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Fermionic molecular dynamics
Nuclear Physics North-Holland
A515 (1990) 147-172
FERMIONIC
MOLECULAR
DYNAMICS
H. FELDMEIER GSI, Gesellschaft fiir Schwerionenforschung mbH, Postfach 110552, D-6100 Darmstadt, Fed. Rep. Germany Received
8 January
1990
Abstract: A new type of molecular dynamics is proposed to solve approximately the many-body problem of interacting identical fermions with spin $. The interacting system is represented by an antisymmetrized many-body wave function consisting of single-particle states which are localized in phase space. The equations of motion for the parameters characterizing the many-body state (e.g. positions, momenta and spin of the particles) are derived from a quantum variational principle. The proposed fermionic molecular dynamics (FMD) model is illustrated with the help of two examples.
1. Introduction Early in heavy-ion research it has been proposed to describe the collision of two nuclei by molecular dynamics in which classical nucleons interact via two-body forces ‘*2). These ideas have been pursued since then 3-6). The major problem with these models which are appropriate for distinguishable classical particles is the fact that nucleons in nuclei and in nuclear collisions are close enough in phase space that the Pauli principle becomes important. The most common approach to improve on that is to mock the influence of the Pauli correlations by an additional momentumdependent potential which becomes strongly repulsive if the fermions get too close in phase space 7-9). Other approaches add fluctuating forces (collision term) which randomly change the momenta of pairs of nucleons ‘0~‘3). Here the Pauli exclusion principle is explicitly taken into account by not allowing scattering into occupied phase space. The motion in between able particles. One knows from thermodynamics
collisions
other
it may make
quantum
effects
are absent,
is, however,
again that of distinguish-
that even for macroscopic a big difference
objects,
where
all
in macroscopic
quantities like the specific heat whether one has Boltzmann, Bose-Einstein or Fermi-Dirac statistics. Not only for that reason, it seems quite important to include the correct statistics also in molecular dynamics, even if other quantum effects are neglected. This paper presents for the first time a non-relativistic model called fermionic molecular dynamics (FMD) which combines from the very beginning Fermi-Dirac statistics with a semi-quanta1 trajectory picture. In the limit where the system becomes dilute in phase space (low density and/or high temperature) it turns into classical 0375-9474/90/$03.50 (North-Holland)
@ Elsevier
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148
H. Feidmeier / Fermionic rn~leeuIar dynamics
molecular
dynamics.
not describe however
The FMD
all wave-mechanical
included an analogue
model
is not fully quantum
interference
on the many-body
In sect. 2 the model is defined bosom
model
mechanical
effects. The Fermi-Dirac
as it does statistics
is
level.
in general terms for identical
may be defined
fermions.
by using symmetrized
For identical
wave functions.
Sect. 3 demonstrates the ideas of the model in two examples. The first is a single fermion moving on a trajectory in an external electromagnetic field. There it is shown how the dynamics in the spin degree of freedom can be described in a classical language although it is in accordance with quantum mechanics in the two-component spinor space. The second example considers the scattering of two fermions with equal spin. Sect. 4 explores further the many-body system.
2. The fermionic molecular dynamics model In the fermionic principle
molecular
dynamics
(FMD)
model
the quantum
variational
14-16)
(2.1) is applied to an antisymmetrized many-body state. This state is parametrized by a set of time-dependent parameters Q(t) = {qY(t), v = 1,2,. . .} which are regarded as the generalized coordinates of the system. The variation has to be performed with respect to all qy( t) keeping the end points fixed, i.e. 8qv( t,) = 8qy( t2) = 0. @ denotes the total hamiltonian of the system and t the time. The variation leads to EulerLagrange equations which are the equations of motion for the parameters {q”(t)}. The general idea is to find a parametrization which contains the essential degrees-offreedom of the system. Usually one takes advantage ameters which have a semiclassical meaning. Classical molecular dynamics describes the motion
of intuition of point-like
and selects parparticles
under
the influence of mutual interactions. Fermionic molecular dynamics extends this picture in a natural way towards quantum mechanics by introducing for each molecule (here each nucleon) a wave packet Is(t)> which is localized in phase space. For convenience one may take a gaussian form
(xi_bi(l))Au(t)(x,i_bj(t))+77(f) Ix(f),cb(r))+
(2.2)
The time-dependent parameters q(t) = (b(t), A,(t), q(t), x(t), #(t)} define the wave packet uniquely. The parameters b(t), Al,(f) and n(t) are complex. li( t) contains the information on the mean position and the mean momentum. The symmetric matrix Aij( t) describes the spreading and its real part has to be positive. n(t) takes care of the normalization and contains a time-dependent phase. This phase, which is in our application of no consequence for the dynamics, is dropped in the following.
Ii.
The spin of the fermion
Feldmeier / Fermionic
is represented
149
rnofecMi~r dynamics
by a two-component
spinor
Ix(t), $(t))
(2.3) where the real polar angles
x(t)
and #(t)
denote
the direction
of the spin.
For identical fermions the ansatz for the parameterized many-body IQ(t)) is an antisymmetrized state formed out of the single-particle given in eq. (2.2). The most simple N-body state is a normalized determinant built from N different single-particle wave packets
wave function states Iqk( t)) single Slater
(2.4) where
As the single-particle states are in general not orthogonal, the many-body state in eq. (2.5) has to be normalized. Nevertheless it is completely antisymmetric with respect to particle exchange and thus a proper wave function for identical fermions. Define a “Lagrange function” by 3(Q(r),
$(t)):=(QW$
in which e(t) = {& = dq,/dt, equations of motion resulting Euler-Lagrange equations
-HIQW)=%(QW,
Q(t))-WQO))
3
(2.6)
v = 1,2, . . . } is the set of generalized velocities. The from the variational principle (2.1) are given by the
d a3 --=dt 84”
a3 aqv
with ~={k,j},
(2.7)
where k = 1,2,. . . , N enumerates the particles and j denotes the different parameters for each single-panicle state. These equations may be non-linear and they need not resemble at all a Schrodinger equation. They can be regarded as semi-quanta1 equations of motion which for special cases may turn into classical equations. If IQ(t)) spans the whole Hilbert space then the set of equations of motion (2.7) is just the Schrodinger equation written in terms of {q,(t)}. If for certain initial states the exact solution of the Schrodinger equation can be expressed at all times by the parametrized state IQ(t)> then the resulting equations of motion for {q”(t)} are also exact. An example is the interaction-free evolution of a gaussian wave if the exact packet. The equations of motion for {qz,( t)} will prove meaningful solution can be approximated well by (Q(t)). -Y( Q( t), o(f)) is not the usual Lagrange function known from classical mechanics. It depends on both spatial coordinates and momenta, and contains the generalized
150
H. Feldmeier 1 Fermionic
velocities
Q1,(t) = dq,(t)/dt
the time derivative
only linearly.
moiec~Iar dynamics
Because of this homogeneity
one can write
part as
(2.8) The hamilton
function
is defined
as the expectation
value of the hamiltonian
Since X and ~~*/~~~ do not depend on the generalized of motion (2.7) can be written in the following general
@
velocities gy, the equations form r5,16) (2.10)
where &,, velocities
is a skew-symmetric
matrix
which
depends
on {q_(f)}
but not on the
(2.11)
The classical Hamilton equations of motion can be written in form of eqs. (2.10) by setting q = {r, p}, where r and p denote conjugate variables, and i2p,, = &2=Q,Vti,2=-&,=-1, (2.12)
Before discussing present
the model
further
in general
terms let us be more specific and
examples.
3, Applications In the following subsections we shall illustrate the ideas of fermionic molecular dynamics with the help of two examples. First we consider a single fermion in an external electromagnetic field. We treat explicitly the coupling of the spin variables with the field and study the influence of the spin on the trajectory. In the second example, which is about two identical fermions, one can see how the Pauli exclusion principle enters the equations of motion for the parameters corresponding to the relative distance and relative momentum. To simplify the equations let us restrict the freedom in the possible shapes of the coordinate space part in eq. (2.2) by assuming A,(f) = Sjjfa(t) so that each singleparticle wave packet is spherical at all times.
H. Feldmeier / Fermionic molecular dynamics 3.1. A CHARGED
FERMION
IN AN ELECTROMAGNETIC
151
FIELD
To get a feeling for how the proposed model for fermionic motion works, let us consider the time evolution of a single charged fermion in an external time-independent electromagnetic
field.
For the sake of simplicity
we assume
in this section
a real and time-independent
width parameter a, so that the wave packet is not allowed evolves in time. We denote the single-particle state by Is(r)) = Ir(t), p(r), x(t), where the coordinate
to spread
4(t)> = Ir(r), p(~))Olx(r),
4(r)),
or shrink
as it
(3.1)
space part Ir( t), p(t))
(xlr(t),p(f))=--&iexp
-(x~~‘))2+ip(t)*x I
(Tao)-
(3.2)
0
I
is expressed with the mean position r(t) and the mean momentum with the complex parameter b(t). The non-relativistic hamiltonian is
FJ=&(-&A(x)c
- )’
p( 6) rather than
+;A”(&-f/qB(_x,k),
(3.3)
where e is the charge, m the mass, and pB = he/(2mc) the magnetic moment of the fermion. ,k and ,x denote the momentum and coordinate space operator, respectively. The charge of the particle interacts with the external vector potential (A’, A) while the interaction of the spin with the electromagnetic field is obtained from a FoldyWouthuysen transformation “) as &(ZB(X The term with the electric
,k) = L +&O’A”(z)
x k + gsV x A(x)).
field VA0 is responsible
The other term is the energy
for the spin-orbit
of the spin in the magnetic
(3.4) interaction.
field V x A (g, = 1, gs-
2
for electrons). q are the Pauli spin matrices which act on the spin part Ix, 4) of the wave function (3.1). In order to apply the variational principle (2.1) one has to express the time derivative part ir,P,x,~li~lr,p,X,~)=-pr-~(sinfx)* and the Hamilton
(3.5)
function x(r,
p,
x, 4) := hi4 x, 4lHlr, p, x, 4)
(3.6)
as a function of the variables. In the spirit of a semi-classical approximation let us assume for the moment that the width a, of the wave packet is small compared to typical variations in A”(x). This implies that we use only the first term of the Taylor expansion (A”(x))=A”(r)+$AA”(r)T+.
...
(3.7)
H. Feldmeier / Fermionic molecular dynamics
152
With that the Hamilton
function
becomes
a(r,p,x.~)=~(p-~A(r))‘+~A’(r)+~ 0
-hdsinx (&(cp) cos $+B2(r,p) sin+)+cosxNr,~)l, where (B, , B2, I&) = B denote the three components of the magnetic by the spin of the moving fermion. From the Lagrange function
field as seen
~=-_li.r-~(sin~x)‘-~(r,p,~,~), one gets the following
Euler-Lagrange
(3.9)
equations
(+A EC?= ++fE dt ai
ap
(3.8)
(3.10)
ap
(3.11)
($=-
O=$-:->z=
-&sinX+FF
or
(3.12)
k=,
.’
The first two equations (3.10) and (3.11) are the usual Hamilton for position and momentum. One sees, however, that inclusion x and C#Jgives an addition to the electrostatic interaction of the spin with the electromagnetic
(3.13) equations of motion of the spin variables
and the Lorentz force, namely, the field. It is also interesting to note
that the equations (3.12) and (3.13) for the spin degrees of freedom Hamilton equations if one chooses as conjugate variables 4
and
s:=ihcosx.
With the projection
of the spin on the 3-axis,
equations
(3.12) and (3.13) transform
of motion
can be cast into
(3.14)
s, and the azimuthal
angle,
d, the
to (3.15)
and (3.16) Unlike a classical angular momentum, s is not a vector and its range is limited between -4h and +$h. Furthermore, there is no kinetic energy of the classical type s2/(2M). By replacing sin x and cos x with s in the Hamilton function (3.8) the h
H. Feldmeier / Fermionic molecular dynamics
in pB drops
out and there is thus no h anymore
like a classical system, spin with the varying Ehrenfest’s
theorem
the spin degree
153
so that also in this respect
it looks
but it is not, Thomas precession and the interaction of the electric and magnetic fields is included. Application of
would
not lead to these equations.
of freedom
is regarded
In so-called
A expansions
to be first order in h and usually
dropped.
