Statistical Properties of Many Particle Eigenfunctions
Eric J. Heller1, 2, ∗ and Brian R. Landry2
Department of Physics, Harvard University, Cambridge, MA 02138
Department of Chemistry and Chemical Biology, Harvard University, Cambridge, MA 02138
(Dated: November 12, 2018)
Wavefunction correlations and density matrices for few or many particles are derived from the
properties of semiclassical energy Green functions. Universal features of fixed energy (microcanoni-
cal) random wavefunction correlation functions appear which reflect the emergence of the canonical
ensemble as N → ∞. This arises through a little known asymptotic limit of Bessel functions.
Constraints due to symmetries, boundaries, and collisions between particles can be included.
I. INTRODUCTION
The standard tools of quantum chaos investigations include random matrix theory and periodic orbit theory
(Gutzwiller trace formula), the Van Vleck-Morette-Gutzwiller propagator, and many techniques and phenomena
derived from these approaches. Standing somewhat to the side as an inspired insight is Berry’s conjecture, which
loosely stated is the idea that as ~ → 0 eigenstates will be indistinguishable from superpositions of infinitely many
(local) plane waves with random amplitude, direction, and phase, but with fixed wavelength appropriate to the local
kinetic energy. In two dimensions, these assumptions result in strictly Gaussian statistics of the eigenfunctions and
the autocorrelation function 〈ψ∗(~x)ψ(~x+ ~R)〉 = J0(ka) where k is the local wavenumber and |~R| = a.
The Berry random plane wave (RPW)[1] hypothesis is free of any specific dynamical information, except fixed
total energy, which defines the “ensemble” (i.e. microcanonical). The perspective developed here suggests that by
extending the RPW hypothesis we can conveniently accommodate many other constraints, incorporating information
about real systems. In fact this program has already begun, with Berry’s inclusion of the presence of nearby hard
walls[2], and Bies and Heller’s soft boundary results[3], and multiple hard walls[4]. Related work by Urbina and
Richter[5] and one of us [6] may also be viewed in this light.
The idea of random waves subject to constraints is not confined to one particle in two dimensions. Indeed Berry
gave the N - dimensional formula for free particles in his 1977 paper[7]. Since the underlying idea in the RPW
hypothesis is uniform randomness within a quantum context, i.e. the underpinning of quantum statistical mechanics,
we must encounter some familiar territory as the RPW hypothesis is extended to the large N limit. In 1994, Srednicki
had suggested that the Berry random wave hypothesis was indeed a foundation for quantum statistical mechanics[8],
and showed that the appropriate canonical ensemble was reached for large N , depending on particle statistics. The
present paper shows more specifically what happens as the number of particles increases, through a nonstandard
and apparently unpublished asymptotic form for Bessel functions (we have not been able to find it in the literature,
although it “ought” to be there), which encodes the equivalence of the canonical and microcanonical ensembles of
statistical mechanics. In making the connections to quantum statistical mechanics one also needs procedures for
incorporating constraints, which are an essential aspect of the theory. Thus our procedures for generalizing the RPW
to include constraints, mentioned above, is an essential new feature, since the constrained eigenstates are no longer
random in Berry’s (and Srednicki’s) original sense.
Given a continuum at energy E, such as in an enclosure with walls very far away, we can perform the average over
all random waves as a trace, i.e.
〈ψ∗(~x)ψ(~x′)〉 = Tr [δ(E −H)|~x〉〈~x′| ] , (1)
which immediately yields Berry’s result, apart from normalization which we choose differently here. However a trace
over a basis is independent of any unitary transformation on that basis, so it does not matter whether we use a trace
over a complete set of random waves or simple local plane waves; both give J0(ka) for the case of one free particle in
two dimensions. In this way the imaginary part of the retarded Green’s function − 1
Im [G+(E)] = δ(E−H) becomes
central, formally convenient, and equivalent to Berry’s RPW hypothesis.
∗Electronic address: heller@physics.harvard.edu
http://arxiv.org/abs/0704.1881v1
mailto:heller@physics.harvard.edu 
II. PRELIMINARIES
We begin by reviewing well known formalism to establish context and notation. The Green function completely
characterizes a quantum system, whether it is interacting or not, or has few or many degrees of freedom. The retarded
Green function G+, i.e.
G+ = P
− iπδ(E −H), (2)
where P stands for the principal value of the integral, is the basis for wavefunction statistics and density matrix
information, through the follow relations, with a convenient choice of normalization:
< ψ(x)ψ∗(x′) > = − 1
Im〈x|G+|x′〉/ρ(E) (3)
= 〈x|δ(E −H)|x′〉/ρ(E) (4)
where
ρ(E) = Tr[δ(E −H)] (5)
and where < · · · > stands for the average over the degeneracies. We take these degeneracies to be of dimension up
to ND − 1, where N is the number of particles and D the spatial dimension each particle lives in. (We use boldface
notation, e.g. x for the N ∗ D degrees of freedom.) If true degeneracies do not exist in a particular system, we
can artificially open the system up to a continuum. For example, a two dimensional closed billiard does not have a
degeneracy, but it acquires one if we open a hole in it and let it communicate with the outside unbounded 2D space.
