A model study of two interacting fermion gases

Nuclear Physics A394 (1983) 334-348 (~) North-Holland Publishing Company

A MODEL STUDY OF TWO INTERACTING FERMION GASES t P. BUCK and H. FELDMEIER t* Institut fffr Kernphysik, Technische Hochschule Darmstadt, FRG

Received 27 May 1982 (Revised 19 July 1982) Abstracl: To gain a better understanding of the mass distribution in heavy-ion collisions we solve numerically the time-dependent many-body Schr6dinger equation for a fermion system. The model describes the motion of A 1+A 2 interacting fermions in one dimension. Initially the particles are separated into two subsystems. Then they are brought into contact and particle transfer between the systems causes the relative mass number to dissipate and fluctuate. Special attention is focussed on the variance of the mass distribution. The strong influence of the Pauli exclusion principle is demonstrated and conclusions are drawn concerning the extended time-dependent Hartree-Fock theories which include collision terms.

1. Introduction

C h a r g e a n d mass d i s t r i b u t i o n s have been of special interest in h e a v y - i o n physics. M a n y different m o d e l s have been a p p l i e d to explain the e x p e r i m e n t a l data. A m o n g the m i c r o s c o p i c theories the t i m e - d e p e n d e n t H a r t r e e - F o c k t h e o r y ( T D H F ) takes up a special position, because it p r o v i d e s a q u a n t u m m e c h a n i c a l d e s c r i p t i o n in terms of a wave function. W h i l e the m e a n values of the mass d i s t r i b u t i o n s a n d o t h e r o n e - p a r t i c l e quantities can be r e p r o d u c e d by T D H F , the widths of the d i s t r i b u t i o n s - c o n t a i n i n g i n f o r m a t i o n a b o u t the t w o - p a r t i c l e c o r r e l a t i o n s - are u n d e r e s t i m a t e d by an o r d e r of m a g n i t u d e ~'2). Therefore T D H F has been e x t e n d e d to include explicitly t w o - b o d y c o r r e l a t i o n s ( E T D H F ) 3 - 7 ) . Regardless of the difficult question c o n c e r n i n g the validity of the a p p l i e d statistical a p p r o x i m a t i o n s , n u m e r i c a l results are only a v a i l a b l e for simplified e q u a t i o n s 8 - ~2). It therefore seems interesting to study the d y n a m i c s of a small fermion system in a simple q u a n t u m m e c h a n i c a l model. The d y n a m i c s in the m o d e l are m a i n l y g o v e r n e d by a o n e - p a r t i c l e h a m i l t o n i a n , but t w o - b o d y collisions are explicitly included by a t w o - p a r t i c l e interaction. O u r m o d e l is thus c o n s t r u c t e d in the spirit of an e x t e n d e d t i m e - d e p e n d e n t H a r t r e e - F o c k model, where the particles are m o v i n g in a selfconsistent p o t e n t i a l well with a residual t w o - b o d y interaction. The o n e * Work supported by Gesellschaft fiir Schwerionenforschung mbH, Darmstadt. ** Heisenberg Fellow. 334

P. Buck, H. Feldmeier / Two interactino fermion 9ases

335

particle potential in our model, however, is not selfconsistent, and the model is strongly simplified to allow an exact numerical treatment for solving the timedependent many-particle Schr6dinger equation. Thus within our model system we are able to compare the exact solution with different approximations. We hope that some general features can be carried over to more complicated systems such as heavy ions. This paper is organized as follows: In sect. 2 we introduce our model. Subsect. 3.1 contains a survey of the numerical results. In subsect. 3.2 we compare them with what one would expect from a classical picture. We will show the strong influence of the Pauli exclusion principle. It will turn out that the Pauli principle in connection with one mean potential keeps the variance of the mass distribution small, except for very high excitation energies or very low densities. We therefore do not expect that short-ranged two-body collisions are able to account for the measured mass dispersion. In subsect. 3.3 we discuss a product ansatz for the two-particle density used in the literature 13, 14). It will be shown that this approach is not reliable for calculating the mass dispersion. In cases other than that of a single Slater determinant, where it is exact, the variance is overestimated. Sect. 4 finally gives a summary of our results.