The equations of motion (3.10)-(3.13) describe in a natural way the coupling between the spin and the classical trajectory of a fermion. In the following section it is shown how the trajectories of two fermions are influenced by the Pauli principle. 3.2. SCATTERING
OF
TWO
IDENTICAL
FERMIONS
As another example let us apply the ideas of FMD to the scattering of two identical fermions. For the two-body state we make an ansatz in which the center-of-mass part \q,.,.) separates from the part 14) for the relative motion and spins:
14c.m., s>=l4c.m.)olq)~ The center-of-mass
wave function
is parameterized
(3.17)
’
by qc.m.= {B, A}
as
(x-B(t))2 (Xlqc.m.t~))=d& exp I
(3.18)
,
2A(t)
c.m.
where X is the center-of-mass
{
coordinate
and N,.,.(t)
the norm (3.19)
Here and throughout this section the real and imaginary are denoted by the indices R and I, respectively, like B=&+tiB,
or
part of complex
A=A,+iA,.
The wave packet (q) for the relative motion contains {b, a, x, , 4, , x2, 4~~).The parametrized form in coordinate IXlW,
numbers
(3.20) also the spins, thus q = space takes the form
4d~nolx,(~),
42(t))
(3.21) with the time-dependent
normalization
u’c;R;;‘cr,)3’2[eXlJ {$} -exp { -z}$2], where b(t) = bR( t) + ib,( t) and a(t) = uR( t) + ia,( t). The spin overlap
(3.22) is abbreviated
by S12 s,2=
l(XIW,
W)lXz(f),
~&))I2
= f( 1 + sin x, sin x2 cos (4, - 42) + cos x, cos x2) .
(3.23)
H. Feldmeier / Fermionic molecular dynamics
154
The two-body wave packet (3.17) is in general not a single Slater determinant as given in eq. (2.5) because the center-of-mass motion for a Slater determinant does not separate
if the single-particle
packets
have different
section
we let the width A(t) for the center-of-mass
packet
be independent
parameters
mass degree of freedom. The relation between b(t) and the more intuitive “relative momentum” p(t) is Mt)
= r(r) -al(t)
motion
in order to obtain
and
width parameters.
In this
and a(t) for the relative
free motion quantities
in the center-of-
“distance”
b,(r) = a,(r)p(r)
,
r(t) and
(3.24)
respectively. It has to be kept in mind that r(t) and p(t) have their classical meaning only if the two particles are far away in phase space. This is demonstrated in fig. 1, where the real part of the wave function (3.21) is displayed for several choices of the parameters. The spins are in all cases assumed to be parallel. In the first case p = 0, a, = 0 and r= Irl is taken to be large compared to 6. This is a situation where the Pauli principle is not active and r has the classical meaning of the mean distance between the particles. However, that is not completely correct as the particles are indistinguishable and the physical situation is the same for r and -r, thus, the sign of r cannot be measured. The second example is for jr1 <&. From the figure
Fig. 1. Real part of the wave function for the relative coordinate as a function axis defined by r. For details see text.
of x which is along the
H. Feldmeier 1 Fermionic molecular dynamics
it is evident
that now the mean
distance
classical interpretation completely. which specifies the state. Analogue wave function
is for r = 0, p = 216
the same distance
[ in phase
is of the order
155
of X& and r has lost its
Nevertheless, r may still serve as a parameter statements hold for the parameter p. The third and a, = 2.5aR. In this state the particles have
space as in the first example
although
they are very
close in coordinate space (the measure [ is defined in eq. (3.38)). Also here the Pauli principle is not active since a wave function without the exchange part would give the same expectation values. In summary, r and p have their classical meaning only when the particles are far away in phase space. In this case the Pauli principle is also not of importance. One should always carefully distinguish between parameters and physical observables. All measurable quantities have to be calculated from the parametrized state as the expectation value of the corresponding operator. For example, the measured mean distance (q[,x* Iq)“* between two identical fermions is not zero even if r = 0. The total hamiltonian H,,, consists of the center-of-mass kinetic energy and a hamiltonian @ for the relative motion: (3.25)
H In the model
there
=&!c’+Vk, 4, a(l),
is no restriction
in the choice
o(2)).
(3.26)
of the two-body
interaction
V(x &, a(l), o(2)). It may depend on the relative distance z, relative momentum ,k and on the spins via the Pauli spin matrices ~(1) = ~01 and cr(2) = 1 @a which act in the spin space of particle 1 and 2, respectively. 3.2.1. Center-of-mass motion. As a little exercise and as an example for an exact solution of the variational principle, are given in this subsection:
all expressions
for the center-of-mass
motion
(3.27)
,
(3.28)
(3.29) In the center-of-mass wave function R(t) and P(t) have always the classical meaning center-of-mass position and total momentum, respectively. They are related to B(t) by I&(t) = R(t) -A,(t)P( t) and B,(t) = AR( t)P( t). The width parameter A(t) is taken as a dynamical variable and the wave packet is not assumed to be narrow. Therefore the kinetic energy contains besides P’/(2M) the zero-point energy
156
H. Feldmeier / Fermionic molecular dynamics
3/(4MA,). The last term in eq. (3.28) is a total time derivative the Euler-Lagrange equations given below:
and drops
out in
(3.30)
d
()=---__=
a=%.,. a=%,.
d a=%,. ()=----=_ dt a&
a=%.,. aAR
BR
___
(3.31)
(3.32)
A’,
(3.33) It is easy to verify that the solution the expected
of these eight coupled
equations
of motion
is
exact result &=O,
B,=o,
&=O,
A, = l/M,
(3.34)
or P=o,
k=P/M,
&=O,
A, = l/M,
which represents the free motion of a gaussian wave packet. 3.2.2. Relative motion. The Lagrange function for the relative
(3.35)
motion (3.36)
-Y=(41i~14)-(41~k214)(qjvJq)=~~-~-.I’. is more involved
as it contains
exchange
“Ir are given in the appendix. help of the kinetic
energy
contributions
In the following
are discussed
with
9:
1 p2+((r-a,p)2/a’,) =----1 - e-5S,2 2p where 6 is an abbreviation
and spin. The parts X0 and
the novel aspects
e6S,,+
3
(3.37)
4paR ’
for bi+bf [=aR=
(r-aIp)2 aR
+tp2a,
(3.38)
and S,> is the spin overlap given in eq. (3.23). Comparing eq. (3.37) for the relative kinetic energy with eq. (3.29) for the center-of-mass kinetic energy, one sees the influence of the antisymmetrization. If the two fermions have opposite spin, which means S,2 = 0, there is no Pauli principle and the relative kinetic energy is the same as for a single gaussian packet which is not antisymmetrized. For not orthogonal
157
H. Feldmeier / Fermionic molecular dynamics
spins, i.e. S,2 # 0, one recognizes 5 as a measure for the distance in phase space. If 6% 1 the fermions are far in phase space so that the Pauli principle is not acting anymore.
It is interesting
(3.38) while a, together to the measure
to note that G
plays the role of a length
with p modify the naive picture that r2/a,
4 which comes from the spatial
scale in eq.
is the contribution
distance.
For the case of free motion (2r = 0) one can show explicitly that the Euler-Lagrange equations resulting from the Lagrange function eq. (3.36) give the exact result &=O, i,=o,
b,=o,
&=O,
A=%
b, = 1IcL > (3.39)
A=%
%2=0,
or in terms of r and p i=pIpL,
@=O.
(3.40)
Here, like for the center-of-mass motion, the parametrized wave packets are the exact solution of the Schrodinger equation. Without an interaction two identical fermions do not know about each other and move like distinguishable particles. The wave function is nevertheless always antisymmetric so that a measurement of their distance is of course effected by the Pauli principle. It is also interesting to note that for free motion the phase space distance 5 defined above is a constant in time because the solution of the Euler-Lagrange equations (3.39) is r(f) = r,+pt/p and a,(t) = t/p while p and aR do not change in time. Consider as an initial condition two fermions well separated in space such that r2 9 aR which means according to eq. (3.38) 5s 1. Now give them a small momentum p such that they are heading towards each other. After some time the two wave packets will reach each other and completely overlap in space. Nevertheless, they are still very far from each other in phase space since the time-independent combination (r(t) - a,( t)p)’ enters the measure 5. The interaction
energy
V’”= (91 _V\q) is given
in an analytic
potentials which have a gaussian or quadratic dependence The expectation value of a(l)cr(2) which appears for spin also calculated. Since the exact expression is lengthy the discussed here in the limit of narrow wave packets. Suppose
form
for central
on x in the appendix. exchange potentials is interaction energy is the functional form of
I/’ is given by V(g), then
V(q) = (41V(,x)ld~
V(r)-
V(i(ru,/aR-p(a2,+a:)/a,)) 1 -epcS,,
e-‘S,,
(3.41)
Like in the expression for the kinetic energy the classical result is obtained for S,2=0 or e -5=0 . When the particles are close in phase space the interaction is modified by the exchange term which is momentum dependent. The imaginary i =&i does not concern us since V(x) contains only even powers of x so that V(q) remains real. One should keep in mind that in ground-state nuclei the wave packets are not narrow I’). Their spatial width cannot be smaller than about the inverse of the Fermi
H. Feldmeier / Fermionic molecular dynamics
158
momentum because
which is l/kr
each particle
Let us summarize the relative packet.
= 0.7 fm, otherwise
contributes at this point
coordinate
energy becomes
to the kinetic
the differences
of two identical
First, in the classical
the kinetic
at least 3/(4ma,)
between
fermions
the fermionic
and the motion
case the matrix elements
too large
energy. motion
in
of a single wave
of the skew-symmetric
matrix
dik are just (0, 1; -1, 0} and the parameters qj can be grouped in pairs of canonical variables. In the fermionic motion dik( q) becomes q-dependent and variables which were canonical pairs in the classical limit are not canonical any longer. The reason is that the parameters (r, p, oa, a,) loose their physical meaning if the fermions are close in phase space. For example, for r-0 the parameter r is not at the position of the peak in the density anymore; actually the wave packet varies very little as a function of r when the particles are spatially near as displayed in the center part of fig. 1. s&(q) takes care of the varying influence of the parameters q on the wave function in a similar fashion as a mass tensor does for generalized coordinates. Thus dik (q) plays the role of a metric tensor. Second, the generalized forces M’/dqi cannot be replaced by classical forces when the fermions are close in phase space. The exchange term introduces an r-dependence in the kinetic energy Y and a momentum dependence in the interaction even if the classical V(r) depends only on the spatial distance. 3.2.3. Relative motion with a time-independent width parameter. This section investigates briefly how the equations of motion change if the shape of the relative wave packet (eq. (3.21)) is restricted further by removing the width degree of freedom a(t) = aR(t)+ ia, as a dynamical variable 19). For simplicity only parallel spins are considered, i.e. x, =x2 =x and 4, = & = 4. By setting ua( t) = ao, a,(t) = 0 and (iR( t) = 0, ci,( t) = 0 in the previous section we obtain -_li.r+i.pe-c
To=
-2$(sin
1 -e-c 2
y^=lp 2P
2
2
3
-E
:‘-‘,“_of”
is)‘,
+-
(3.43)
%a0
’
where t=
r2/ao+p2ao,
L!!=_Yo-F-Zr.
The potential energy Zr is not given here explicitly not act on the spins. The Euler-Lagrange equations
(3.44)
but it is assumed that y does for the spin variables x and 4
are
()=dL~_G=_~sinx dt &#J
a+
or
$=o,
(3.45)
or
i=o.