Of course this changes the billiard properties, and the size of the hole might be problematic, but in fact we shall
never really have to open a system up in this way. The quantity δ(E−H) then implies the average over all scattering
wavefunctions at fixed energy E.
There are other interpretations which can be put on the average correlation < ψ(x)ψ∗(x′) >; for example we can
imagine a large number of potentals which differ in some far away place, and in a way so as to all have an eigenvalue at
a particular energy. Then, the average has the interpretation of the average over this “disorder” ensemble. A slightly
different procedure is advocated by Richter et. al., wherein an energy average is taken[5]. Another interpretation can
be applied to individual eigenstates in a closed system, assuming they are at least locally uniform in their properties,
by taking the average over different points of origin x. This is particularly appropriate when the analogous classical
system is chaotic, as mentioned above [1]. We will be evaluating the Green functions semiclassically in what follows,
restricting the time over which the contributing trajectories propagate.
The wavefunction correlation is equal to the coordinate space matrix element of the constant energy density matrix:
< ψ(x)ψ∗(x′) >= 〈x|δ(E −H)|x′〉/ρ(E) = ρ(x,x′, E) (6)
Reduced density matrices can also be derived from wavefunction correlations ; e.g.
ρ̃(~x1, ~x
1, E) =
d~x2d~x3 · · · d~xN ρ(~x1, ~x2, · · · ; ~x′1, ~x2, · · · ;E), (7)
the one particle reduced density matrix.
We can approach the correlations via Fourier transform from the time domain, since
δ(E −H) = 1
eiEt/~e−iHt/~ dt. (8)
Thus the statistics, density matrices and correlations are derivable without further averaging by knowing the time
propagator.
In the following, we define the Green function propagator G(x,x′, t) and the retarded Green function propagator
G+(x,x′, t) as
G(x,x′, t) = 〈x|e−iHt/~|x′〉
G+(x,x′, t) =
Θ(t)〈x|e−iHt/~|x′〉 (9)
where Θ(t) is the Heavyside step function Θ(t) = 0, t < 0, Θ(t) = 1, t > 0. It is very rewarding to expand the
propagator in semiclassical terms, involving short time (zero length) and longer trajectories. We take Gdirect(x,x +
r, t) = 〈x| exp[−iHt/~]|x+r〉, the very short time semiclassical propagator, which for N particles each in D dimensions
reads
Gdirect(x,x + r, t) ≈
2πi~t
)ND/2
2/2~t−iV (x+ r
)t/~ (10)
where r2 = |r|2.
It is not difficult to cast the Fourier transform of this short time version to fit the definition of a Hankel function,
G+cl(x,x + r, E) =
2πi~t
)ND/2
2/2~t−iV (x+ r
)t/~eiEt/~ dt = − im
d (kr) (11)
where d = ND/2−1, k = k(x+r/2, E) andH(1)d (kr) = Jd(kr)+iNd(kr) is the Hankel function of order d, and Jd is the
regular Bessel function of order d. The wavevector k varies with the local potential, i.e. ~2k(x, E)2/2m = E − V (x).
Here, using only the extreme short time version of the propagator, we must suppose r is not large compared to
significant changes in the potential, but this restriction can be removed by using the full semiclassical propagator
rather than the short time version. For the case of one particle in two dimensions, d = 0, and we recover Berry’s
original result for one particle in 2D, 〈ψ∗(~x)ψ(~x + ~r)〉 ∝ J0(kr).
According to the short time approximation, for any N ,
< ψ(x)ψ∗(x+ r) >≈ −
G+cl(x,x + r, E)
Jd(kr) (12)
where k = k(x, E). This result includes interparticle correlations through the potential V (x) and the spatial depen-
dence of k = k(x, E); the diagonal r = 0 limit (following section) is equivalent to classical statistical mechanics. The
implications of this for the nondiagonal short time Green’s function are intriguing. The way r is defined, it does not
matter whether one particle is off diagonal (xi 6= xi′ ) or several or all of them. For given r, the Green’s function will
be the same, apart from changes in the potential V (x + r/2).
It is interesting that although the short time Green function is manifestly semiclassical, the energy form, e.g. Eq. 12
is obtained by exact Fourier transform of the semiclassical propagator, rather than by stationary phase.
