a = t(2s)-+(~+ +c), (18) hll = i(2s)*(C+ -C), is conveniently defined as the eigenvector lc) of C corresponding eigenvalue (2s)+ c, Clc) = (2s)*clc).
to a complex (19)
’ p recisely such a wave packet lc), which is It has been shown 1, “) that 4( x,. c ) is usually referred as a coherent state. It is clear that the expectation value of the Hamiltonian 2 = E,,+AC+C+3B(C+2+C2), (20)
appearing in eq. (6), is given by (clJ?lc) = &? = E,,+2s[Ac*c+3B(c*2+cZ)],
(21)
i.e.,%? is the same as given by eq. (17a). The expectation value of2 for the wave packet lc) coincides with the expectation value of H for the wave function 4(x; c). Since 2 is a representation of H restricted to the subspace spanned by the wave functions 4(x; c), it is obvious that 4(x; c) coincides with lc), which, by construction, possesses definite average values a and pa of the operators d and h,. These observations partially justify the usual quantization procedure for a semiclassical Hamiltonian X, which consists in the replacement of classical conjugate variables a and pa by operators il and 8,. However, such a simple-minded quantization of* would replace in eq. (20) the term AC+ C by 3 A(C+C+ CC+) and would lead to an incorrect correlation energy. The complex method described may also be useful for obtaining anharmonic collective Hamiltonians, which are still quadratic inp, but may contain an u-dependent mass parameter. We find indeed that (Oia210) = w/(8s(A+B)), <016210) =
362
J.
DA PROVIDENCIA
References I) D. M. Brink and A. Weiguny, Phys. L&t. 26B (1968) 494; Nucl. Phys. A12@:(1968) 59 2) H. Ui and L. C. Biedenham, Phys. Lett. 26B (1968) 608 3) Chun Wa Wong, Nucl. Phys. A147 (1970) 545 4) J. da Providencia, Portugaliae Physica 4 (1965) 171; 5 (1968) 71