(3.46)
H. Feldmeier / Fermionic molecular dynamics
As anticipated motion
it turns
out that the spins will not change
159
in time. The equations
of
for r and p are O=dG_G dt a@
_+2e” 1 _,-s
(
r;-
r(r- i)/a,+p(p1 _e-5
ap
Or
+)a0 >
2 e-“a0 +(I
-e-“)’
(r(p*ti)-p(r.$))
(3.47) and
O=-$F-f$ p+
26 l-e-5
. p(
r(r-Ei)lh+zdp-$h 1 -e-E
or 2 ePg
) +(l-e-‘)2ao
(r(pe
4 -p(r*
4)
In this example the columns of the skew-symmetric matrix dik are contained in the second line of eqs. (3.47) and (3.48), respectively. It has a complicated form as function of r and p but reduces to (0,1; -1,O) if the phase space distance .$ becomes large compared to one in which case one gets back the classical equations of motion. For the reduced set of variables we were able to solve eqs. (3.47) and (3.48) for i and fi. The result is (3.49)
(3.50) where
(Y,(5) and CY~( 5) are functions
of 5 = r2/ a,+p*a,
and given by (3.51)
2e-‘( 1 -e-*) a2(5)
Since (~~(5% 1) = 5% 1 means that implies that they or close by with identical fermions
=
(l+e-*)2(1-e-s)-2~e-~(l+e-5)’
(3.52)
1 and (~~(5% 1) = 0 one recognizes for 5~ 1 Hamilton equations. the two fermions are far from each other in phase space, which could be far in coordinate space with small relative momentum large relative momentum (in any case 5% 1). In that limit the behave like classical distinguishable particles although their wave
160
H. Feldmeier / Fermionic mo~ecuiar dynamics
function is of course still antisymmetrized. (,$< 1) cy,([)+$t and 1y~(5)+3/5~ which (3.49) and (3.50) vanish In this example
like $6 but the remaining
one sees that,
from the parameterization are regarded as canonical Hamilton
function
When they get close in phase space means that the Hamilton-like parts in
for CsZ,
parts increase
the equations
like 3/f2.
of motion
which
result
(3.21) cannot be cast into Hamilton form when r and P variables. To prove this statement let us suppose that a
Xpauri exists such that (3.53)
where i denotes the three spatial directions. Let us now disprove the existence of ~~,“,i by calculating the mixed derivatives of the equations of motion (3.49) and (3.50). It is easy to verify that (3.54) If eq. (3.53) were true, the mixed derivatives
should
be the negative
of each other,
which is not the case here. This disproves the existence of a hamiltonian aipPauli(r, p) which would describe the fermionic dynamics derived from the ansatz (3.21) for the wave function. It is, however, possible to find a pair of canonical variables (p, m), which are then nonlinear functions of the original variables (r, pf [refs. ‘““)] with the elements so that the skew-symmetric matrix dfi,, becomes tri-diagonal -1, 0,l. With these new variables the equations of motion assume the form (3.53) where %&,,i(p, n) = X(r, p) is the total energy expressed in the new canonical variables. Eq. (3.53) represents
an approach
often used in literature
to incorporate
the effects
of the Pauli principle 7-9). D’ff I erent forms of momentum-dependent potentials have been added to the classical hamiltonian. Here we see that this method differs in several aspects from fermionic dynamics. Firstly, as discussed above the dynamical behaviour of P and p need not be of Hamilton replace (V(x)) simply by V(r) because even for exchange term which is not small for t< 1. Since (FMD) a quantum variational principle based on
type. Secondly, one should not narrow wave packets there is an in fermionic molecular dynamics wave functions is involved these
quantum effects are included. The equations of motion simplify appreciably for a time-independent width parameter a,, but the price to be paid is that now the free motion without interaction is not exact anymore. The equations of free motion are (3.55)
161
H. Feldmeier / Fermionic molecular dynamics
For 5 < 1 they differ essentially
from the expected
+ = p/p
result
and $ = 0. The
shape of the wave packet is restrained too much. The particles scatter even if there is no interaction. This deficiency is also present in the so called Pauli potentials 7-9). Inclusion
of the complex
width
has been demonstrated
a(t)
as a dynamical
in the previous
variable
cures this problem
as
section.
3.2.4. Conservation laws. Considering the symmetries of the Lagrange function Zttot= &,,,+Z [eqs. (3.27) and (3.36)] one constructs with Noether’s theorem the following conserved quantities: - Energy in the center-of-mass and relative motion (Zc,.,. and 2 do not explicitly depend on time) -=-
(3.57)
and (3.58) - Total momentum
(Yte,,, is translationally
invariant) (3.59)
$P=-$K)=O. - Center-of-mass
angular
- Total angular
momentum
momentum
(Xc.,. is rotational
in the relative
motion
(2
l-e-< (rxp)+
invariant)
is rotational
invariant)
n,+n2
l-e-‘&,
2
> (3.61)
=&x_k+$T(l)+@J(2)))=0, where nk
=
(xk,
+k
idxk,
is the spin direction of particle orbital angular momentum
4k)
=
(sin
xk
cm
4k,
sin
xk
k. If the interaction
sin
@k,
cos
xk)
does not depend
(3.62) on spin the
(3.63) and the spin ($((r(l)+cr(2)))=
v
I ‘,:t 12
are conserved
separately.
(3.64)
H. Feldmeier / Fermionic molecular dynamics
162
All conservation for the relative to the additional
laws can be proven
angular factors.
momentum
by using the equations
Please note, that in the general
3.2.5. Numerical results. As a first application motion
of two fermions
of motion.
and the spin are somewhat
which are bound
together
The proofs
more involved
due
case r xp is not conserved.
of FMD
we study
by an attractive
the relative
gaussian
potential
(see eq. (A.2)) with a strength of V,= -0.5 hc/fm and a range of r,,= 4 fm. The reduced mass is taken to be p =2.5 h/(c. fm). In the first and second line of fig. 2 the time evolution of the density probability to find the particles with a certain relative distance in the x-y plane is shown for parallel and anti-parallel spins. The fermions are initially at rest with a mean distance of 2 fm and the initial width parameter is taken to be a(t = 0) = 1 fm2, so that they have only little overlap in phase space (5 = 4 or e-‘S,, = 0.018). For both cases the density plot shows a coupled oscillation of the distance and the width of the wave packet. The two markings “+” indicate the actual position of the parameter r and -r. Although during the time evolution the distance r(t) stays around the initial large value of about 4 the motion of the identical fermions differs substantially from the motion of two fermions with opposite spin. The reason is that for the latter case the spin overlap S12 = 0 so that all exchange contributions vanish. The third line in fig. 2 starts with the same initial wave packet but this time the mass of the particles as well as the strength of the potential is multiplied with a factor of 100. The trajectories for r(t) which result from the classical equations of motion are invariant under this transformation; however, the spreading of the wave packet is reduced substantially and one is closer to the classical limit. The detailed time evolution of the measurable distance d and the parameters which define the relative wave function is displayed in fig. 3. The distance d between the fermions is defined as the root of the quadrupole moment along the r-direction. (3.65) The direction r/lrl is given by the direction of the major axis corresponding to the largest eigenvalue of the second moment tensor. For particles with a distance larger than the width of the packet one gets d = (rl ( c 1assical limit). In the upper part of fig. 3 one sees that as a result of the Pauli principle the fermions with equal spins (solid line) do not get close in space, while the particles with opposite spin (dashed line) oscillate through d = 0. At variance with d the parameter r(t) goes through zero even for equal spins. The dotted line is for the case where the mass and the potential strength are hundred times larger. Here, the time dependence of r(t) is almost identical to the purely classical trajectory which is not shown here. The lower part of fig. 3 displays the oscillations in the real and imaginary part of the width parameter a(t). In the nonrelativistic energy regime the scattering of nucleons cannot be described by classical equations of motion. The reason is the short range of the nucleon-nucleon
H. Feldmeier
/ Fermionic
molecular
dynamics
N
Q
-$
+ mj c.4
c-4
0
0
-t
X
+
II
c-4
0 -cli $
II
c-4
F----b
OJ/A
163
164
H. Feldmeier / Fermionic molecular dynamics
I
I
I
I
I
I
I
I
/
0
5
10
15
20
25
30
35
LO
time cth, Fig. 3. Measurable distance and parameters of the wave packet as a function of time for two fermions oscillating in an attractive potential (same case as in fig. 2). Full lines are for parallel spins, dashed lines for antiparallel spins (no Pauli principle), and dotted lines are for heavy particles with antiparallel spins. See text for details and choice of the parameters.
interaction, which is only a few times longer than the Compton wavelength of the nucleon. Therefore, a wave packet well localized with respect to the range of the interaction spreads too fast during the interaction time. The following estimate illustrates this. Let us take a rather large width initially and assume uR = 1 fm2 and a, = 0 just before the collision and let us allow a, to increase during the collision time by 1 fm2. The collision time and the increase in a, we estimate from the eq. (3.39) for free motion. The time a nucleon needs to travel for about the range of the interaction, R = 5 fm, is t = Rp/p. During this time the imaginary part of the
H. Feidmeier
/ Fermionic
molecular
dynamics
165
width increases by a, = t/g, = R/p. If we allow for a, c 1 fm’ this means that p&5 fm-’ or p b 1 CeV/c. This small exercise shows that a nonquantal description in terms of trajectories can be used only in the relativistic regime. Thus, a nonrelativistic nonquantal trajectory picture cannot be applied for the scattering of nucleons 18). In fig. 4 the density for the scattering from a strongly repuisive potential is shown for particles with equal spin and opposite spin. We include the free motion, which
,Q -10 X/Tg
-5
Q
x/r0
5
10 -IO
x/r*
-5
0
x/r0
5
x/r0
x/r0
Fig. 4. Time evolution of the two-body density probability as a function of the relative distance between two fermions which are scattering from a repulsive potential. The contour plot is for z = 0 and the center-of-mass coordinates have been integrated over. First line: free motion of fermions with parallel spin. Second line: scattering for parallel spins. Third line: scattering for antiparalle~ spins (no Pauli principle). Fourth line: scattering of heavy particles with antiparallel spins. Fifth line: free motion of heavy particles with antiparallel spins. The positions of the parameter I and --r are marked by f. Contour tines are at (0.5, 0.2, 0.05, 0.01) of the maximum density. See text for details and choice of the parameters.
166
H. Feldmeier / Fermionic molecular dynamics
is the exact solution of the wave packet some resemblance
of the Schrodinger equation, in absence of an interaction. with a trajectory
to illustrate the amount of spreading In order to get a picture which has
we are forced to choose almost relativistic
initial
momenta. For the upper three lines of fig. 4 we took pft = 0) = (-4,0,0) h/fm, r(t=0)=(3,0.5,0) fm, a(t =0)=06fm’ and as reduced mass p =2.5 h/(c* fm). The potential
has V, = 5
A/(c . fm) and r, = 1 fm. The first line for the free scattering
shows the fast spreading which has been discussed above. The second line is also for parallel spins but the repulsive potential is present. Besides the nonzero scattering angle one sees that the potential is also blowing up the wave packet so that a classical trajectory approximation is not possible. For the third and fourth line of fig. 4 the reduced mass, the initial momentum and the strength of the potential has been increased by a factor of 100, so that the classical trajectory remains the same, but the free spreading is reduced by a factor of 100. The last line, which is for free motion, shows almost no spreading anymore, but the repulsive potential also blows up the wave function of the heavy particles, as can be seen from the third line. Nevertheless this case resembles more a classical trajectory picture. The classical scattering angle is larger because, due to the finite width of the packet, the expectation value (q/J’(q) is weaker than the corresponding V(r). There is yet another condition which cannot be fulfilled in the scattering case, namely, in order to keep approximately the gaussian shape of the packet it has to be narrow compared to variations in the potential ‘*). More precisely d’V/dr’
“aag+ai:
I d’V/dr”
1
3
(3.66)
aR
where the right-hand side measures the spatial width of the wave. In our case the ratio of the derivatives is of the order of the range r. which is 1 fm and therefore inequality (3.64) is not really fulfilled for the cases shown in fig. 4. The situation should improve in the many-body case where each particle feels a smoother mean field produced
by its many
neighbours. 4. The many-body system
In nuclei one has two kinds of fermions, protons and neutrons, many-body wave function splits into two antisymmetric parts IQ> = IQprotonJO IQ”e”tTO”S) *
therefore
the
(4.1)
The hamiltonian @= T+ y=;
k-
T(k)+
; k
V(k, I) -
(4.2)
consists of the kinetic energies _T(k) and the two-body interaction _V(k, I) between the particles. There is no restriction on the form of the interaction, it may for example be spin and isospin dependent. All degrees of freedom for nonrelativistic nucleons are present in the many-body wave function (4.1).
167
H. Feldmeier / Fermionic molecular dynamics 4.1. THE
GROUND
STATE
OF NUCLEI
In the fermionic molecular dynamics model, as usual in quantum mechanics, the ground state is defined as the parametrized state in which the energy X= (H) has an absolute
minimum
Thus 2 is stationary
in the parameter
Q = {qk,j ; k = 1, . . . , A; j = 1,. . . , n}.
space
with respect to variations
of qk,j around
for all -&(QltrlQ)=O
qk,J
their ground-state
values (4.3)
.