III. DIAGONAL LIMIT
The diagonal (r → 0) N body Green function is obtained using the asymptotic form
Jd(kr) =
Γ(d+ 1)
we obtain
G+cl(x,x, E)
Γ(d+ 1)
where the second form uses Stirling’s approximation, n! ∼ nne−n
2πn, and is appropriate below when we consider
large N . We note that this behaves as k2d ∼ (E−V (~x))d. This factor is familiar from the computation of the classical
density of states. Tracing over all ~x results in
Γ(d+ 1)
δ(E −Hcl(p,x)) = ρcl(E) (15)
i.e. the classical density of states. The association of the short time propagator with the classical Hamiltonian and
classical density of states is well known. The Berry RPW hypothesis, the short time propagator, and the classical or
Weyl (sometimes called Thomas-Fermi) term in the quantum density of states are all closely related.
The quantum spacial integral is over all coordinates, so how does the classical partition function emerge if the
classical integral is only over classically allowed coordinates? For forbidden positions, k is imaginary and can be
written as say iκ. An identity for Hankel functions can then be used (in+1H
n (ix) =
Kn(x)) to show that the
green function is real so that the imaginary part is zero, explaining why the integral is only over classically allowed
positions.
As long as r = 0 (i.e. diagonal Green’s function) the results obtained within the short time propagator approximation
for any quantity in the presence of a potential (including interparticle potentials such as atom-atom interactions) will
be purely classical. Since we will be discussing the equivalence of the results from the different ensembles for r 6= 0,
it is useful to recall how the classical coordinate space densities in the different ensembles can be shown to coincide
since this corresponds to the r = 0 case.
The normalized phase space density in the microcanonical ensemble and the phase space density in the canonical
ensemble are given by
ρcl(p,x, E) =
ρcl(E)
δ(E −Hcl(p,x)) (16)
ρcl(p,x, β) =
Qcl(β)
e−βHcl(p,x) (17)
respectively. The density of states and partition function are of course the normalization factors so that
ρcl(E) =
dxdp δ(E −Hcl(p,x)) (18)
Qcl(β) =
dxdp e−βHcl(p,x) (19)
Integrating each phase space density over momentum space allows us to compare the coordinate space densities:
ρcl(x, E) =
dx p2d
ρcl(x, β) =
e−βV (x)
dx e−βV (x)
with p =
2m(E − V (x)).
Using the relationship between E and β, E − 〈V 〉 = ND
, where 〈V 〉 is the ensemble average of the potential in one
of the statistical ensembles, the coordinate space density becomes
p2d = (2m(d+ 1)/β)d
(〈V 〉 − V (x)) β
In the limit N → ∞ ( d→ ∞) this is
p2d = (2m(d+ 1)/β)de(〈V 〉−V (x))β (23)
dx p2d
e−V (x)β
dx e−V (x)β
This is one of the standard ways of establishing a connection between the ensembles[9].
Since the diagonal Green’s function gives classical results we can use it to study classical properties. For example,
we can inquire about the average two particle spacing distribution ρE(r12) or the probability density for a single
particle PE(~x1) starting with the short time semiclassical Green’s function and the results will coincide with classical
microcanonical statistical mechanics. This statement holds for all N . Similarly, in the large N limit the canonical
ensemble results for these quantities must emerge. This point becomes more interesting for the non-diagonal case,
considered next.
IV. LINK TO THE CANONICAL ENSEMBLE
A. Bessel functions become Gaussians
As yet we have found nothing too surprising or useful beyond standard classical statistical mechanics. This changes
when we consider the large N limit for the non-diagonal Green’s function, r 6= 0. Taking the large N limit of Eq. 12,
2 4 6 8
2 4 6 8
2 4 6 8
2 4 6 8
FIG. 1: As N increases, the combination 1
Jd(x), where d = ND/2 − 1, approaches a Gaussian. This is the key link between
the quantum microcanonical and canonical ensembles.
we are confronted with a new question about Bessel functions. The large d limit of Jd(x) is indeed well known, but
this is not yet sufficient for our purposes. It reads
Jd(kr)
(kr)d
2d Γ(d+ 1)
This is the standard formula given in the usual references. Eq. 25 should be the first term in a power seres for Jd(kr)
in kr. Another standard result is the power series expansion, valid for all d and kr:
Jd(kr) =
(−1)m
m!Γ(m+ d+ 1)
)2m+d
We actually require a different asymptotic result. What make our demands unusual is that, assuming we want the
energy to increase in proportion to the number of particles (appropriate to many applications of the large N limit),
then k ∼
d; this means that for fixed r the combination (kr) is increasing as
d as d → ∞. If the
argument of the Bessel function increases without bound along with it’s order, some new considerations come into
play. We find the desired form using Eq. 26, after summing a series recognized as that of a Gaussian Taylor expansion,
(kr)d
Jd(kr) =
2d d!
−k2r2
4(d+ 1)
2d d!
