Volume 119B, number 1,2, 3
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16 December 1982
NUCLEAR COLLECTIVE COORDINATES AS QUANTUM VARIABLES AND A SELF-CONSISTENT DESCRIPTION OF DAMPED VIBRATIONS "~ Helmut HOFMANN
Physics Department and Cyclotron Institute, Texas A&M University, College Station, TX 77843, USA and Physik.Department der Technischen Untversit#t M~nchen, 8046 Garehing, West Germany Received 15 June 1982 Revised manuscript received 9 September 1982
We use the method of Bohm and Pines to introduce nuclear collectivevariables on a quantum basis, and to derive a hamiltonlan which is suitable for describing dissipative motion self-consistently. As a first application, we study vibrations of the mean collective coordinate and its momentum and obtain a generalized RPA dispersion relation.
1. Introductory remarks. In nuclear physics it has been useful to introduce collective degrees of freedom Qu as extended variables [1 ]. A great advantage of such a procedure is that the definition of the nucleonic degrees of freedom is unchanged. This is very important for practical applications, but clearly unsatisfactory from a principle point of view. Usually, the Qu are treated as classical objects, which requires requantization if typical quantum effects are to be described. The quantization poses problems in itself. In principle, the generator coordinate method [2] can overcome these difficulties by (a) integrating over the collective parameters and (b) trying to derive a Schr6dinger-type equation for the weighting functions. However, the derivation of this equation is by no means easy and becomes possible only by posing strong conditions like the gaussian overlap assumption. Furthermore, this method is probably not adequate for deriving equations for collective motion in the presence of dissipative behavior. The method of functional integrals [3] is a generalization of the generator-coordinate method, with fields instead of coordinates. In principle, this method could also provide a satisfactory treatment of dissipative collective motion. In practice the techniques associated Supported in part by the U.S. National Science Foundation under the grant NSF PHY-8109019 and the Deutsche Forschungsgemeinschaft. 0 031-9163 [82/0000-0000/$02.75 © 1982 North-Holland
with this method are very complicated even for reversible mechanics, and dissipative motion has not yet been studied. To deal with such problems we propose to use a method which was invented originally by Bohm and Pines [4] to describe the electron gas. To introduce collective variables, the original phase space (xi, Pi, i = 1..... 3A} is extended by adding collective variables Qta and their conjugate momentaPu,/a = 1..... n. One easily can obtain a hamiltonian ~ of the total system having a comparatively simple structure. The price one has to pay is that a set o f n subsidiary conditions (for n collective variables) must be full'died simultaneously with the equation of motion defined by the ~7. But the Q~, P~ need not be restricted to classical objects. We shall now describe the principles of the method in the way we find suitable nuclear physics problems.
2. Introducing collective variables. We begin by supposing that we know the original many-body hamiltonian H(Sci, [oi) for the A-nucleon system. Hdepends on the 3A cartesian coordinates xi and their conjugate momenta which, like H are operators in an infinite-dimensional Hilbert space L. The coordinates and momenta obey the canonical commutation relations [xi, [~j] = i~Sij. For simplicity, we will not discuss spin coordinates, which can be included by an obvious generalization. We now introduce another Hilbert space L C span7
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ned by all well-behaved functions of a new variable Q. For the sake of simplicity we will restrict the discussion here to the case of just one variable. The extension to more degrees of freedom is straightforward. In LC, we have the corresponding operator Q and their conjugate momentum/S with the canonical commutation relation [Q,P] = ih. Their motion is governed by a hamiltonian operator/tC in LC, which we are free to choose at our convenience as long as it is hermitean. As the notation suggests, the Q will be used as collective variables. Our theory of collective motion will be formulated in the outerproduct space L tot = L ® LC, consisting of all well-behaved functions of both the x i and the Q. Among the operators inLto t are the extensions of the original 2i, Pi, Qu, Pu, H, and/tC, which we will denote by the same symbols. The commutation relations among the )?i, i°i and the Q,/6 remain valid for the extensions in Lto t and we also have additional commutation relations in Lto t:
[&, Q]
= [~j,P] = [ ~ , Q] = l~i,P] = o .
(1)
The physically meaningful observables, defined in L are identified with their extensions in Lto tThe extended space carries too much information as there are too many coordinates. To remedy this situation we have to require that the solutions of the equations of motion fulfill a subsidiary condition. The original many body dynamics takes place in the x ispace which may be expressed as a hypersurface of the extended space. By an appropriate choice of the origin of the O-variable this hypersurface can be defined by 0 = O.