.J
In this notation, the previously used index Y has been decomposed into two indices where the first, k, numbers the wave packet and the second, j, the parameters within each packet. Looking at the equations of motion (2.10) we see that the ground state is stationary since all derivatives of 2 vanish and hence all &, = 0. In particular, +,, = 0, $k = 0 which means that the wave packets do not move around. Nonetheless the nucleons are not at rest, the Fermi motion is present in the zero-point motion of each gaussian wave packet. The FMD ground state is completely time-independent as it should be. Here, the FMD model differs conceptually from all other molecular dynamics models for heavy-ion collisions. There, one usually gives the distinguishable classical particles a random motion to mock the Fermi motion. One then has to fight against particle evaporation from a state which is considered to be the ground state. Even if one succeeds with some tricks to avoid disintegration of the “ground state” nucleus the classical model is still only for Boltzmann particles obeying Boltzmann statistics and not Fermi-Dirac statistics. The FMD model compleys with Fermi-Dirac statistics since it is defined with antisymmetrized states. Eq. (4.3) provides not only the ground-state parameters for the centroids but also the COmpkX width parameter ak and the spin directions Xk and & for each single-particle state lqk). Since the two-body interaction invariant a corresponding number for the minimum in the energy.
4.2. THE
LAGRANGE
The single-particle the overlap matrix,
_V(k, I) is translational, rotational and Galilei of parameters has to be kept fixed while searching
FUNCTION
states are in general not orthogonal. Therefore the inverse ok,, appears when calculating the expectation values: (Q1’)l,, := (&).
The Lagrange
function
(2.6) contains
the time-derivative
of
(4.4) part (4.5)
168
the kinetic
H. Feldmeier / Fermionic molecular dynamics
energy
(4.6) and the two-body
interaction
where iqk, q,) = jqk)O]q,). The overlaps (ye 1qr) and the matrix and (qk 1T/q,) are given in an anafytic form in the appendix.
elements
(qk Idq,/dt)
5. Summary and outlook Fermionic molecular dynamics (FMD) describes a system of fermions by an antisymmetrized many-body state built from single-particle wave-packets of gaussian shape. Each single-panicle state is parametrized by its mean spatial position, mean momentum, two angles for the spin direction and a complex width parameter. The equations of motion for the parameters are derived from the quantum variational principle. They are in general not Hamilton-like equations for the selected parameters. By a nonlinear transformation among the parameters it is, however, possible to find pairs of canonical variables which then obey Hamilton-Iike equations 2’). The expectation value of the hamiltonian expressed in these new variables would then be the proper Pauli potential %r,auii. In sect. 3.1 this transformation has been explicitly given for the spin variables of a single fermion. For the two- and many-body cases the transformation is much more involved 2’). The FMD model incorporates the quanta1 Pauli principle into a trajectory picture for particles. This has been achieved by giving up the correspondence principle in its historical form which may be formulated as follows: The classical coordinates of the particles are the expectation values of the corresponding quantum mechanical observables calculated with narrow wave packets. The equations of motion are obtained by taking expectation values of the Heisenberg equation for the operators. In contrast to that the FMD model defines the wave functions in terms of parameters which represent the allowed degrees of freedom of the system. The equations of motion for the parameters are the Euler-Lagrange equations of a quantum variation in the Schrijdinger picture. In the classical limit of narrow wave packets, FMD contains parameters like the mean position which correspond to classical variabIes in the sense of the correspondence principle. But there are also degrees of freedom, like the spin or the width, which have no analogue in classical mechanics. Furthermore, the parameters are always defined even in the pure quanta1 limit. As a simple example for this, let us denote by the parameter Y the position of the maximum in the relative wave function
H. Feldmeier / Fermionic molecular dynamics
for two particles
and by x the operator
particles
one
fermions
or bosons
which
implies
correspondence
has that
r=(g)
and
the wave function (x)=0.
principle
for the relative
Ehrenfest’s
This
distance.
theorem
holds.
is antisymmetric
example
cannot be applied
shows anymore
169
that
For distinguishable
For
indistinguishable
or symmetric,
respectively,
for identical
particles
to the relative distance
the
operator
s. One might think of replacing it by 0 or the operator used in eq. (3.65) but expectation values of these operators give only the magnitude of the distance not the direction. At variance with these complications the parameter r and equation of motion for it can always be used. Actually r is exactly the quantity has in mind when one thinks of the relative distance of two particles which
the but the one are
described by a well-localized wave packet (see fig. 1). In the quanta1 limit where the single-particle wave functions are closely packed in phase space the FMD equations are expected to give results similar to the time-dependent Hartree-Fock equations. Thus, we believe that FMD could bridge in a natural way the gap between the purely quanta1 ground state of a nucleus and two colliding nuclei where the energy per particle is high enough to allow for a molecular dynamics picture. Especially in the expansion phase when the system cools down and forms fragments it is important to study a model like FMD which treats the Pauli principle and the uncertainty relation with concepts of quantum mechanics rather than with mocked up classical mechanics. After all, nuclei at energies below the threshold for particle emission are quanta1 objects, where the main properties are governed by Fermi-Dirac statistics and the quantum uncertainty due to the small size of the system. First numerical results for the two-body case show the strong influence of the Pauli principle on the trajectory when the fermions come close. An open task which has not been addressed in this paper interaction which gives reasonable ground-state properties
is to find a two-body as well as adequate
nucleon-nucleon cross sections for nonrelativistic energies. Another project is to investigate the numerical effort for solving the FMD equations for two colliding heavy nuclei. Due to the nonorthogonality of the wave packets the particles do not interact by two-body interactions only. The skew-symmetric matrix &,, correlates all the particles in a N-body fashion. The reason is that the projection (2.5) on an antisymmetric many-body state (or the Pauli principle) induces N-body correlations. A possible approximation to reduce computer time, if necessary, could be to assume a low density limit in which the particles are described by pairs of antisymmetric two-body states. One would then deal with a superposition of two-body interactions which are, however, modified by the Pauli principle as derived in sect. 3.2. This might be a good approximation as the density in phase space has an upper limit of (2~/ h)3. Therefore, the ground state is the worst case for this approximation while it will become more and more valid during the collision where, due to the increasing entropy, the phase-space density is decreasing, even during the time when the spatial density is rising.
H. Feldmeier / Fermionic molecular dynamics
170
I
would
like to acknowledge
and P. Manakos.
stimulating
I wish to thank
their help in solving
and helpful
Christof
the equations
Gattringer
of motion
discussions
with L. Wilets
and Thorsten
for the scattering
Schwander
for
problem.
Appendix The time-derivative
(qli$
Iq>
part .& of sect. 3.2.2 is given by
=k ‘R’ “Il~~‘psR e-*S”+i f!. I2 I
d,
bf+bi
R
e-FS,2
3 d -2 ; Im (In a)
1 - e-5S,2
2aZ,
41co~x1+~2co~x2_~
+i 2
d
2 dr(4kf42)
1 - emcS,,
-- e-’ (i2 sin x, -Xi sin x2) sin (4, - +2) + (cos x, +cos x2)($, + &) 4 1 - e-{S,, (A.1) The expectation
value of a potential
with the radial
Iq) is defined
&J,@R-~R‘~ The normalization or by
’
N is given
S,z=I(x,,
(A.2)
in eq. (3.21) and bRaR + ha,
B=
a;+a:
is given by
(;;-)“’ e-c.(eBZ'A-ee-DZ'AS,2),
(qlexp (-d2/r3)lq)=it; where the wave function
shape exp (-,x2/$
c=
’
&+a:
a,(b’R-
6;)+2a,b,.
6, (A.3)
&+a:
in eq. (3.22) and the spin overlap
411X2,42>12=~(~+~l
S,, in eq. (3.23)
*n2) >
(A.4)
where nk = (sin xk cos &, sin X~ sin &, cos x~) denotes the direction of the spin in the three-dimensional coordinate space. For the spin exchange potentials one needs the matrix elements (Xl, 4, I’IIXI 3 ~IXX2,42kTlX2,~2)
= nl - n2 7
(A.9
(Xl,
=%3 - nl - n2) .
(A.61
4, lalxr,
The following expressions packet is parametrized as
42XX2.42IQIXI,
$1)
apply to the many-body
case. The single-particle
wave
(A.7)
H. Feldmeier / Fermionic
where
b, = bRI+ ib,, and a, = a,,+
molecular
ia,, are complex.
171
dynamics
The overlap
matrix
is given by
with hk,
In the calculation
+k
IXI,
6)
=
cos
$,yk
cos
of the time-derivative
fx,
+
sin fxk sin ix, ePi(*h-d” .
(A-9)
part L&, one encounters
with (A.ll)
;b?kld=; ($T$-g+(~)2)Glklql),
(A.12)
;
(A.13)
I
h
b?k19,) = i sin
&k
hk
sin ix, e -1(+1-+,) (Xk,
The single-particle
matrix
element
+k
h) (XI,
for the kinetic
41)
’
energy
is (A.15)
(4ilT/41)=(4kl~.k’~qi)=~(~;li2;1;(~)’)(q,~. k
I
k
I
References 1) L. Wilets, A.D. MacKellar and G.A. Rinker Jr., Proc. IV Int. Workshop on gross properties of nuclei and nuclear excitations (Hirschegg, Austria, 1976) p. 111 2) A.R. Bodmer and C.N. Panos, Phys. Rev. Cl5 (1977) 1342; A.R. Bodmer, C.N. Panos and A.D. MacKellar, Phys. Rev. C22 (1980) 1025 3) J.J. Molitoris, J.B. Hoffer, H. Kruse and H. Stticker, Phys. Rev. Lett. 53 (1984) 899 4) T.J. Schlagel and V.R. Phandharipande, Phys. Rev. C36 (1987) 162 5) E. Betak, preprint DUBNA E-86-701 6) R. Schmidt, B. Klmpfer, H. Feldmeier and 0. Knospe, Phys. Lett. B229 (1989) 197 7) L. Wilets, E.M. Henley, M. Kraft and A.D. MacKellar, Nucl. Phys. A282 (1977) 341; L. Wilets, Y. Yariv and R. Chestnut, Nucl. Phys. A301 (1978) 359 8) C. Dorso, S. Duarte and J. Randrup, Phys. Lett. B188 (1987) 287; C. Dorso and J. Randrup, Phys. Lett. 9215 (1988) 611; Phys. Lett. B232 (1989) 29 9) D.H. Boa1 and J.N. Glosli, Phys. Rev. C38 (1988) 1870 IO) J. Aichelin, A. Rosenhauer, G. Peilert, H. Stiicker and W. Greiner, Phys. Rev. Lett. 58 (1987) 1926; Phys. Rev. C37 (1988) 2451
172
H. Feldmeier / Fermionic molecular dynamics
11) G. Peilert, H. Stocker, W. Greiner, A Rosenhauer, A. Bohnet and J. Aichehn, Phys. Rev. C39 (1989) 1402 12) G.E. Beauvais, D.H. Boa1 and J.C.K. Wong, Phys. Rev. C35 (1987) 545 13) D.H. Boa1 and J.N. Glosli, Phys. Rev. C38 (1988) 2621 14) A.K. Kerman and S.E. Koonin, Ann. of Phys. 100 (1976) 332 15) E. Caurier, B. Grammaticos and T. Sami, Phys. Lett. B109 (1982) 150; S. Droidi, J. Okolowicz and M. Ploszajczak, Phys. Lett. B109 (1982) 145 16) M. Saraceno, P. Kramer and F. Fernandez, Nucl. Phys. A405 (1983) 88 17) J.D. Bjorken and S.D. Drell, Relativistische Quantenmechanik (B.I., Mannheim 1966) 18) Jutta Kunz, Klassisches Model1 mit Pionenfreiheitsgrad zur Beschreibung hochenergetischer Schwerionenstoge, PhD thesis (1981), Giessen D26 19) H. Feldmeier, Proc. NASI conference on the nuclear equation of state, Peniscola, Spain (1989), ed. W. Greiner (Plenum, New York, 1989) 20) E.J. Heller, J. Chem. Phys. 62 (1975) 1544 21) C. Coriano, R. Parwani and H. Yamagishi, preprint, Dept. of Physics, Stony Brook 1989
A515 (1990) 147-172
FERMIONIC
MOLECULAR
DYNAMICS
H. FELDMEIER GSI, Gesellschaft fiir Schwerionenforschung mbH, Postfach 110552, D-6100 Darmstadt, Fed. Rep. Germany Received
8 January
1990
Abstract: A new type of molecular dynamics is proposed to solve approximately the many-body problem of interacting identical fermions with spin $. The interacting system is represented by an antisymmetrized many-body wave function consisting of single-particle states which are localized in phase space. The equations of motion for the parameters characterizing the many-body state (e.g. positions, momenta and spin of the particles) are derived from a quantum variational principle. The proposed fermionic molecular dynamics (FMD) model is illustrated with the help of two examples.