2r2/(4(d+1)), (27)
where again ~2k2/2m = E − V (x). Note that as d → ∞, the argument of the Gaussian holds fixed because of the
factor of d + 1 in the denominator of that argument. Figure 1 illustrates the convergence to the Gaussian as N
increases. The asymptotic limit in Equation 27 is not in the usual references, although related results have been given
for N-bead polymer random chain end-to-end distributions[10]. The connection between the path integral for the
propagator and polymer chains is well known[11].
It is interesting that a Gaussian emerges from Bessel functions in the large N limit. We can put Eq. 27 together
with Eq. 12 and Eq. 4, and express the result, as N → ∞,
< ψ(x)ψ∗(x+ r) > = ρ(x,x′, E) → 1
2π~2d!
2r2/4(d+1). (28)
For noninteracting particles moving in zero potential but confined to volume V the short time approximation becomes
exact and k is constant. For this system the wavefunction correlation becomes
< ψ(x)ψ∗(x + r) > = ρ(x,x′, E) → 1
2r2/4(d+1). (29)
Something familiar is emerging, here derived in the unfamiliar context of fixed energy (microcanonical ensemble). For
comparison we recall the standard result for the ideal gas at temperature T [12]:
〈x|e−βH |x+ r〉
Tr[e−βH ]
= ρ(x,x′, β) =
2/λ2 (30)
where λ = h/
2πmκT is the thermal wavelength. Indeed for the free particle case, k is fixed by E and 〈K〉 =
D/2NκT = ~2k2/2m, where K is the kinetic energy and κ is Boltzmann’s constant,
2r2/4(d+1) = e−πr
2/λ2 . (31)
The canonical ensemble result for the propagator has “dropped out” of the asymptotic large N limit of a micro-
canonical Green function, at least for noninteracting particles, and an unusual asymptotic form for the Bessel function
has emerged as the link. With some caveats, the statement
δ(E −H) ∼ e−βH (32)
has meaning in the large N limit, where it is understood E grows as N , and a temperature extracted. At a qualitative
level, Eq. 32 merely expresses the known equivalence of the ensembles. In the case of an interaction potential, the
relation between E and temperature is of course problematical.
B. Interacting Particles - Short Time Limit
We can say more about interacting particles using only the short time propagator introduced above. Longer time
events will be discussed in Sec. VI. The short-time approximation to the correlation function for large N , which is
equal to the matrix elements of the density operator in coordinate space using our normalization, (Eq. 28) is given by
ρcl(x,x
′, E) =
2π~2d!
2r2/4(d+1) (33)
with ~k =
2m(E − V (x+x′
)) and r = |x − x′|. Again, the Gaussian form of this expression arises from the
asymptotic limit of the Bessel function. In the interacting case this can again be brought into the same form as the
equivalent expression at constant temperature:
ρcl(x,x
′, β) =
2πβ~2
+V ( x+x
In order to make the connection we must identify the energy with a certain temperature. This relationship between
E and β is
E − 〈V 〉 = ND
where 〈V 〉 is the ensemble average of the potential in one of the statistical ensembles. Using this relationship in Eq. 33
gives
ρcl(x,x
′, E) =
2π~2d!
2~2β e
m(〈V 〉−V )r2
2~2(d+1) (36)
In order for Eq. 36 to be equivalent to Eq. 34 the term with 〈V 〉−V must be negligible. This is true for configurations
of particles which possess the typical (and vastly most probable) sum total kinetic energy for all the particles. Since
the typical total kinetic energy is by far the most probable, nearly all points in configuration space lead to small
values of 〈V 〉 − V , and that term is negligible almost always. The remaining terms in Eq. 36 and Eq. 34 are shown
to be the same by the equivalence of the classical ensembles as shown in Sec. III.
It is also telling to trace over the coordinates of all but one of the interacting particles, given by a coordinate ~y.
We thus seek the reduced density matrix, diagonal or off diagonal in ~y. The trace will over many coordinates be
overwhelmingly dominated (in the large N limit) by the most probable total kinetic energy for all the particles. Then
we find
G(~y, ~y′, β) ∼ λ−3N−2e−πr
2/λ2 (37)
where r2 = |~y − ~y′|2 and λ = h/
2πmκT . Thus the quantum mechanical single particle Green function and density
matrix make sense as their imaginary time counterparts in the N → ∞ limit, in accordance with well known results
for the canonical ensemble.
C. Large N limit and Boltzmann averaged Green functions
Even though it is a necessary consequence of the equivalence of the ensembles, it is interesting to establish the
generality of the Boltzmann average over the energy of a noninteracting subsystem in the following way. Suppose
N −M particles are no longer interacting with the remaining M particles, but their states are correlated by having
been in contact in the past with the total energy fixed at E. In the time domain and in an obvious notation we have
G+N (y, z;y
′, z′, t) = i~ G+N−M (y,y
′, t)G+M (z, z
′, t) (38)
Then the Fourier convolution theorem can be applied to the Fourier transform into the energy domain, i.e.