2. The model

Our model has been constructed in imitating the mechanism of particle exchange in heavy-ion reactions. There one can distinguish three stages: First the nuclei approach each other but are still separated, so that no particles can pass from one nucleus to the other. Then the ions come into contact, and particle transfer sets in. Finally they separate again, with no further particle exchange being possible. The analogue in our model is as follows: A fermions are kept in a onedimensional square-well potential with infinitely strong walls and length L. The system is divided by a wall into two regions of equal length ("left" and "right", see fig. 1). Up to time t = 0 we have sharp particle numbers A1 on the left- and A 2 o n the right-hand side (A1 + A 2 = A). The wall inhibits the particles to go from one side to the other (separated nuclei). At time t = 0 the wall is being removed, and the particles can cross over to the other side and move in the whole container (nuclei in contact). Finally the wall is reinstalled so that no further particle transfer is possible. We define an hermitian operator N L which counts the particles on the left side of our system. In coordinate representation N L is given by A

NL ---- Z [1-- 0(x,)], i=1

(2.1)

336

P. Buck, H. Feldmeier / Two interactin# fermion yases "left

.

.

.

.

right"

i 'qF

O

'qW

//2

Fig. 1. Schematic picture of the model. A wall prevents particle transfer at times t < 0.

where A is the total number of particles of the system, and O(x) is the step function: 0(x) = { ;

if otherwise. x relates to the right-hand side

The variance 0-1 of the particle number on one side is defined as a 2 = ( N 25 - (NL) 2.

(2.2)

Up to time t = 0 0 -2 is zero. The particles interact with each other via a two-body potential which consists of a repulsive core and an attractive tail as shown in fig. 2. The core has a strength of 56 MeV and a range of 2 fm, and the attractive part has a depth of - 7 MeV and reaches out to 8 fm. The hamiltonian of the system without the wall is given by

[2m + Vo(xi) + ~ V(Ix,-xil),

H = i=1

i
Ix, - x j l

Fig. 2. Shape of the two-body potential as a function of the relative distance.

(2.3)

P. Buck, H. Feldmeier / Two interactino fermion 9ases

337

where m is the nucleon mass, Vo is the square-well one-body potential, and V denotes the two-body interaction. A measure for the strength of the two-body interaction used is the decay time of the time-time correlation function between a state evolved with the one-body hamiltonian and the same state evolved in time with the one-body plus two-body interaction. We calculate l( q'(to)l e i " " -

VVhl~(to + t))l z,

(2.4)

where I~(t)) is the wave function in the Schr6dinger representation and H - V is the one-body part of the hamiltonian. Numerical computations for cases discussed in sect. 3 lead to a decay time ~ ~ 10 -22 s rather independent of the starting time to. r may be compared with 7"1 = L/vv (vv = Fermi velocity) which gives the time scale for the one-particle motion. T1 turns out to be about 10 - z l s, so that T~/r ,,~ 10. This means that the two-body collisions violently destroy the one-body motion. The simplest way to introduce a wall would be to add to the hamiltonian a delta function. But this definition did not work satisfactorily, because even a very strong delta-function potential is not completely impenetrable for particles. We therefore choose a different procedure to define the wall: The hamiltonian H 0 which includes the wall has to commute with the operator N e which counts the particles on the left-hand side. Only then the particle number on each side and all higher moments of NL are conserved. We represent therefore the hamiltonian H of the system without the wall (2.3) in the eigenbasis of N e and cancel all matrix elements which connect states of different particle numbers on one side. Taking this operator as Ho, the wall is then defined as the difference between Ho and H. Thus by construction it is impenetrable for particles. Before giving the exact definition of the wall operator we should like to discuss the temptation to represent all operators in the eigenbasis of NL, as for example done by Randrup 15). A description of the time evolution of t)ae system in this socalled left-right basis, however, is not possible. The reason being that exact eigenstates of NL are either identically zero on one side (x < ½L) or on the other (x > ½L). This implies that the kinetic energy as well as any local operator has vanishing matrix elements between left and right states. The exchange of particles cannot take place in a vector space spanned by a finite number of exact eigenstates of Ne. This peculiar behaviour can be understood better by considering the eigenvalues and eigenstates of NL in a truncated vector space spanned by the big-box basis states. For illustration we diagonalized N L in the one-body space consisting of the nd lowest big-box single-particle states. The resulting eigenstates and eigenvalues are displayed in fig. 3 for n a = 18. Only one-half of them are shown, as the other states can be obtained by reflection at x = ½L. It turns out that the left-right basis can be represented in the