1. Introduction Early in heavy-ion research it has been proposed to describe the collision of two nuclei by molecular dynamics in which classical nucleons interact via two-body forces ‘*2). These ideas have been pursued since then 3-6). The major problem with these models which are appropriate for distinguishable classical particles is the fact that nucleons in nuclei and in nuclear collisions are close enough in phase space that the Pauli principle becomes important. The most common approach to improve on that is to mock the influence of the Pauli correlations by an additional momentumdependent potential which becomes strongly repulsive if the fermions get too close in phase space 7-9). Other approaches add fluctuating forces (collision term) which randomly change the momenta of pairs of nucleons ‘0~‘3). Here the Pauli exclusion principle is explicitly taken into account by not allowing scattering into occupied phase space. The motion in between able particles. One knows from thermodynamics
collisions
other
it may make
quantum
effects
are absent,
is, however,
again that of distinguish-
that even for macroscopic a big difference
objects,
where
all
in macroscopic
quantities like the specific heat whether one has Boltzmann, Bose-Einstein or Fermi-Dirac statistics. Not only for that reason, it seems quite important to include the correct statistics also in molecular dynamics, even if other quantum effects are neglected. This paper presents for the first time a non-relativistic model called fermionic molecular dynamics (FMD) which combines from the very beginning Fermi-Dirac statistics with a semi-quanta1 trajectory picture. In the limit where the system becomes dilute in phase space (low density and/or high temperature) it turns into classical 0375-9474/90/$03.50 (North-Holland)
@ Elsevier
Science
Publishers
B.V.
148
H. Feidmeier / Fermionic rn~leeuIar dynamics
molecular
dynamics.
not describe however
The FMD
all wave-mechanical
included an analogue
model
is not fully quantum
interference
on the many-body
In sect. 2 the model is defined bosom
model
mechanical
effects. The Fermi-Dirac
as it does statistics
is
level.
in general terms for identical
may be defined
fermions.
by using symmetrized
For identical
wave functions.
Sect. 3 demonstrates the ideas of the model in two examples. The first is a single fermion moving on a trajectory in an external electromagnetic field. There it is shown how the dynamics in the spin degree of freedom can be described in a classical language although it is in accordance with quantum mechanics in the two-component spinor space. The second example considers the scattering of two fermions with equal spin. Sect. 4 explores further the many-body system.
2. The fermionic molecular dynamics model In the fermionic principle
molecular
dynamics
(FMD)
model
the quantum
variational
14-16)
(2.1) is applied to an antisymmetrized many-body state. This state is parametrized by a set of time-dependent parameters Q(t) = {qY(t), v = 1,2,. . .} which are regarded as the generalized coordinates of the system. The variation has to be performed with respect to all qy( t) keeping the end points fixed, i.e. 8qv( t,) = 8qy( t2) = 0. @ denotes the total hamiltonian of the system and t the time. The variation leads to EulerLagrange equations which are the equations of motion for the parameters {q”(t)}. The general idea is to find a parametrization which contains the essential degrees-offreedom of the system. Usually one takes advantage ameters which have a semiclassical meaning. Classical molecular dynamics describes the motion
of intuition of point-like
and selects parparticles
under
the influence of mutual interactions. Fermionic molecular dynamics extends this picture in a natural way towards quantum mechanics by introducing for each molecule (here each nucleon) a wave packet Is(t)> which is localized in phase space. For convenience one may take a gaussian form
(xi_bi(l))Au(t)(x,i_bj(t))+77(f) Ix(f),cb(r))+
(2.2)
The time-dependent parameters q(t) = (b(t), A,(t), q(t), x(t), #(t)} define the wave packet uniquely. The parameters b(t), Al,(f) and n(t) are complex. li( t) contains the information on the mean position and the mean momentum. The symmetric matrix Aij( t) describes the spreading and its real part has to be positive. n(t) takes care of the normalization and contains a time-dependent phase. This phase, which is in our application of no consequence for the dynamics, is dropped in the following.
Ii.
The spin of the fermion
Feldmeier / Fermionic
is represented
149
rnofecMi~r dynamics
by a two-component
spinor
Ix(t), $(t))
(2.3) where the real polar angles
x(t)
and #(t)
denote
the direction
of the spin.
For identical fermions the ansatz for the parameterized many-body IQ(t)) is an antisymmetrized state formed out of the single-particle given in eq. (2.2). The most simple N-body state is a normalized determinant built from N different single-particle wave packets
wave function states Iqk( t)) single Slater
(2.4) where
As the single-particle states are in general not orthogonal, the many-body state in eq. (2.5) has to be normalized. Nevertheless it is completely antisymmetric with respect to particle exchange and thus a proper wave function for identical fermions. Define a “Lagrange function” by 3(Q(r),
$(t)):=(QW$
in which e(t) = {& = dq,/dt, equations of motion resulting Euler-Lagrange equations
-HIQW)=%(QW,
Q(t))-WQO))
3
(2.6)
v = 1,2, . . . } is the set of generalized velocities. The from the variational principle (2.1) are given by the
d a3 --=dt 84”
a3 aqv
with ~={k,j},
(2.7)
where k = 1,2,. . . , N enumerates the particles and j denotes the different parameters for each single-panicle state. These equations may be non-linear and they need not resemble at all a Schrodinger equation. They can be regarded as semi-quanta1 equations of motion which for special cases may turn into classical equations. If IQ(t)) spans the whole Hilbert space then the set of equations of motion (2.7) is just the Schrodinger equation written in terms of {q,(t)}. If for certain initial states the exact solution of the Schrodinger equation can be expressed at all times by the parametrized state IQ(t)> then the resulting equations of motion for {q”(t)} are also exact. An example is the interaction-free evolution of a gaussian wave if the exact packet. The equations of motion for {qz,( t)} will prove meaningful solution can be approximated well by (Q(t)). -Y( Q( t), o(f)) is not the usual Lagrange function known from classical mechanics. It depends on both spatial coordinates and momenta, and contains the generalized
150
H. Feldmeier 1 Fermionic
velocities
Q1,(t) = dq,(t)/dt
the time derivative
only linearly.
moiec~Iar dynamics
Because of this homogeneity
one can write
part as
(2.8) The hamilton
function
is defined
as the expectation
value of the hamiltonian
Since X and ~~*/~~~ do not depend on the generalized of motion (2.7) can be written in the following general
@
velocities gy, the equations form r5,16) (2.10)
where &,, velocities
is a skew-symmetric
matrix
which
depends
on {q_(f)}
but not on the
(2.11)
The classical Hamilton equations of motion can be written in form of eqs. (2.10) by setting q = {r, p}, where r and p denote conjugate variables, and i2p,, = &2=Q,Vti,2=-&,=-1, (2.12)
Before discussing present
the model
further
in general
terms let us be more specific and
examples.
3, Applications In the following subsections we shall illustrate the ideas of fermionic molecular dynamics with the help of two examples. First we consider a single fermion in an external electromagnetic field. We treat explicitly the coupling of the spin variables with the field and study the influence of the spin on the trajectory. In the second example, which is about two identical fermions, one can see how the Pauli exclusion principle enters the equations of motion for the parameters corresponding to the relative distance and relative momentum. To simplify the equations let us restrict the freedom in the possible shapes of the coordinate space part in eq. (2.2) by assuming A,(f) = Sjjfa(t) so that each singleparticle wave packet is spherical at all times.
H. Feldmeier / Fermionic molecular dynamics 3.1. A CHARGED
FERMION
IN AN ELECTROMAGNETIC
151
FIELD
To get a feeling for how the proposed model for fermionic motion works, let us consider the time evolution of a single charged fermion in an external time-independent electromagnetic
field.
For the sake of simplicity
we assume
in this section
a real and time-independent
width parameter a, so that the wave packet is not allowed evolves in time. We denote the single-particle state by Is(r)) = Ir(t), p(r), x(t), where the coordinate
to spread
4(t)> = Ir(r), p(~))Olx(r),
4(r)),
or shrink
as it
(3.1)
space part Ir( t), p(t))
(xlr(t),p(f))=--&iexp
-(x~~‘))2+ip(t)*x I
(Tao)-
(3.2)
0
I
is expressed with the mean position r(t) and the mean momentum with the complex parameter b(t). The non-relativistic hamiltonian is
FJ=&(-&A(x)c
- )’
p( 6) rather than
+;A”(&-f/qB(_x,k),
(3.3)
where e is the charge, m the mass, and pB = he/(2mc) the magnetic moment of the fermion. ,k and ,x denote the momentum and coordinate space operator, respectively. The charge of the particle interacts with the external vector potential (A’, A) while the interaction of the spin with the electromagnetic field is obtained from a FoldyWouthuysen transformation “) as &(ZB(X The term with the electric
,k) = L +&O’A”(z)
x k + gsV x A(x)).
field VA0 is responsible
The other term is the energy
for the spin-orbit
of the spin in the magnetic
(3.4) interaction.
field V x A (g, = 1, gs-
2
for electrons). q are the Pauli spin matrices which act on the spin part Ix, 4) of the wave function (3.1). In order to apply the variational principle (2.1) one has to express the time derivative part ir,P,x,~li~lr,p,X,~)=-pr-~(sinfx)* and the Hamilton
(3.5)
function x(r,
p,
x, 4) := hi4 x, 4lHlr, p, x, 4)
(3.6)
as a function of the variables. In the spirit of a semi-classical approximation let us assume for the moment that the width a, of the wave packet is small compared to typical variations in A”(x). This implies that we use only the first term of the Taylor expansion (A”(x))=A”(r)+$AA”(r)T+.
...
(3.7)
H. Feldmeier / Fermionic molecular dynamics
152
With that the Hamilton
function
becomes
a(r,p,x.~)=~(p-~A(r))‘+~A’(r)+~ 0
-hdsinx (&(cp) cos $+B2(r,p) sin+)+cosxNr,~)l, where (B, , B2, I&) = B denote the three components of the magnetic by the spin of the moving fermion. From the Lagrange function
field as seen
~=-_li.r-~(sin~x)‘-~(r,p,~,~), one gets the following
Euler-Lagrange
(3.9)
equations
(+A EC?= ++fE dt ai
ap
(3.8)
(3.10)
ap
(3.11)
($=-
O=$-:->z=
-&sinX+FF
or
(3.12)
k=,
.’
The first two equations (3.10) and (3.11) are the usual Hamilton for position and momentum. One sees, however, that inclusion x and C#Jgives an addition to the electrostatic interaction of the spin with the electromagnetic
(3.13) equations of motion of the spin variables
and the Lorentz force, namely, the field. It is also interesting to note
that the equations (3.12) and (3.13) for the spin degrees of freedom Hamilton equations if one chooses as conjugate variables 4
and
s:=ihcosx.
With the projection
of the spin on the 3-axis,
equations
(3.12) and (3.13) transform
of motion
can be cast into
(3.14)
s, and the azimuthal
angle,
d, the
to (3.15)
and (3.16) Unlike a classical angular momentum, s is not a vector and its range is limited between -4h and +$h. Furthermore, there is no kinetic energy of the classical type s2/(2M). By replacing sin x and cos x with s in the Hamilton function (3.8) the h
H. Feldmeier / Fermionic molecular dynamics
in pB drops
out and there is thus no h anymore
like a classical system, spin with the varying Ehrenfest’s
theorem
the spin degree
153
so that also in this respect
it looks
but it is not, Thomas precession and the interaction of the electric and magnetic fields is included. Application of
would
not lead to these equations.
of freedom
is regarded
In so-called
A expansions
to be first order in h and usually
dropped.