G+N (y, z;y
′, z′, E) =
G+N−M (y,y
′, E − E′)G+M (z, z
′, E′) dE′ (39)
which incidentally leads to some rather unlikely looking identities for Bessel functions; the reader may easily generate
them. Our purpose is served if, focussing on the subsystem of M particles, we trace over the N −M y coordinates.
This gives
Try[G
N−M (E − E
′)] ∼ lim
Γ(dN−M + 1)
)dN−M
Γ(dN−M )
πdN−M+1|y′ − y|2dN−M
times a volume factor, in the case of an ideal gas. The second term is not a function of E′. Therefore the integral of
it times GM (z, z
′, E) is proportional to δ(z′ − z). So long as z 6= z′ that term is zero. Neglecting all unimportant (for
this argument) factors this leaves
Try[G
N−M (E − E
′)] ∝ (E − E′)dN−M = EdN−M
)dN−M
∼ EdN−M e−βE
with of course β = 1/κT . In arriving at Eq. 41 we used E = D
NκT for the case of particles embedded in D
dimensions. Finally we arrive at
Try[G
N (E)] ∝
G+M (z, z
′, E′) dE′ = G+M (z, z
′, β) (42)
in the large N limit. This establishes the generality of the Boltzmann average over the subsystem energy for large N .
This discussion establishes again the connection between the canonical and microcanonical ensembles, however in a
way not involving the Bessel functions and their asymptotic form, so it is less general than other results in this paper
valid for any N .
D. Stationary phase canonical limit
It is also possible to recover the Gaussian form in Eq. 28 by carrying out the integral in Eq. 11 by stationary phase,
provided the real factor involving t in the denominator is taken into the exponent, as −ND/2 log t i.e.
G+cl(x,x+ r, E) =
)ND/2
2/2~t−iV (x+ r
)t/~+iEt/~−ND/2 log t dt. (43)
The complex stationary phase point t∗ in the large N limit becomes t∗ = −iND~/(2(E−V )) , yielding the same result
as in Eq. 28, with ~2k(x, E)2/2m = E − V (x), and making this another route between the quantum microcanonical
and canonical ensembles. Since the positions are arbitrary we cannot however identify the average kinetic energy with
E − V , and thus without further averaging we cannot associate t∗ with any inverse temperature. It is interesting
nonetheless that there is a complex time t∗ appropriate to every position x, even if that time is not related to the
temperature. For an ideal gas the stationary phase time is t∗ = −i~/κT = −iβ~, after making the identification
E = ND/2kT . A discussion about traces over most of the coordinates and the recovery of the usual temperature
through 〈K〉 = D/2NkT proceeds as in Sec. IVB.
V. CONSTRAINTS
In the large N limit the ergodic hypothesis is strongly motivated, but statistical mechanics does not pre-suppose
that ergodicity is unchecked; rather constraints are always present, such as walls and boundaries which control volume.
Ergodicity is then defined with respect to these constraints. The guiding idea in this paper, i.e. the extended Berry
RPW hypothesis, is that eigenstates of the full system are “as random as possible, subject to prior constraints”. In
this way thermodynamic constraints arise naturally. The real time, real energy (microcanonical ) semiclassical Green
function approach not only automatically generates the averages required to get appropriate wavefunction statistics,
it also provides a natural way to include many constraints such as walls, symmetries, and even the existence of
collisions between particles by going beyond the short time limit term to include returning (not necessarily periodic)
trajectories. The semiclassical Ansatz for these extended problems in the presence of constraints is
G(x,x′, t) ≈ Gdirect(x,x′, t) +
Gj(x,x
′, t) (44)
where Gj(x,x + r, t) is a semiclassical (Van Vleck-Morette-Gutzwiller) Green function,
Gj(x,x
′; t) =
)ND/2 ∣
∂2Sj(x,x
′; t)
∂x∂x′
iSj(x,x
′; t)/~−
corresponding to the jth trajectory contributing to the path from x to x+r, and Gdirect(x,x+r, t) is given by Eq. 10.
The short time term Gdirect(x,x+ r, t), is singled out as the shortest contributing trajectory: supposing r to be small
compared to distances to walls etc., we still have a short time, ballistic trajectory as quite distinct from trajectories
which have traveled some distance away and come back. There are cases where this separation is not clean; for such
cases we can adjust notation accordingly. Note that since a trace over all position is not being taken, there is no
appearance semiclassically of periodic orbits as the only surviving contributors. “Closed” orbits however can play a
large role semiclassically, a fact recognized long ago by Delos[13].