338

P. Buck, H. l:eldmeier / Two interacting,/ermion gases

NO 1 000005

i

i

NO 2 0000 t 8

No 3 .000041


N o ,4 000074

r

NO 5 000 1]7

.

L NO 6

NO7

.000172

0002`45

I

L/2

NO

I

8

I

L

Fig. 3. Eigenstates of the operator N L in the one-particle space, obtained by a diagonalization in a truncated vector space of 18 big-box states. Only one-half of them are shown, because states 10 to 18 can be obtained from states 9 to 1 by reflection at x = ½L. The numbers are the appropriate eigenvalues.

big-box vector space very well except for two states (nos. 9 and 10). These have a large number of nodes and thus high energy, and they are not localized on one side or the other. The same behaviour prevails in all cases, even if we increase the dimension nd up to 100. There are always only two or four states which make the difference between a (nd - 2 ) - or (n d - 4)-dimensional left-right basis and the n ddimensional big-box basis. A numerical test confirmed the expectation that omission of these two or four states hinders almost completely the free motion of particles in the box. Thus a transition between left and right states can only proceed through intermediate occupation of these "doorway" states and is therefore at least of second order. Therefore the transition operator used in ref. 15) has to be regarded as an effective one-body operator of a rather complicated structure. After these remarks about the properties of two almost equal vector spaces we want to define the actual construction of the wall operator and the initial states. As it is not possible to include all Slater determinants for A particles distributed over the chosen big-box single-particle states, we truncate the A-body vector space by taking only the lowest 300 eigenstates of the one-particle part of H. Let us denote this truncation by the projection operator P. Then we find the eigenvalues n 2 of P N ~ P in the A-body space. Without truncation N 2 would have the square numbers O, 1, 4 ..... A 2 as eigenvalues. In our case the condition (k-~)2
k = 0 , 1 ..... A,

(2.5)

defines subspaces ~k spanned by the eigenstates belonging to the eigenvalues n~2. e is a parameter which is taken to be around 0.1. The subspaces ffk contain states with nearly sharp particle number n i ~ k on the left-hand side. If we are dealing with a system of definite parity, P N L P is already diagonal, if P also projects on states of a given parity. This is the reason to use PNzeP. For symmetric systems

P. Buck, H. Feldmeier / Two interacting fermion gases

339

condition (2.5) is replaced by ( ½ A ) 2 + k 2 _ n 2 < ~2,

k = 0, 1..... ½A.

(2.6)

Let the Pk be projection operators onto the subspaces ~k. By means of the Pk we define the hamiltonian H 0 of the separated system (wall installed) as follows:

Ho =

PkHPR+ k=O

f(ni) i=

1--

Pk •

(2.7)

k=O

H is the hamiltonian of the system without wall (2.3), N is the dimension of the vector space, in our case N = 300, and f(ni) is a strength function which shifts upwards the energies of those states which do not fulfil the above condition (2.5) or (2.6) on the eigenvalues hi. f(nl) ranges between 0 and 100 MeV depending on how well ni approaches an integral number. The second term provides for transitions into the subspaces of nearly sharp particle numbers on each side. Thus all other states are depopulated. Ho therefore drives the system into asymptotic states with good particle numbers. The wall W is defined as the difference between Ho and H : W = H o - H.