The equations of motion (3.10)-(3.13) describe in a natural way the coupling between the spin and the classical trajectory of a fermion. In the following section it is shown how the trajectories of two fermions are influenced by the Pauli principle. 3.2. SCATTERING
OF
TWO
IDENTICAL
FERMIONS
As another example let us apply the ideas of FMD to the scattering of two identical fermions. For the two-body state we make an ansatz in which the center-of-mass part \q,.,.) separates from the part 14) for the relative motion and spins:
14c.m., s>=l4c.m.)olq)~ The center-of-mass
wave function
is parameterized
(3.17)
’
by qc.m.= {B, A}
as
(x-B(t))2 (Xlqc.m.t~))=d& exp I
(3.18)
,
2A(t)
c.m.
where X is the center-of-mass
{
coordinate
and N,.,.(t)
the norm (3.19)
Here and throughout this section the real and imaginary are denoted by the indices R and I, respectively, like B=&+tiB,
or
part of complex
A=A,+iA,.
The wave packet (q) for the relative motion contains {b, a, x, , 4, , x2, 4~~).The parametrized form in coordinate IXlW,
numbers
(3.20) also the spins, thus q = space takes the form
4d~nolx,(~),
42(t))
(3.21) with the time-dependent
normalization
u’c;R;;‘cr,)3’2[eXlJ {$} -exp { -z}$2], where b(t) = bR( t) + ib,( t) and a(t) = uR( t) + ia,( t). The spin overlap
(3.22) is abbreviated
by S12 s,2=
l(XIW,
W)lXz(f),
~&))I2
= f( 1 + sin x, sin x2 cos (4, - 42) + cos x, cos x2) .
(3.23)
H. Feldmeier / Fermionic molecular dynamics
154
The two-body wave packet (3.17) is in general not a single Slater determinant as given in eq. (2.5) because the center-of-mass motion for a Slater determinant does not separate
if the single-particle
packets
have different
section
we let the width A(t) for the center-of-mass
packet
be independent
parameters
mass degree of freedom. The relation between b(t) and the more intuitive “relative momentum” p(t) is Mt)
= r(r) -al(t)
motion
in order to obtain
and
width parameters.
In this
and a(t) for the relative
free motion quantities
in the center-of-
“distance”
b,(r) = a,(r)p(r)
,
r(t) and
(3.24)
respectively. It has to be kept in mind that r(t) and p(t) have their classical meaning only if the two particles are far away in phase space. This is demonstrated in fig. 1, where the real part of the wave function (3.21) is displayed for several choices of the parameters. The spins are in all cases assumed to be parallel. In the first case p = 0, a, = 0 and r= Irl is taken to be large compared to 6. This is a situation where the Pauli principle is not active and r has the classical meaning of the mean distance between the particles. However, that is not completely correct as the particles are indistinguishable and the physical situation is the same for r and -r, thus, the sign of r cannot be measured. The second example is for jr1 <&. From the figure
Fig. 1. Real part of the wave function for the relative coordinate as a function axis defined by r. For details see text.
of x which is along the
H. Feldmeier 1 Fermionic molecular dynamics
it is evident
that now the mean
distance
classical interpretation completely. which specifies the state. Analogue wave function
is for r = 0, p = 216
the same distance
[ in phase
is of the order
155
of X& and r has lost its
Nevertheless, r may still serve as a parameter statements hold for the parameter p. The third and a, = 2.5aR. In this state the particles have
space as in the first example
although
they are very
close in coordinate space (the measure [ is defined in eq. (3.38)). Also here the Pauli principle is not active since a wave function without the exchange part would give the same expectation values. In summary, r and p have their classical meaning only when the particles are far away in phase space. In this case the Pauli principle is also not of importance. One should always carefully distinguish between parameters and physical observables. All measurable quantities have to be calculated from the parametrized state as the expectation value of the corresponding operator. For example, the measured mean distance (q[,x* Iq)“* between two identical fermions is not zero even if r = 0. The total hamiltonian H,,, consists of the center-of-mass kinetic energy and a hamiltonian @ for the relative motion: (3.25)
H In the model
there
=&!c’+Vk, 4, a(l),
is no restriction
in the choice
o(2)).
(3.26)
of the two-body
interaction
V(x &, a(l), o(2)). It may depend on the relative distance z, relative momentum ,k and on the spins via the Pauli spin matrices ~(1) = ~01 and cr(2) = 1 @a which act in the spin space of particle 1 and 2, respectively. 3.2.1. Center-of-mass motion. As a little exercise and as an example for an exact solution of the variational principle, are given in this subsection:
all expressions
for the center-of-mass
motion
(3.27)
,
(3.28)
(3.29) In the center-of-mass wave function R(t) and P(t) have always the classical meaning center-of-mass position and total momentum, respectively. They are related to B(t) by I&(t) = R(t) -A,(t)P( t) and B,(t) = AR( t)P( t). The width parameter A(t) is taken as a dynamical variable and the wave packet is not assumed to be narrow. Therefore the kinetic energy contains besides P’/(2M) the zero-point energy
156
H. Feldmeier / Fermionic molecular dynamics
3/(4MA,). The last term in eq. (3.28) is a total time derivative the Euler-Lagrange equations given below:
and drops
out in
(3.30)
d
()=---__=
a=%.,. a=%,.
d a=%,. ()=----=_ dt a&
a=%.,. aAR
BR
___
(3.31)
(3.32)
A’,
(3.33) It is easy to verify that the solution the expected
of these eight coupled
equations
of motion
is
exact result &=O,
B,=o,
&=O,
A, = l/M,
(3.34)
or P=o,
k=P/M,
&=O,
A, = l/M,
which represents the free motion of a gaussian wave packet. 3.2.2. Relative motion. The Lagrange function for the relative
(3.35)
motion (3.36)
-Y=(41i~14)-(41~k214)(qjvJq)=~~-~-.I’. is more involved
as it contains
exchange
“Ir are given in the appendix. help of the kinetic
energy
contributions
In the following
are discussed
with
9:
1 p2+((r-a,p)2/a’,) =----1 - e-5S,2 2p where 6 is an abbreviation
and spin. The parts X0 and
the novel aspects
e6S,,+
3
(3.37)
4paR ’
for bi+bf [=aR=
(r-aIp)2 aR
+tp2a,
(3.38)
and S,> is the spin overlap given in eq. (3.23). Comparing eq. (3.37) for the relative kinetic energy with eq. (3.29) for the center-of-mass kinetic energy, one sees the influence of the antisymmetrization. If the two fermions have opposite spin, which means S,2 = 0, there is no Pauli principle and the relative kinetic energy is the same as for a single gaussian packet which is not antisymmetrized. For not orthogonal
157
H. Feldmeier / Fermionic molecular dynamics
spins, i.e. S,2 # 0, one recognizes 5 as a measure for the distance in phase space. If 6% 1 the fermions are far in phase space so that the Pauli principle is not acting anymore.
It is interesting
(3.38) while a, together to the measure
to note that G
plays the role of a length
with p modify the naive picture that r2/a,
4 which comes from the spatial
scale in eq.
is the contribution
distance.
For the case of free motion (2r = 0) one can show explicitly that the Euler-Lagrange equations resulting from the Lagrange function eq. (3.36) give the exact result &=O, i,=o,
b,=o,
&=O,
A=%
b, = 1IcL > (3.39)
A=%
%2=0,
or in terms of r and p i=pIpL,
@=O.
(3.40)
Here, like for the center-of-mass motion, the parametrized wave packets are the exact solution of the Schrodinger equation. Without an interaction two identical fermions do not know about each other and move like distinguishable particles. The wave function is nevertheless always antisymmetric so that a measurement of their distance is of course effected by the Pauli principle. It is also interesting to note that for free motion the phase space distance 5 defined above is a constant in time because the solution of the Euler-Lagrange equations (3.39) is r(f) = r,+pt/p and a,(t) = t/p while p and aR do not change in time. Consider as an initial condition two fermions well separated in space such that r2 9 aR which means according to eq. (3.38) 5s 1. Now give them a small momentum p such that they are heading towards each other. After some time the two wave packets will reach each other and completely overlap in space. Nevertheless, they are still very far from each other in phase space since the time-independent combination (r(t) - a,( t)p)’ enters the measure 5. The interaction
energy
V’”= (91 _V\q) is given
in an analytic
potentials which have a gaussian or quadratic dependence The expectation value of a(l)cr(2) which appears for spin also calculated. Since the exact expression is lengthy the discussed here in the limit of narrow wave packets. Suppose
form
for central
on x in the appendix. exchange potentials is interaction energy is the functional form of
I/’ is given by V(g), then
V(q) = (41V(,x)ld~
V(r)-
V(i(ru,/aR-p(a2,+a:)/a,)) 1 -epcS,,
e-‘S,,
(3.41)
Like in the expression for the kinetic energy the classical result is obtained for S,2=0 or e -5=0 . When the particles are close in phase space the interaction is modified by the exchange term which is momentum dependent. The imaginary i =&i does not concern us since V(x) contains only even powers of x so that V(q) remains real. One should keep in mind that in ground-state nuclei the wave packets are not narrow I’). Their spatial width cannot be smaller than about the inverse of the Fermi
H. Feldmeier / Fermionic molecular dynamics
158
momentum because
which is l/kr
each particle
Let us summarize the relative packet.
= 0.7 fm, otherwise
contributes at this point
coordinate
energy becomes
to the kinetic
the differences
of two identical
First, in the classical
the kinetic
at least 3/(4ma,)
between
fermions
the fermionic
and the motion
case the matrix elements
too large
energy. motion
in
of a single wave
of the skew-symmetric
matrix
dik are just (0, 1; -1, 0} and the parameters qj can be grouped in pairs of canonical variables. In the fermionic motion dik( q) becomes q-dependent and variables which were canonical pairs in the classical limit are not canonical any longer. The reason is that the parameters (r, p, oa, a,) loose their physical meaning if the fermions are close in phase space. For example, for r-0 the parameter r is not at the position of the peak in the density anymore; actually the wave packet varies very little as a function of r when the particles are spatially near as displayed in the center part of fig. 1. s&(q) takes care of the varying influence of the parameters q on the wave function in a similar fashion as a mass tensor does for generalized coordinates. Thus dik (q) plays the role of a metric tensor. Second, the generalized forces M’/dqi cannot be replaced by classical forces when the fermions are close in phase space. The exchange term introduces an r-dependence in the kinetic energy Y and a momentum dependence in the interaction even if the classical V(r) depends only on the spatial distance. 3.2.3. Relative motion with a time-independent width parameter. This section investigates briefly how the equations of motion change if the shape of the relative wave packet (eq. (3.21)) is restricted further by removing the width degree of freedom a(t) = aR(t)+ ia, as a dynamical variable 19). For simplicity only parallel spins are considered, i.e. x, =x2 =x and 4, = & = 4. By setting ua( t) = ao, a,(t) = 0 and (iR( t) = 0, ci,( t) = 0 in the previous section we obtain -_li.r+i.pe-c
To=
-2$(sin
1 -e-c 2
y^=lp 2P
2
2
3
-E
:‘-‘,“_of”
is)‘,
+-
(3.43)
%a0
’
where t=
r2/ao+p2ao,
L!!=_Yo-F-Zr.
The potential energy Zr is not given here explicitly not act on the spins. The Euler-Lagrange equations
(3.44)
but it is assumed that y does for the spin variables x and 4
are
()=dL~_G=_~sinx dt &#J
a+
or
$=o,
(3.45)
or
i=o.