(2)
We will therefore require that in the extended space the solutions become compatible with this equation. A physical motivation for this particular choice will be given below. Next we have to choose a hamiltonian ~ in the extended space. Our criteria for choosing ~ are twofold: (i) The dynamics of the original problem in L, governed by H, must be embedded in Lto t in such a way that the results for observables in L are easily and exactly recoverable. (ii) The form of ~ must be such that it facilitates obtaining approximate solutions which tell about the collective motion. To satisfy the second criterion of facilitating ap-
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proximate solutions we have to know something about H. Motivated by many successful studies of nuclear collective motion [1,5] we choose/t to be of the form:
# = # o - ~, KPP
(3)
where/5 is a one-body operator. It might be identified with multipole distortions of the single particle potential. In this case/-)0 would be chosen to be a spherical shell model hamiltonian. This would be a suitable form to describe small amplitude vibrations. In this paper we will restrict ourselves to this harmonic case but will not go into the details of specifying/5 and/~0. The interested reader may be referred to ref. [6] where possible choices of,# and/t0 will be discussed even for cases away from the minimum. Let us now come to the choice of the hamiltonian ~ in the extended space. As mentioned before, we will restrict ourselves to describe collective motion in harmonic order only. For this reason it will be sufficient to find the collective part of ~ to quadratic order in 0 and 16. Thus we are led to choose '
SO
(4)
where & and (~ may depend only on the intrinsic coordinates and momenta. Thus far we have not gained anything from introducing the collective coordinates. It is well known that the collective motion is related to the time dependence of (/5). We would thus like to see our collective variables 0 take on the dynamics of/5. To do this, we have to make a canonical transformation which mixes the collective and intrinsic coordinates. An appropriate choice for the transformation is
= exp[-(i/h)P/5] ,
(5)
which is unitary because P commutes with/5. This transformation is a translation operator with respect to the collective coordinates, and changes the subsidiary condition (4) into
O r = J'Q j ' + = 0 - / 5 = 0 .
(6)
The right-hand side of this equation establishes the relation between 0 and/5. It should be noted that any choice of the original condition other than (2) must be accompanied by a change in the definition of the transformation it. Under the transformation T, the hamiltonian (4) becomes
Volume 119B, number 1,2,3 r =
PHYSICS LETTERS
+
=¢B0f+
_ fi1 K FA. / ~ + ~ ( Q _ jO) + _~&(Q - p ) 2 (7)
assuming that F commutes both with C and &, which will be justified when we specify C and f below. Now, at last, we see how we can use our freedom in the choice of f and C to simplify the dynamics: If we choose f a c-number, and C a one-body operation linear in/~, then all the two-body terms in eq. (7) can be made to cancel. Specifically, the required choice is =
-
(8)
With this choice, eq. (7) simplifies to 1 ^2 ~'T = TI210 ~+ -- [30-p + ~aQ ,
(9)
where we have introduced the notation =
+
(10)
We now have to simplify the first term TH0 ~'+ in eq. (9). Expanding the exponentials of eq. (5) in powers of P, we obtain the the familiar commutator series ¢/101"+ =/10 - ( i / ] i ) [ / ~ , / t 0 ] / ~ - (1/2/~2)
[P, IP, £r011P 2 + ....
(11)
We shall neglect the terms of third and higher order in the collective momentum. They would definitely introduce anharmonic effects. But we may expect, indeed, these terms to be small. Suppose both the potential in /t0 as well as/~ are independent of the particle momenta/~i-In this case also [P,[P, = -(I/m) x ~k(~F/~Xk) 2 becomes independent of the Pi. This would imply the higher order terms to become both identically zero. In realistic models both/40 and/6 may contain spin orbit terms and eventually an effective mass, and thus the series might terminate at some higher order. However, it seems reasonable to expect the effect of such terms to be small. Thus we finally arrive at the approximate hamiltonian obtained by combining eqs. (9) and (11):
H0]]
~ A = f l o + I ~ P - - ~ O , p + I B p 2 + ~ a1Q ~2 ,
(12)
[,~,Pl
within the subspace specified by the subsidiary condition (2) (or (6)). It is important to note that ~A does indeed allow motion within this subspace, since [QT, ~ A ] = --(1/2h) [ ~ , P l P 2 ~_ O,
(15)
applying the approximations used in deriving eq. (12). In fact, when we attempt to solve ~ A we shall approximate/~ by a c-number, which will insure that [QT, ~TA] = 0 exactly. Thus our ultimate approximation scheme is completely consistent with the subsidiary condition, and we can be sure that the extra degrees of freedom do not lead to spurious results.