A. N particles and a wall
A very useful example is provided by a plane Dirichlet wall felt by all the particles (e.g. ψ(~x1, ~x2, · · ·~xN ) = 0 for
yi = 0, i = 1, · · ·N), as in a gas confined by a rigid container. The Green function and eigenfunctions must vanish
if one or more particles approaches this wall. We can use the method of images, generalized to N particles, if the
particles are noninteracting. (The interacting case can in principle be handled by semiclassical trajectory techniques
which we bring up in the next section.)
The Green function Gwall(x,x
′) will consist of the shortest distance contribution for which all particles take a direct
path from x to x′, plus paths where one particle has bounced off the wall, paths where two particles have, etc. These
histories are included automatically if we apply the symmetrization operator which imposes the image reflections.
This operator can be written
(1 −Ri) = 1−
RiRj − · · · (46)
where Ri is the operator for reflection about the y = 0 axis for the i
th particle. Applied to the Green function
G(x,x + r, t), considered as a function of the coordinates in x in the absence of the wall, R yields the series
Gwall(x,x
′, t) = Gdirect(x,x
′, t)−
Gi(x,x
′, t) +
Gij(x,x
′, t)− · · · (47)
where Gi(x,x
′, t) corresponds to the ith particle getting from ~xi to ~x
i by bouncing off the wall while the others
take direct paths, etc. The Fourier transform gives an analogous equation for Gwall(x,x
′, E). The effect of the
symmetrization is to create Green function sources reflected across the wall and given proper sign, in the manner
familiar from the method of images. The short time path is shown by the direct path solid line in Fig 2, corresponding
to the term Gst(x,x
′, t). The bounce path is equivalent to a source reflected across the wall with an opposite sign,
i.e. the method of images. Define
G+st(x,x+ r, E)
Jd(kr)
(kr)d
≡ a(k)Fd(kr) (48)
FIG. 2: A short and a bouncing path for a particle propagating near a wall. The bounce contribution, if viewed by the image
method, is equivalent to a contribution of opposite sign coming from the reflected point ~xR with the wall removed.
G+wall(x,x
′, E)
= a(k)
Fd(kr)−
Fd(kri) +
Fd(krij)− · · ·
 . (49)
This is the general result for any N . It would appear to be difficult to take it further, since all the distances, e.g.
rij =
m 6=i,j
|~xm − ~x′m|2 + |~xRi − ~x′i|2 + |~xRj − ~x′j |2, (50)
where ~xRj is the reflected j
th particle coordinates, involve square roots. However if we use the large N asymptotic
form, we find, using Fd(kr) → exp[−k2r2/4(d+ 1)]/2dd!,
Im [Gwall(x,x
′, E)] =
i − e−γ(r
1− e−γ∆
where γ = k2/4(d + 1) = π/λ2 and ∆2i = (r
2 − r2i . Since ri is the “direct” distance from ~xi to ~x′i, (see Fig 2),
∆2i records the distance change upon reflection of the i
th particle. We note that ∆2i (and thus the Green function)
vanishes as any particle approaches a wall in either x or x′. It is also simple to see that the single particle density
ρ(~x) in this noninteracting case becomes, for large N ,
ρ(~x) = ρ0(1− e−4γx
) (52)
where x is the distance to the wall and ρ0 is the density far from the wall.
The formulas Eq. 49 and Eq. 51 generalize Berry’s result[14] for the wavefunction squared of one particle in two
dimensions near a wall, namely
〈|ψ(~x)|2〉 =
1− J0(k|~xR − ~x|)
d~x (1− J0(k|~xR − ~x|))
. (53)
The Gaussian we get for large N has a very simple interpretation. First we note that for noninteracting systems in
the canonical ensemble we can write the total density matrix as a product of one particle density matrices. This is
essentially the form of Eq. 51, since we can write each one particle density matrix as
ρ(~x, ~x′, β) = e−γ|~x−~x
′|2/N
1− e−γ(|~xR−~x′|2−|~x−~x′|2)
1− e−γ|~xR−~x|2
1− e−γ|~xR−~x|2
1− e−γ|~xR−~x|2
) (54)
FIG. 3: The particle symmetry or antisymmetry condition is equivalent to requiring mirror symmetry or antisymmetry across
the ~xi = ~xj (hyper)plane. This corresponds to having additional contributions from the images of the particles reflected over
the symmetry planes.
where the second form is the diagonal element. However Eq. 54 also arises as the density matrix obtained from the
Boltzmann average of Berry’s result; i.e. averaging the fixed energy results over a canonical distribution of energies,
as can be seen from the integral
1− J0(k|~xR − ~x|)
2k2/2mdk
k e−β~
2k2/2mdk
1− e−m|~x
R−~x|2/2β~2
For D = 2 and N = 1 a Boltzmann average yields the Gaussian. Indeed this necessarily holds in any number
of dimensions; i.e. the appropriate Boltzmann average of Jd(kr)/(kr)
d must yield a Gaussian for any d. In the
thermodynamic N → ∞ limit for noninteracting particles, each particle separately is Boltzmann distributed over
energy, so the result must be the same as a Boltzmann average of the one particle results for any dimension D and
for any constraints.