(3.46)
H. Feldmeier / Fermionic molecular dynamics
As anticipated motion
it turns
out that the spins will not change
159
in time. The equations
of
for r and p are O=dG_G dt a@
_+2e” 1 _,-s
(
r;-
r(r- i)/a,+p(p1 _e-5
ap
Or
+)a0 >
2 e-“a0 +(I
-e-“)’
(r(p*ti)-p(r.$))
(3.47) and
O=-$F-f$ p+
26 l-e-5
. p(
r(r-Ei)lh+zdp-$h 1 -e-E
or 2 ePg
) +(l-e-‘)2ao
(r(pe
4 -p(r*
4)
In this example the columns of the skew-symmetric matrix dik are contained in the second line of eqs. (3.47) and (3.48), respectively. It has a complicated form as function of r and p but reduces to (0,1; -1,O) if the phase space distance .$ becomes large compared to one in which case one gets back the classical equations of motion. For the reduced set of variables we were able to solve eqs. (3.47) and (3.48) for i and fi. The result is (3.49)
(3.50) where
(Y,(5) and CY~( 5) are functions
of 5 = r2/ a,+p*a,
and given by (3.51)
2e-‘( 1 -e-*) a2(5)
Since (~~(5% 1) = 5% 1 means that implies that they or close by with identical fermions
=
(l+e-*)2(1-e-s)-2~e-~(l+e-5)’
(3.52)
1 and (~~(5% 1) = 0 one recognizes for 5~ 1 Hamilton equations. the two fermions are far from each other in phase space, which could be far in coordinate space with small relative momentum large relative momentum (in any case 5% 1). In that limit the behave like classical distinguishable particles although their wave
160
H. Feldmeier / Fermionic mo~ecuiar dynamics
function is of course still antisymmetrized. (,$< 1) cy,([)+$t and 1y~(5)+3/5~ which (3.49) and (3.50) vanish In this example
like $6 but the remaining
one sees that,
from the parameterization are regarded as canonical Hamilton
function
When they get close in phase space means that the Hamilton-like parts in
for CsZ,
parts increase
the equations
like 3/f2.
of motion
which
result
(3.21) cannot be cast into Hamilton form when r and P variables. To prove this statement let us suppose that a
Xpauri exists such that (3.53)
where i denotes the three spatial directions. Let us now disprove the existence of ~~,“,i by calculating the mixed derivatives of the equations of motion (3.49) and (3.50). It is easy to verify that (3.54) If eq. (3.53) were true, the mixed derivatives
should
be the negative
of each other,
which is not the case here. This disproves the existence of a hamiltonian aipPauli(r, p) which would describe the fermionic dynamics derived from the ansatz (3.21) for the wave function. It is, however, possible to find a pair of canonical variables (p, m), which are then nonlinear functions of the original variables (r, pf [refs. ‘““)] with the elements so that the skew-symmetric matrix dfi,, becomes tri-diagonal -1, 0,l. With these new variables the equations of motion assume the form (3.53) where %&,,i(p, n) = X(r, p) is the total energy expressed in the new canonical variables. Eq. (3.53) represents
an approach
often used in literature
to incorporate
the effects
of the Pauli principle 7-9). D’ff I erent forms of momentum-dependent potentials have been added to the classical hamiltonian. Here we see that this method differs in several aspects from fermionic dynamics. Firstly, as discussed above the dynamical behaviour of P and p need not be of Hamilton replace (V(x)) simply by V(r) because even for exchange term which is not small for t< 1. Since (FMD) a quantum variational principle based on
type. Secondly, one should not narrow wave packets there is an in fermionic molecular dynamics wave functions is involved these
quantum effects are included. The equations of motion simplify appreciably for a time-independent width parameter a,, but the price to be paid is that now the free motion without interaction is not exact anymore. The equations of free motion are (3.55)
161
H. Feldmeier / Fermionic molecular dynamics
For 5 < 1 they differ essentially
from the expected
+ = p/p
result
and $ = 0. The
shape of the wave packet is restrained too much. The particles scatter even if there is no interaction. This deficiency is also present in the so called Pauli potentials 7-9). Inclusion
of the complex
width
has been demonstrated
a(t)
as a dynamical
in the previous
variable
cures this problem
as
section.
3.2.4. Conservation laws. Considering the symmetries of the Lagrange function Zttot= &,,,+Z [eqs. (3.27) and (3.36)] one constructs with Noether’s theorem the following conserved quantities: - Energy in the center-of-mass and relative motion (Zc,.,. and 2 do not explicitly depend on time) -=-
(3.57)
and (3.58) - Total momentum
(Yte,,, is translationally
invariant) (3.59)
$P=-$K)=O. - Center-of-mass
angular
- Total angular
momentum
momentum
(Xc.,. is rotational
in the relative
motion
(2
l-e-< (rxp)+
invariant)
is rotational
invariant)
n,+n2
l-e-‘&,
2
> (3.61)
=&x_k+$T(l)+@J(2)))=0, where nk
=
(xk,
+k
idxk,
is the spin direction of particle orbital angular momentum
4k)
=
(sin
xk
cm
4k,
sin
xk
k. If the interaction
sin
@k,
cos
xk)
does not depend
(3.62) on spin the
(3.63) and the spin ($((r(l)+cr(2)))=
v
I ‘,:t 12
are conserved
separately.
(3.64)
H. Feldmeier / Fermionic molecular dynamics
162
All conservation for the relative to the additional
laws can be proven
angular factors.
momentum
by using the equations
Please note, that in the general
3.2.5. Numerical results. As a first application motion
of two fermions
of motion.
and the spin are somewhat
which are bound
together
The proofs
more involved
due
case r xp is not conserved.
of FMD
we study
by an attractive
the relative
gaussian
potential
(see eq. (A.2)) with a strength of V,= -0.5 hc/fm and a range of r,,= 4 fm. The reduced mass is taken to be p =2.5 h/(c. fm). In the first and second line of fig. 2 the time evolution of the density probability to find the particles with a certain relative distance in the x-y plane is shown for parallel and anti-parallel spins. The fermions are initially at rest with a mean distance of 2 fm and the initial width parameter is taken to be a(t = 0) = 1 fm2, so that they have only little overlap in phase space (5 = 4 or e-‘S,, = 0.018). For both cases the density plot shows a coupled oscillation of the distance and the width of the wave packet. The two markings “+” indicate the actual position of the parameter r and -r. Although during the time evolution the distance r(t) stays around the initial large value of about 4 the motion of the identical fermions differs substantially from the motion of two fermions with opposite spin. The reason is that for the latter case the spin overlap S12 = 0 so that all exchange contributions vanish. The third line in fig. 2 starts with the same initial wave packet but this time the mass of the particles as well as the strength of the potential is multiplied with a factor of 100. The trajectories for r(t) which result from the classical equations of motion are invariant under this transformation; however, the spreading of the wave packet is reduced substantially and one is closer to the classical limit. The detailed time evolution of the measurable distance d and the parameters which define the relative wave function is displayed in fig. 3. The distance d between the fermions is defined as the root of the quadrupole moment along the r-direction. (3.65) The direction r/lrl is given by the direction of the major axis corresponding to the largest eigenvalue of the second moment tensor. For particles with a distance larger than the width of the packet one gets d = (rl ( c 1assical limit). In the upper part of fig. 3 one sees that as a result of the Pauli principle the fermions with equal spins (solid line) do not get close in space, while the particles with opposite spin (dashed line) oscillate through d = 0. At variance with d the parameter r(t) goes through zero even for equal spins. The dotted line is for the case where the mass and the potential strength are hundred times larger. Here, the time dependence of r(t) is almost identical to the purely classical trajectory which is not shown here. The lower part of fig. 3 displays the oscillations in the real and imaginary part of the width parameter a(t). In the nonrelativistic energy regime the scattering of nucleons cannot be described by classical equations of motion. The reason is the short range of the nucleon-nucleon
H. Feldmeier
/ Fermionic
molecular
dynamics
N
Q
-$
+ mj c.4
c-4
0
0
-t
X
+
II
c-4
0 -cli $
II
c-4
F----b
OJ/A
163
164
H. Feldmeier / Fermionic molecular dynamics
I
I
I
I
I
I
I
I
/
0
5
10
15
20
25
30
35
LO
time cth, Fig. 3. Measurable distance and parameters of the wave packet as a function of time for two fermions oscillating in an attractive potential (same case as in fig. 2). Full lines are for parallel spins, dashed lines for antiparallel spins (no Pauli principle), and dotted lines are for heavy particles with antiparallel spins. See text for details and choice of the parameters.
interaction, which is only a few times longer than the Compton wavelength of the nucleon. Therefore, a wave packet well localized with respect to the range of the interaction spreads too fast during the interaction time. The following estimate illustrates this. Let us take a rather large width initially and assume uR = 1 fm2 and a, = 0 just before the collision and let us allow a, to increase during the collision time by 1 fm2. The collision time and the increase in a, we estimate from the eq. (3.39) for free motion. The time a nucleon needs to travel for about the range of the interaction, R = 5 fm, is t = Rp/p. During this time the imaginary part of the
H. Feidmeier
/ Fermionic
molecular
dynamics
165
width increases by a, = t/g, = R/p. If we allow for a, c 1 fm’ this means that p&5 fm-’ or p b 1 CeV/c. This small exercise shows that a nonquantal description in terms of trajectories can be used only in the relativistic regime. Thus, a nonrelativistic nonquantal trajectory picture cannot be applied for the scattering of nucleons 18). In fig. 4 the density for the scattering from a strongly repuisive potential is shown for particles with equal spin and opposite spin. We include the free motion, which
,Q -10 X/Tg
-5
Q
x/r0
5
10 -IO
x/r*
-5
0
x/r0
5
x/r0
x/r0
Fig. 4. Time evolution of the two-body density probability as a function of the relative distance between two fermions which are scattering from a repulsive potential. The contour plot is for z = 0 and the center-of-mass coordinates have been integrated over. First line: free motion of fermions with parallel spin. Second line: scattering for parallel spins. Third line: scattering for antiparalle~ spins (no Pauli principle). Fourth line: scattering of heavy particles with antiparallel spins. Fifth line: free motion of heavy particles with antiparallel spins. The positions of the parameter I and --r are marked by f. Contour tines are at (0.5, 0.2, 0.05, 0.01) of the maximum density. See text for details and choice of the parameters.
166
H. Feldmeier / Fermionic molecular dynamics
is the exact solution of the wave packet some resemblance
of the Schrodinger equation, in absence of an interaction. with a trajectory
to illustrate the amount of spreading In order to get a picture which has
we are forced to choose almost relativistic
initial
momenta. For the upper three lines of fig. 4 we took pft = 0) = (-4,0,0) h/fm, r(t=0)=(3,0.5,0) fm, a(t =0)=06fm’ and as reduced mass p =2.5 h/(c* fm). The potential
has V, = 5
A/(c . fm) and r, = 1 fm. The first line for the free scattering
shows the fast spreading which has been discussed above. The second line is also for parallel spins but the repulsive potential is present. Besides the nonzero scattering angle one sees that the potential is also blowing up the wave packet so that a classical trajectory approximation is not possible. For the third and fourth line of fig. 4 the reduced mass, the initial momentum and the strength of the potential has been increased by a factor of 100, so that the classical trajectory remains the same, but the free spreading is reduced by a factor of 100. The last line, which is for free motion, shows almost no spreading anymore, but the repulsive potential also blows up the wave function of the heavy particles, as can be seen from the third line. Nevertheless this case resembles more a classical trajectory picture. The classical scattering angle is larger because, due to the finite width of the packet, the expectation value (q/J’(q) is weaker than the corresponding V(r). There is yet another condition which cannot be fulfilled in the scattering case, namely, in order to keep approximately the gaussian shape of the packet it has to be narrow compared to variations in the potential ‘*). More precisely d’V/dr’
“aag+ai:
I d’V/dr”
1
3
(3.66)
aR
where the right-hand side measures the spatial width of the wave. In our case the ratio of the derivatives is of the order of the range r. which is 1 fm and therefore inequality (3.64) is not really fulfilled for the cases shown in fig. 4. The situation should improve in the many-body case where each particle feels a smoother mean field produced
by its many
neighbours. 4. The many-body system
In nuclei one has two kinds of fermions, protons and neutrons, many-body wave function splits into two antisymmetric parts IQ> = IQprotonJO IQ”e”tTO”S) *
therefore
the
(4.1)
The hamiltonian @= T+ y=;
k-
T(k)+
; k
V(k, I) -
(4.2)
consists of the kinetic energies _T(k) and the two-body interaction _V(k, I) between the particles. There is no restriction on the form of the interaction, it may for example be spin and isospin dependent. All degrees of freedom for nonrelativistic nucleons are present in the many-body wave function (4.1).
167
H. Feldmeier / Fermionic molecular dynamics 4.1. THE
GROUND
STATE
OF NUCLEI
In the fermionic molecular dynamics model, as usual in quantum mechanics, the ground state is defined as the parametrized state in which the energy X= (H) has an absolute
minimum
Thus 2 is stationary
in the parameter
Q = {qk,j ; k = 1, . . . , A; j = 1,. . . , n}.
space
with respect to variations
of qk,j around
for all -&(QltrlQ)=O
qk,J
their ground-state
values (4.3)
.