(2.8)

The wall can be removed and reinstalled by a strength function g(t), which varies between zero and one. The initial state must have sharp particle number on each side. It is chosen to be an eigenstate of PkHP k with k particles on the left. It includes already the two-body correlations of H. The particles are identical fermions. In particular they all possess equal spin and isospin quantum-numbers.

3. Results

For the above-described model we solve numerically the time-dependent manybody Schr6dinger equation. The system consists of 10 identical nucleons in a box which is 44 fm long. Then the Fermi energy corresponds to about one-third (one dimension) of the Fermi energy of a nucleus. Herewith we get a time scale typical for nuclear dynamics. For the initial state we take the lowest eigenstate of the (wall included) hamiltonian H o given in eq. (2.7) with the desired particle numbers on each side. At time t = 0 we remove the wall continuously until t = 10 -22 s. Thereafter the dynamics are governed by the hamiltonian H without the wall [eq. (2.3)], until we reinsert the wall at t = 46 x 10- 22 s. At t = 47 x 10- 22 s the wall is completely reinstalled, and we calculate the dynamics with H o until 60 x 10-22 s have elapsed, thus giving the system time to equilibrate.

340

P. Buck, H. Feldmeier / Two interactingJermion gases

The influence of the t w o - p a r t i c l e i n t e r a c t i o n ("real gas") is s t u d i e d in c o m p a r i s o n with the i n t e r a c t i o n free case ("ideal gas").

3.1. SURVEY Fig. 4 shows the m e a n value
5 4 o

2

1.o 0.5

,, .. ~, .. ,, ~ ',

,, ,

/

g(t)

o

i , i 10 20 t

.

I 30

, 410

I 50

L 60

(10-22s)

Fig. 4. Mean particle number (NL) on the left-hand side and the variance 0"2 for a symmetric system as a function of time. g(t) is the strength of the wall. The full line refers to the real gas, whereas the dashed line denotes the ideal Fermi gas.

341

P. Buck, H. Feldmeier Two interactingfermion gases 6



'

'

'

'

I

:

I

5

4

02

'

',

:

', :

I

:

I

41 : I . ~, . I . I . I .

:

1 .O

0.5

0.5

0

', : I

i

O

10

:

0

'. I :

i

20

-

,

30

40

I

50

60

0

10 20

t (10"~2s)

30

I

40

,

50

60

t (10-22S)

Fig. 5. and a 2 for an asymmetric ideal Fermi gas as a function of time. Initially there are 6 particles on the left- and 4 particles on the right-hand side.

Fig. 6. Same as fig. 5, but for the real gas,

oscillation is strongly d a m p e d and the frequency is about a factor of two greater than in the interaction free case. The ideal gas wobbles back and forth with a period which is just the time for a nucleon to cross the system. It takes a b o u t 0.9x 10 -21 s until reaches its first minimum, which means that the particles have crossed to the other side. In the real gas case the particles do not reach the opposite wall, because they undergo collisions on their way and therefore bounce to and fro. This explains qualitatively both the higher frequency and the strong d a m p i n g of the oscillation. Also shown in figs. 5 and 6 is the time dependence o f a 2 for the above system. The curves are very similar to the symmetric case (fig. 4). 3.2. CLASSICAL EXPECTATION In order to conceive the o u t c o m e of the calculations we c o m p a r e the results with a classical model, where we replace the fermions by distinguishable classical particles. Following Beck et al. ,6) the classical mass variance tr~ for a symmetric system with A particles is given by tr¢l = ¼A 1 - e x p

2

I(z)dz

.

(3.1)

Here l ( z ) denotes the one-sided current t h r o u g h the window between the two subsystems. The n u m b e r of exchanged particles is gex(t) = 2

;o

I(t)dz.