3. Solution of dynamical equations of motion. Eq. (12) is a central result. It serves as a basis for studying collective dynamics. Several ways are possible to achieve this goal. First, we could follow Bohm and Pines [4] and perform a second canonical transformation. The latter may be chosen in such a way as to remove the coupling terms in {~ and P. In this way the parts for intrinsic and collective motion get renormalized. Unfortunately, this canonical transformation can be defined in the case of no damping only. This will be discussed in ref. [6] where the relation of this procedure to the conventional RPA will be demonstrated as well. As we are mostly interested in obtaining dynamical equations which include dissipative and eventually fluctuating forces we must choose another way. The procedure which we will suggest in ref. [6] is to apply the technique of Nakajima and Zwanzig [7]. In this way we will end up with an equation for the reduced density operator/gcoll(t) for the collective motion. This can be obtained without ever leaving the regime of quantal physics. Deriving and studying this equation for t5coil is beyond the scope of the present letter. So in order to demonstrate the possibility of damped collective motion we will restrict ourselves to the equations for the mean values Q(t) = (O-)t, P(t) = (P)t of the coordinate and momentum respectively. These equations may be deduced directly from Ehrenfest's equations: d/Q(t)/dt = (ifl0 ([J£A, Q.] )t = (BP)t + (if)t,
where we have introduced the notation /$ =(i/h) [/40,/~] , /~ = (i//i)
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,
(13,14)
The solutions of our original hanailtonian/4 are found
dP(t)/dt = (i/hX[~A,/6 ] )t = -t~Q(t) + fl(l~) t .
(16)
In order to close this set we need to calculate all
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terms on the right as functionals of P ( t ) and Q(t). In general this is not possible. In our case, however, we are able to do so as we assume the motion to be harmonic. Therefore, we only need to evaluate the forces to linear order in Q and P. But to this order we may compute ( ~ ) t and (F)t by linear response theory and replace (BP) t by B P ( t ) with B being defined as the unperturbed expectation value of/~: B = (/~)0. Besides this latter point, the procedure is identical to the one used in ref. [8] for the case where the coupling was just given by a term like/3Q/7. Here there is another coupling term,/~F. Therefore, we will encounter not only the familiar response function ~((t) = ~(FF(t) = i0(t) ([F(t), F] ),
(17)
but the functions ~F,~(t), X~-F(t) and ~ / ~ ( t ) as well (their definitions are obvious). Fortunately, all these three functions can be expressed by linear combinations of the first and second derivatives of,~(t) with respect to time and B = (/~)0 = X f ~ ( w = 0). (For more details we refer to ref. [6] .) Finally, one ends up with the following equations: dQ(t)/dt = s e ( t ) + f ds [(a2/as 2) ~(s)] e ( t - s) 0
QL(z) = POXL(Z)/[ 1 -- KXL(Z)] ,
(20)
PL(Z) =P0[1 -/3XL(Z)]/z[1 -- KXL(Z)] .
(21)
where P0 is the initial value P ( t = O) = PO. The initial value for the coordinate Q(t = 0) has been put equal to zero in accordance with the subsidiary condition (6) and our assumption of O6)t=0 being zero. These solutions exhibit that the eigenmodes are to be found among the solutions of the dispersion relation 1 - K XL(Z) = 0.
(22)
This equation is a generalization of the usual RPA dispersion relation, which for temperature zero would read 2l12(fn-Eo) ~2_(En_Eo)
2
- x'(,~),
(23)
to the case of damped vibrations. Eq. (23) involves the reactive part X' of the response function. Its solutions are real frequencies. The Laplace transform XL(Z), on the other hand, gets contributions from the dissipative part as well, which implies that (22) leads to complex solutions. Notice that ×L(Z) is related to the Fourier transform ×(60) by
t
÷/3fo
To solve for the set of integro-differential equations it is simplest to perform a Laplace-transformation, say from Q(t) to QL(Z) and similarly for the other time dependent quantities. Then eq. (18) turns into two algebraic equations. After using eq. (10) their solutions read
K 1 ~ - = n
t
16 December 1982
dsI(a/as)~((s)] Q(t - s) , t
dP(t)/dt = - a Q ( t ) +/32 f ds ~(s) Q(t - s) 0 t
+/3
f
ds[(a/Os) ~(s)] P ( t - s) .