.J
In this notation, the previously used index Y has been decomposed into two indices where the first, k, numbers the wave packet and the second, j, the parameters within each packet. Looking at the equations of motion (2.10) we see that the ground state is stationary since all derivatives of 2 vanish and hence all &, = 0. In particular, +,, = 0, $k = 0 which means that the wave packets do not move around. Nonetheless the nucleons are not at rest, the Fermi motion is present in the zero-point motion of each gaussian wave packet. The FMD ground state is completely time-independent as it should be. Here, the FMD model differs conceptually from all other molecular dynamics models for heavy-ion collisions. There, one usually gives the distinguishable classical particles a random motion to mock the Fermi motion. One then has to fight against particle evaporation from a state which is considered to be the ground state. Even if one succeeds with some tricks to avoid disintegration of the “ground state” nucleus the classical model is still only for Boltzmann particles obeying Boltzmann statistics and not Fermi-Dirac statistics. The FMD model compleys with Fermi-Dirac statistics since it is defined with antisymmetrized states. Eq. (4.3) provides not only the ground-state parameters for the centroids but also the COmpkX width parameter ak and the spin directions Xk and & for each single-particle state lqk). Since the two-body interaction invariant a corresponding number for the minimum in the energy.
4.2. THE
LAGRANGE
The single-particle the overlap matrix,
_V(k, I) is translational, rotational and Galilei of parameters has to be kept fixed while searching
FUNCTION
states are in general not orthogonal. Therefore the inverse ok,, appears when calculating the expectation values: (Q1’)l,, := (&).
The Lagrange
function
(2.6) contains
the time-derivative
of
(4.4) part (4.5)
168
the kinetic
H. Feldmeier / Fermionic molecular dynamics
energy
(4.6) and the two-body
interaction
where iqk, q,) = jqk)O]q,). The overlaps (ye 1qr) and the matrix and (qk 1T/q,) are given in an anafytic form in the appendix.
elements
(qk Idq,/dt)
5. Summary and outlook Fermionic molecular dynamics (FMD) describes a system of fermions by an antisymmetrized many-body state built from single-particle wave-packets of gaussian shape. Each single-panicle state is parametrized by its mean spatial position, mean momentum, two angles for the spin direction and a complex width parameter. The equations of motion for the parameters are derived from the quantum variational principle. They are in general not Hamilton-like equations for the selected parameters. By a nonlinear transformation among the parameters it is, however, possible to find pairs of canonical variables which then obey Hamilton-Iike equations 2’). The expectation value of the hamiltonian expressed in these new variables would then be the proper Pauli potential %r,auii. In sect. 3.1 this transformation has been explicitly given for the spin variables of a single fermion. For the two- and many-body cases the transformation is much more involved 2’). The FMD model incorporates the quanta1 Pauli principle into a trajectory picture for particles. This has been achieved by giving up the correspondence principle in its historical form which may be formulated as follows: The classical coordinates of the particles are the expectation values of the corresponding quantum mechanical observables calculated with narrow wave packets. The equations of motion are obtained by taking expectation values of the Heisenberg equation for the operators. In contrast to that the FMD model defines the wave functions in terms of parameters which represent the allowed degrees of freedom of the system. The equations of motion for the parameters are the Euler-Lagrange equations of a quantum variation in the Schrijdinger picture. In the classical limit of narrow wave packets, FMD contains parameters like the mean position which correspond to classical variabIes in the sense of the correspondence principle. But there are also degrees of freedom, like the spin or the width, which have no analogue in classical mechanics. Furthermore, the parameters are always defined even in the pure quanta1 limit. As a simple example for this, let us denote by the parameter Y the position of the maximum in the relative wave function
H. Feldmeier / Fermionic molecular dynamics
for two particles
and by x the operator
particles
one
fermions
or bosons
which
implies
correspondence
has that
r=(g)
and
the wave function (x)=0.
principle
for the relative
Ehrenfest’s
This
distance.
theorem
holds.
is antisymmetric
example
cannot be applied
shows anymore
169
that
For distinguishable
For
indistinguishable
or symmetric,
respectively,
for identical
particles
to the relative distance
the
operator
s. One might think of replacing it by 0 or the operator used in eq. (3.65) but expectation values of these operators give only the magnitude of the distance not the direction. At variance with these complications the parameter r and equation of motion for it can always be used. Actually r is exactly the quantity has in mind when one thinks of the relative distance of two particles which
the but the one are
described by a well-localized wave packet (see fig. 1). In the quanta1 limit where the single-particle wave functions are closely packed in phase space the FMD equations are expected to give results similar to the time-dependent Hartree-Fock equations. Thus, we believe that FMD could bridge in a natural way the gap between the purely quanta1 ground state of a nucleus and two colliding nuclei where the energy per particle is high enough to allow for a molecular dynamics picture. Especially in the expansion phase when the system cools down and forms fragments it is important to study a model like FMD which treats the Pauli principle and the uncertainty relation with concepts of quantum mechanics rather than with mocked up classical mechanics. After all, nuclei at energies below the threshold for particle emission are quanta1 objects, where the main properties are governed by Fermi-Dirac statistics and the quantum uncertainty due to the small size of the system. First numerical results for the two-body case show the strong influence of the Pauli principle on the trajectory when the fermions come close. An open task which has not been addressed in this paper interaction which gives reasonable ground-state properties
is to find a two-body as well as adequate
nucleon-nucleon cross sections for nonrelativistic energies. Another project is to investigate the numerical effort for solving the FMD equations for two colliding heavy nuclei. Due to the nonorthogonality of the wave packets the particles do not interact by two-body interactions only. The skew-symmetric matrix &,, correlates all the particles in a N-body fashion. The reason is that the projection (2.5) on an antisymmetric many-body state (or the Pauli principle) induces N-body correlations. A possible approximation to reduce computer time, if necessary, could be to assume a low density limit in which the particles are described by pairs of antisymmetric two-body states. One would then deal with a superposition of two-body interactions which are, however, modified by the Pauli principle as derived in sect. 3.2. This might be a good approximation as the density in phase space has an upper limit of (2~/ h)3. Therefore, the ground state is the worst case for this approximation while it will become more and more valid during the collision where, due to the increasing entropy, the phase-space density is decreasing, even during the time when the spatial density is rising.
H. Feldmeier / Fermionic molecular dynamics
170
I
would
like to acknowledge
and P. Manakos.
stimulating
I wish to thank
their help in solving
and helpful
Christof
the equations
Gattringer
of motion
discussions
with L. Wilets
and Thorsten
for the scattering
Schwander
for
problem.
Appendix The time-derivative
(qli$
Iq>
part .& of sect. 3.2.2 is given by
=k ‘R’ “Il~~‘psR e-*S”+i f!. I2 I
d,
bf+bi
R
e-FS,2
3 d -2 ; Im (In a)
1 - e-5S,2
2aZ,
41co~x1+~2co~x2_~
+i 2
d
2 dr(4kf42)
1 - emcS,,
-- e-’ (i2 sin x, -Xi sin x2) sin (4, - +2) + (cos x, +cos x2)($, + &) 4 1 - e-{S,, (A.1) The expectation
value of a potential
with the radial
Iq) is defined
&J,@R-~R‘~ The normalization or by
’
N is given
S,z=I(x,,
(A.2)
in eq. (3.21) and bRaR + ha,
B=
a;+a:
is given by
(;;-)“’ e-c.(eBZ'A-ee-DZ'AS,2),
(qlexp (-d2/r3)lq)=it; where the wave function
shape exp (-,x2/$
c=
’
&+a:
a,(b’R-
6;)+2a,b,.
6, (A.3)
&+a:
in eq. (3.22) and the spin overlap
411X2,42>12=~(~+~l
S,, in eq. (3.23)
*n2) >
(A.4)
where nk = (sin xk cos &, sin X~ sin &, cos x~) denotes the direction of the spin in the three-dimensional coordinate space. For the spin exchange potentials one needs the matrix elements (Xl, 4, I’IIXI 3 ~IXX2,42kTlX2,~2)
= nl - n2 7
(A.9
(Xl,
=%3 - nl - n2) .
(A.61
4, lalxr,
The following expressions packet is parametrized as
42XX2.42IQIXI,
$1)
apply to the many-body
case. The single-particle
wave
(A.7)
H. Feldmeier / Fermionic
where
b, = bRI+ ib,, and a, = a,,+
molecular
ia,, are complex.
171
dynamics
The overlap
matrix
is given by
with hk,
In the calculation
+k
IXI,
6)
=
cos
$,yk
cos
of the time-derivative
fx,
+
sin fxk sin ix, ePi(*h-d” .
(A-9)
part L&, one encounters
with (A.ll)
;b?kld=; ($T$-g+(~)2)Glklql),
(A.12)
;
(A.13)
I
h
b?k19,) = i sin
&k
hk
sin ix, e -1(+1-+,) (Xk,
The single-particle
matrix
element
+k
h) (XI,
for the kinetic
41)
’
energy
is (A.15)
(4ilT/41)=(4kl~.k’~qi)=~(~;li2;1;(~)’)(q,~. k
I
k
I
References 1) L. Wilets, A.D. MacKellar and G.A. Rinker Jr., Proc. IV Int. Workshop on gross properties of nuclei and nuclear excitations (Hirschegg, Austria, 1976) p. 111 2) A.R. Bodmer and C.N. Panos, Phys. Rev. Cl5 (1977) 1342; A.R. Bodmer, C.N. Panos and A.D. MacKellar, Phys. Rev. C22 (1980) 1025 3) J.J. Molitoris, J.B. Hoffer, H. Kruse and H. Stticker, Phys. Rev. Lett. 53 (1984) 899 4) T.J. Schlagel and V.R. Phandharipande, Phys. Rev. C36 (1987) 162 5) E. Betak, preprint DUBNA E-86-701 6) R. Schmidt, B. Klmpfer, H. Feldmeier and 0. Knospe, Phys. Lett. B229 (1989) 197 7) L. Wilets, E.M. Henley, M. Kraft and A.D. MacKellar, Nucl. Phys. A282 (1977) 341; L. Wilets, Y. Yariv and R. Chestnut, Nucl. Phys. A301 (1978) 359 8) C. Dorso, S. Duarte and J. Randrup, Phys. Lett. B188 (1987) 287; C. Dorso and J. Randrup, Phys. Lett. 9215 (1988) 611; Phys. Lett. B232 (1989) 29 9) D.H. Boa1 and J.N. Glosli, Phys. Rev. C38 (1988) 1870 IO) J. Aichelin, A. Rosenhauer, G. Peilert, H. Stiicker and W. Greiner, Phys. Rev. Lett. 58 (1987) 1926; Phys. Rev. C37 (1988) 2451
172
H. Feldmeier / Fermionic molecular dynamics
11) G. Peilert, H. Stocker, W. Greiner, A Rosenhauer, A. Bohnet and J. Aichehn, Phys. Rev. C39 (1989) 1402 12) G.E. Beauvais, D.H. Boa1 and J.C.K. Wong, Phys. Rev. C35 (1987) 545 13) D.H. Boa1 and J.N. Glosli, Phys. Rev. C38 (1988) 2621 14) A.K. Kerman and S.E. Koonin, Ann. of Phys. 100 (1976) 332 15) E. Caurier, B. Grammaticos and T. Sami, Phys. Lett. B109 (1982) 150; S. Droidi, J. Okolowicz and M. Ploszajczak, Phys. Lett. B109 (1982) 145 16) M. Saraceno, P. Kramer and F. Fernandez, Nucl. Phys. A405 (1983) 88 17) J.D. Bjorken and S.D. Drell, Relativistische Quantenmechanik (B.I., Mannheim 1966) 18) Jutta Kunz, Klassisches Model1 mit Pionenfreiheitsgrad zur Beschreibung hochenergetischer Schwerionenstoge, PhD thesis (1981), Giessen D26 19) H. Feldmeier, Proc. NASI conference on the nuclear equation of state, Peniscola, Spain (1989), ed. W. Greiner (Plenum, New York, 1989) 20) E.J. Heller, J. Chem. Phys. 62 (1975) 1544 21) C. Coriano, R. Parwani and H. Yamagishi, preprint, Dept. of Physics, Stony Brook 1989