(3.2)

342

P. Buck, H. Feldrneier / Two i n t e r a c t i n o j e r m i o n oases

For long times Nex becomes large, and according to (3.1) acz~tends asymptotically to ¼A. To compare this classical model for distinguishable particles with our quantum mechanical calculation for indistinguishable fermions, we calculate the one-sided current I(t) quantum mechanically, and take this as an input for eq. (3.1). To do so we first define the operator for the one-sided current as follows :

j+(x)da

-

l mhfd3kfd3k,lk)½(k+k,)dae_i~k_k,~x(k,i. (2g)3

(3.3)

(k + k'Kta > 0

It counts the number of particles which are passing in one direction through an element da per time unit. Ik) and [k') are momentum eigenstates, and x is the position the current relates to. A detailed derivation is given in the appendix. The so-defined current is a hermitian operator which shows all expected properties. Especially the classical limit is contained, as can be seen by expressing the expectation value (,j+ (x)da) in terms of the Wigner function f(x, k) l-refs. Iv 19)]. With p(1) being the quantum mechanical one-particle density, f(x, k) is defined as

f(x, k) -

1 f d3s eiksp~l~(X+½S, X--½S).

(2/r)3

(3.4)

After some transformation we get

d3kkda.f(x, k).

(,j + (x)da) ---

(3.5)

m dkda > 0

Since the Wigner function corresponds to the classical one-particle phase-space density this is just what one would expect. In the one-dimensional case (.j+ (x)da) reduces to

(J+(x))=hfk

>odkkf(x'k)"

(3.6)

With definition (3.6) we are able to link the classical model with our quantummechanical calculation. We solve the time-dependent many-body Schr6dinger equation as before and calculate (j+(x = ½L))(t) at the border between the two systems. Inserting this into (3.1) for I(t) we get the classical a 2 appropriate to our system. Fig. 7 shows a typical result. We considered the symmetric system with the real gas (see fig. 4). In all examples the variance ac~ 2 for distinguishable particles exceeded cr2 for indistinguishable particles by a factor of 5 to 10, regardless of the special interaction.

P. Buck, H. Feldmeier / Two interacting fermion gases

02

3.0

-~

i

,

i

,

i

,

343

i

2.5 2

Ocl

2.0 1.5 1.0 02

0.5 0 g(t)

:

'of

: I ; I

0

10

20

30

I

40

50

60

t (10-~2S)

Fig. 7. Mass variance for distinguishable particles (a20 and for indistinguishable particles (tr2) for a symmetric 10-particlesystem(realgas). This remarkable result is a consequence of the Pauli exclusion principle and the fact that the particles are enclosed in a mean potential. To explain this we imagine the state of the system Iq'(t)) represented in eigenstates [i, nL) of N L which are Slater determinants, built out of the left-right states (cf. fig. 3), A

I~u(t))=

~C,,,L(t)li, nL).

~ nL=0

(3.7)

i

A great variance in mass number implies a wide distribution in the occupation probability of states with different particle numbers on the left. For a symmetric system the variance is given by (3.8) where ci,,~(t) are the expansion coefficients of the actual wave function, A

cra(t) =

~

~

nL=0

i

Ici,.L(t)12(nL--½A) 2.

(3.8)

Due to the Pauli principle states with n L if: ½A must have on the average (n L -½A) particle-hole excitations compared to states with n L = ½A. They possess therefore also a higher energy and their occupation is suppressed. This is illustrated schematically in the upper part of fig. 8. In the case where the particles are not forced to move within one mean potential but rather each component li, nL) of the wave function is allowed to establish its own selfconsistent potential, states with n L > ½A

344

P. Buck, H. Feldmeier / Two interactinyfermion 9ases

or_

"-

-_

_~

Fig. 8. Schematic picture to illustrate in the upper part the suppression of large mass dispersions by a mean potential. The lower part shows the picture in which different exit channels possess different single-particle potentials.