(18)
0
Here we have assumed that the system is prepared at time t = 0 and we want to follow it for t > 0. For simplicity we have put the static value (16)0 of the operator/~ equal to zero. A generalization to finite values is possible (see ref. [6] ). It can be shown that these equations are compatible with the subsidiary condition (6) if the latter is calculated by only using the same assumptions. This is to say we should take (6) and calculate its time-dependent mean. In doing so we should then express (F)t by its linear response expression. One ends up with (d/d0 [(/~t - Q(t)] = 0 . 10
(19)
)((Re w + i Im ¢~) = XL(Im 6o -- i Re co) . Solutions of the dispersion relation (22) for actual cases will be given in ref. [6]. Calculations of the response functions for the case of dissipative motion are reported in ref. [9] where damped vibrations are treated with the collective coordinate being introduced as a classical parameter. The reader will have noticed that the dispersion relation (22) is independent of the so far unspecified parameter/3. This implies that/3 has no influence on the frequency and the damping time of the vibrations. However, in order to get the full solution Q(t) and P ( t ) from eq. (20) we need to know/3. One possible way to fix it is to look at the equilibrium values Q(t ~ oo) and e ( t ~ oo). They are related to QL(Z) and PL(Z) by
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Hartree energy of our original model hamiltonian/-/ given in eq. (3).
the relation lim
16 December 1982
Q(t) = lim ZQL(Z),
t-~
z-~O
and the corresponding one for P. In this way we obtain from (20) Q ( t ~ o o ) = (P0/[1 - K X L ( Z = 0 ) ] ) lim zXL(Z), z--*0 and from (21)
e(t - ~
oo) = ( p 0 / [ 1
(24)
_ ~×L(Z = 0)] ) [1 -- ¢XL(Z = 0)] .
(25) The right-hand side of (24) vanishes because the response function X(t) goes to zero for t-~ 0% provided that the intrinsic motion is damped (see ref. [6] ). In order that the momentum P as well reaches the equilibrium value zero we obviously have to require
(3-1 = XL(Z = 0) = X'(W = 0 ) .
(26)
The latter equation follows from the relation of the Fourier transform to the Laplace transform mentioned before. This particular choice of/3 opens interesting aspects also in connection to the interpretation of the hamiltonian (12). Recall that we deltved our dynamical equations of motion by treatingthe coupling terms (terms linear in Q and P) in first t~tder time dependent perturbation theory. Suppose we forget for a moment the /;-dependent parts of ~A and suppose we treat the term -/~Q/~ by static perturbation theory under the constraint (/~) = Q. We will then end up with the total static energy:
E(Q) = (H 0 ) + { { [X'(6o = 0)1 - 1 _ ~) Q2 . This expression is nothing else but the constraint
(27)
The author would like to thank P.J. Siemens for innumerable discussions and suggestions made over many years, and for his strong and helpful criticism. Furthermore, he wishes to thank D. Pines for explaining to him details of the original method at an early stage of this work. Thanks are also due to A.S. Jensen. Finally he expresses his gratitude to the Physics Department and the Cyclotron Institute of Texas A&M University for their warm hospitality.
References [1] A. Bohr and B. Mottelson, Nuclear structure (Benjamin, New York, 1975). [2] J.J. Griffin and J.A. Wheeler, Phys. Rev. 108 (1957) 311; see also K. Goeke and P.G. Reinhard, Ann. Phys. (NY) 112 (1978) 328. [3] S. Levit, J.W. Negele and Z. Paltiel, Phys. Rev. 22 (1980) 1979; H. Reinhardt, Nucl. Phys. A346 (1980) 1, and references therein. [4] D. Bohm and D. Pines, Phys. Rev. 32 (1953) 609; see also D. Pines, in: The many body problem (Benjamin, New York, 1962). [5] K. Kumar and M. Baranger, Nucl. Phys. A122 (1968) 273. [6] H. Hofmann, A.S. Jensen, C. Ngo and P.J. Siemens, to be published; see also: Proc. Conf. on Large amplitude collective nuclear motion (Keszthely, Hungary, June 1979). [7] S. Nakajima, Prog. Theor. Phys. 20 (1958) 948; R. Zwanzig, J. Chem. Phys. 33 (1960) 1338. [8] H. Hofmann and P.J. Siemens, Nucl. Phys. A257 (1976) 165. [9] A.S. Jensen, J. Letters, K. Reese, H. Hofmann and P.J. Siemens, to be published.
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