B. Symmetries - Fermions and Bosons
Particle symmetry is an essential part of the many body problem. It’s effect, like other symmetries, is to generate
permutations where the distances have changed due to particle exchange. Figure 3 shows this effect graphically. It is
gratifying to see directly that permutations which induce large new distances (coming from remote pairs of particles,
where “remote” is a relative term depending on the temperature) make little contribution. Consider N noninteracting
Fermions or Bosons; we wish to compute the reduced density matrix for two Fermions or Bosons. This is a well known
result for N → ∞[12]. The symmetric or antisymmetric Green function is
GS/A(x,x+ r, E) =
Hd(krn)
(kr)d
where rn =
| ~x1 − ~xp1
′|2 + · · ·+ | ~xN − ~xpN
′|2, {p1, · · · , pN} is the nth permutation of {1, · · · , N}, and ǫn = 1 if the
parity of the permutation is even and ǫn = ±1 if the parity of the permutation is odd (with the upper sign for bosons
and the lower sign for fermions).
〈ψ∗( ~x1 · · · ~xN )ψ( ~x1 · · · ~xN )〉 = −
GS/A(x,x+ r, E)
ρ(E)N !
Jd(krn)
(kr)d
In the limit that N is large, this becomes
〈ψ∗( ~x1 · · · ~xN )ψ( ~x1 · · · ~xN )〉 =
ρ(E)N !
2π~2d!
2r2n/4(d+1) (58)
The diagonal component of this with the rn’s written out explicitly is
〈ψ∗( ~x1 · · · ~xN )ψ( ~x1 · · · ~xN )〉 =
2ρ(E)N !π~2d!
)d N !
−k2(~x1−~xp1)
2/4(d+1) · · · e−k
2(~xN−~xpN)
2/4(d+1) (59)
Up to the normalization constant this is the constant temperature density matrix for N noninteracting fermions or
bosons:
〈ψ∗( ~x1 · · · ~xN )ψ( ~x1 · · · ~xN )〉 =
2ρ(E)N !π~2d!
)d N !
−m(~x1−~xp1)
2/2β~2 · · · e−m(~xN−~xpN )
2/2β~2 (60)
Again the identification E = D
NκT was used. This can be rewritten as an integral over wavevectors:
〈|ψ(x)|2〉 = A
d~k1 · · · d~kN e−β~
2/2m+i~k1·(~x1−~xp1) · · · e−β~
2/2m+i~kN ·(~xN−~xpN ) (61)
where A = m
2ρ(E)N !π~2d!
is the normalization constant. Rearranging gives
〈|ψ(x)|2〉 = A
d~k1 · · · d~kN e−β~
2+···+kN
2)/mei(
~k1−~kp1)·~x1 · · · ei(~kN−~kpN )·~xN (62)
If the volume that the particles are confined to is large but finite,
〈|ψ(x)|2〉d~x3...d~xN = AV N−2
d~k e−β~
2/2mei(
~k1−~kp1)·~x1ei(
~k2−~kp2)·~x2δ~k3,~kp3 · · · δ~kN ,~kpN (63)
For fermions if the wavevector of any two particles are the same the term is killed by the term with the wavevectors
reversed in accordance with the Pauli principle. This leaves only two terms
〈|ψ(x)|2〉d~x3 · · · d~xN = AV N−2
dk e−β~
2/2mei(
~k1−~kp1)·~x1ei(
~k2−~kp2)·~x2 (64)
For bosons there are also only two types of terms, but each is multiplied by the same factor since like terms are added
together. Either way, carrying out the integral over k,
〈|ψ(x)|2〉d~x3 · · · d~xN =
1± e−m( ~x1− ~x2)2/β~2
d~x1d~x2
1± e−m( ~x1− ~x2)2/β~2
) (65)
This is the well known result for the density of two noninteracting fermions or bosons.
VI. SCATTERING
A hard wall is a potential energy feature which induces a boundary condition, requiring the wavefunction or Green
function to vanish as the wall is approached. Softer potentials do not induce fixed boundary conditions and require
a different treatment. A potential may still however be thought of as a constraint: we consider waves as random as
possible subject to the existence of a potential, be it fixed or interparticle. In practice this means we return to the
Green function formulation used throughout.
Consider a soft repulsive or attractive potential somewhere in a noninteracting gas. Assuming no boundaries,
mutually noninteracting particles can interact with the potential 0 or 1 times. (We assume for simplicity that the
potential is short ranged. Because of the ergodicity assumption inherent to the random wave hypothesis, the presence
of remote walls would actually make no difference.) This circumstance develops along lines very similar to the wall,
except that we cannot use the method of images. It illustrates the use of the full semiclassical propagator within this
formalism.