would have a wider potential and thus their energy would not be much higher than for the symmetric states (lower part of fig. 8). This picture corresponds to the one where the liquid-drop energy-surface of two nuclei for different masses and charges governs the nuclid distribution in heavy-ion collisions2°-23). On the other hand our model calculation is very similar to the time-dependent Hartree-Fock model (TDHF). In both, one mean potential determines the evolution of the system. Although this is selfconsistent in the T D H F model, it reflects only the mean evolution. In a symmetric collision, for example, it has to stay symmetric, and is thus smothering all fluctuations which otherwise would be built up. The reason is that an excess of particles on one side means in the symmetric potential a high excitation energy, very much the same as in our model. States with n L ~ ½A feel the mean potential suited for states with nL = ½A and are thus strongly influenced by what Griffin called spurious cross-channel correlations 24). This behaviour will not be changed essentially by inclusion of two-body collisions as we have seen by comparing the real and the ideal gas in our model. The collision terms will increase dissipation and equilibration in the system but will not yield a significant increase in the mass spread. We therefore believe that extended timedependent Hartree-Fock models 3 - 7) will not be able to describe the large fluctuations in the macroscopic variables observed in experiment, as long as they are utilizing one mean selfconsistent single-particle potential which is not a fluctuating quantity itself. 3.3. P R O D U C T ANSATZ FOR THE TWO-BODY DENSITY

Since we know the full dynamics of our model system we can study explicitly the applicability of different approximations in this model. Here we investigate a product ansatz for the two-body density matrix. Let us define a one-particle basis {1i>} by the creation operators {a~" li> = a~+10>}. Then the one- and two-body density-matrices ,~(1) I~'ik and ,~t2) t i j k l are defined in

P. Buck, H. Feldmeier / Two interactino fermion oases

345

the following way :

pik (= (~]a~ail~), 1 )

(3.9) (3.10)

ijkl

with ]~ ) being the state vector of the system. If l~) can be written as a single Slater determinant, the two-body density and all higher density matrices are completely determined by p~). In this case the matrix elements of p~2~are antisymmetrized products of pc~).

•(2) ijkl :

~(1),~(1) lUik Pjl

,~(1) rdl)

P'il Pjk "

(3.11)

The question arises if the product ansatz (3.11) can also be regarded as an approximation for a system which can not be described by a single Slater determinant. This ansatz has been used to calculate variances in heavy-ion collisions 13). To study the approximation in our model we calculate the one-particle density from the known many-body wave function and insert it into (3.11) to get the approximated two-particle density l~(2). From this we calculate (N2L)p by = tr(fi~2)N2),

(N2)p

(3.12)

where the trace has to be carried out with the two-particle states. ¢r2 is then given by 2

/N2\

O'p ~ N

L/p--\

/N

N2

L/p,

(3.13)

where (NL) p is exact, because it is calculated with the exact one-particle density. For the ideal gas ~r2 is exact if we start with a single Slater determinant, since the system evolves by a one-particle hamiltonian. For the real gas a typical result is shown in fig. 9. As in subsect. 3.2 the figure relates to a symmetric 10-particle system. O'p2 is not a good approximation. It exceeds ~r2 up to a factor of two. This has been a common feature of several examples. Therefore the product ansatz (3.11) seems not to be useful for the description of mass variances in heavy-ion reactions. We guess that it is this enhancement effect of the approximation which leads in the work of Sch6ne et a1.13) to values of a 2 which are of the same order of magnitude as the experimental data. As has been pointed out by Wichmann 25) the product ansatz (3.11) corresponds to a two-particle density of a grand-canonical ensemble. This means that the particle number A of the whole system fluctuates. Only the mean number is fixed. Therefore the fluctuation of A is superposed on the fluctuation of NL. Even if the system has

346

P. Buck, H. Feldmeier / Two interactinyjermion gases 1.5

.

,

•

,

.

,

.

,

,

,

I'01-

.

2 ~ Op

g(t)

O" I

:

J I

, I

, I

/

0 0

10

20

30

40

50

60

t (10-22s) Fig. 9. a~ calculated with a product ansatz for the two-body density in comparison with the exact

a 2 for the real gas.

sharp particle number on one side the product ansatz may therefore yield a nonvanishing variance. This effect is indicated in fig. 9 at time t = 0 but is even more distinct if one starts with an excited state. It is worst if (NL) ~ ½A as in the case of heavy-ion collisions. The grand-canonical ensemble may of course be used for small subsystems with (NL) << A.

4. Summary Although the model we studied is oversimplified with respect to the restriction to one spatial dimension and the use of an external mean potential instead of a selfconsistent one, it provided valuable insight into the microscopic time evolution of two penetrating fermion gases. The consequences of the Pauli principle and the effects of two-body collisions could be especially investigated. We could show that the Pauli principle together with the mean potential reduces drastically the width of the mass distribution if compared to the classical one. The sometimes used approximation of replacing the unknown two-body density by an antisymmetrized product of one-body matrix-elements turned out to be useless for calculating variances if the state is not exactly a single Slater determinant. A deeper reason is that the product ansatz includes a fluctuation in the total mass number which is for two equal systems of about the same size as the fluctuation in the relative mass number. Studying the differences in the dynamics between the real and the ideal gas we saw that the collision terms decrease essentially the equilibration time for the mean value of the mass distribution but increase only slightly its width. The small mass variances are in contrast to measured heavy-ion distributions but are in accord with results from time-dependent Hartree-Fock calculations. More generally speaking,

P. Buck, H. Feldmeier / Two interacting fermion gases

347

we expect that the commonly adopted view of heavy ions as two interacting fermion gases enclosed in a mean single-particle potential cannot account for the large measured mass widths (of the order of ¼A). We even believe that our model calculations indicate that any model which works with one selfconsistent mean potential cannot reproduce the essential characteristics of large fluctuations in heavyion collisions. The inclusion of short-ranged collision terms will contribute to the dissipation and help to equilibrate but will not lead to a fluctuation of the selfconsistent potential about its mean. These long-range correlations, however, are needed to explain the measured data. We acknowledge helpful discussions with Prof. Dr. F. Beck.

Appendix To define the quantum mechanical operator for the one-sided current (3.3) we start with the operator for the full current through a surface element da at position ~ : j(~)da = ~m {p6(x - ¢) + 6(x - ¢)p}da.

(A.1)

With p = hk and ( x l k ) = (2~)-:~e i k ' x this reads in the momentum representation

j(¢~da-

1 h fdak fd3k,lk~½(k+k,)dae_i(k_k,).¢(k,i.

(2~) 3 m

(A.2)

We want to define a current j+(¢)da which relates to particles which traverse da in positive direction and a j_(¢)da for particles which traverse da in the negative direction. Then j÷ (¢)da and j _ (¢)da are connected by the relation j(~,)da = j + (¢)da + j _ (¢)da,

(A.3)

wherej(¢)da is the net current through da. Requiring j+(¢)da and j_(¢)da to be hermitian we are thus lead to the following definition 26) : 1 mhf d3 k f d 3 k , l k ) ½ ( k + k , ) d a e _ , ( k _ k , ) . ¢ ( k , [ . j + (~,)da - (2/t)3

(A.4)

(k +k'kia > 0

For j_(~)da the integration has to be carried out over the remaining hemisphere

P. Buck, H. Feldmeier/ Two interactinoJermion gases"

348 with

(k + k')da

< 0. F o r t h e m a t r i x e l e m e n t s (~02[j + (~)dalq)~) o n e gets

( q o 2 [ / + ( ¢ ) d a l q h ) = ½(qo2~(¢)dal~p, )

mh 2rtl ~ q)~(¢-½x)tpl(¢ + ½x)-q)~(¢)¢,(~ dxda.

(A.5)

H e r e the s e c o n d t e r m is a n o n l o c a l p r i n c i p a l - v a l u e integral. F o r o u r o n e - d i m e n s i o n a l s y s t e m (A.4) a n d (A.5) b e c o m e

j+(Y.) --

m 2~

dk

dk'[k)½(k +k')e-itk-k')~(,k'[, (A.6)

k+k'>O (¢p21J+(~)ltpx)

= y(¢p2[J()lq0,) ' 4

h 1 t~qo~(~-½x)tp,(~+½x)-qo~(~)tpl(()dx ' m 2n lt~

(A.7)

x2

respectively.

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