Eq. 46 and Eq. 47 both hold, with the effect of Ri changed to mean “the i
th particle takes the path from initial
to final coordinates in which it deflects from the potential, if such a path exists classically”. For N particles, there is
a “direct” term in Eq. 47 where no particle interacts with the potential, N terms where one of them does, etc. We
have, in the simple case shown in Fig. 4, and in analogy with Eq. 47,
G(x,x′, t) = Gdirect(x,x
′, t) +
Gbounce,i(x,x
′, t) +
Gbounce,i,j(x,x
′, t) + · · · (66)
with Gdirect(x,x
′, t) given by Eq. 10, and e.g.
Gbounce,i(z,yi, z+ r,y
i, t) ≈
(N−1)D
∂2Si(yi,y
i; t)
∂yi∂y
2/2~t−iV (z+ r
)t/~+iSi(yi,y
;t)/~−
2 (67)
Considering this term where only the ith particle with coordinate yi interacts with the potential, we have N − 1
“spectator” z particles, and the propagator becomes a product of the noninteracting Green function for N − 1
particles and a more complicated Van Vleck semiclassical term for the colliding particle. The noninteracting part
contributes a term (N − 1)D/2 log t in the exponent along with the one particle classical action of the ith particle.
For sufficiently large N , and tracing over the z particles, this factor leads again to the usual time condition t∗ = −iβ~
and a thermal average of the one particle energy Green function under the Fourier transform from time to energy, as
in Equation 42:
G(y,y′, E) ≈ G(y,y′, β) = Gdirect(y,y′, β) +
Gbounce,i(y,y
′, β) +
Gbounce,i,j(y,y
′, β) + · · · (68)
t∗ = −iβ~ becomes the imaginary time over which the action for the y coordinates are evaluated.
VII. CONCLUSION
Starting with Berry’s random plane wave conjecture for chaotic Hamiltonian systems, we have followed it’s impli-
cations for moderate and large numbers of particles N . In the large N limit we have necessarily arrived at some
familiar territory in statistical mechanics. We have adopted a Green function, semiclassical perspective, arriving at a
Gaussian-Bessel function asymptotic result for energy Green functions, providing an analytic connection between the
quantum microcanonical and canonical ensembles. We have extended the incorporation of constraints into the random
wave hypothesis, considering several types of constraints, including walls and interparticle collisions. Indeed the guid-
ing perspective has been to make quantum waves “as random as possible subject to known prior constraints”. This
must ultimately be equivalent to the ergodic hypothesis of quantum statistical mechanics. The nonstandard methods
and perspective used here may possibly lead to new avenues of inquiry, and it is our hope that the semiclassical
approach might permit new ways of treating strongly interacting systems.
The next stage in the development of this approach is to consider short ranged potentials between particles, i.e.
interparticle collisions. The first corrections to the free particle limit involve binary collisions, which can be computed
semiclassically or using a delta potential appropriate to s-wave scatterers. Again the effect of the other particles will
be to provide a thermal reservoir which essentially averages the Green function over a thermal distribution of energies
(if N is sufficiently large). We save this for a future paper, where we hope to examine specific potentials and derive
two particle radial distribution functions.
FIG. 4: A short ballistic and a colliding path both lead to the same final point for a particle propagating near a localized
repulsive potential. The colliding path cannot be treated by the short time approximation; rather, a Van Vleck Green function
is required. In this term, all but the ith particle remain in place.
Acknowledgments We thank Adam Wasserman for helpful discussions, and the National Science Foundation under
grant NSF-CHE- 0073544.
[1] Berry M V 1983 in Chaotic Behaviour of Deterministic Systems ed G Iooss G, R Helleman and R Stora (New York:
North-Holland) p 171
[2] Berry M V 2002 J. Phys. A: Math. Gen. 35 3025
[3] Bies W E and Heller E J 2002 J. Phys. A: Math. Gen. 35 5673
[4] Bies W E, Lepore N, and Heller E J 2003 J. Phys. A: Math. Gen. 36 1605
[5] Urbina J D and Richter K 2003 J. Phys. A: Math. Gen. 36 L495
[6] Heller E J 2006 Mol. Phys. 104 1207
[7] Berry M V 1977 J. Phys. A: Math Gen. 10 2083
[8] Srednicki M 1994 Phys. Rev. E 50 888
[9] Jancel R 1969 Foundations of Classical and Quantum Statistical Mechanics (Oxford: Pergamon)
[10] Kleinert H M 1995 Path Integrals in Quantum Mechanics, Statistics, and Polymer Physics 2nd edition (Singapore: World
Scientific)
[11] Chandler D and Wolynes P G 1981J. Chem. Phys. 74 4078
[12] Pathria R K 1996 Statistical Mechanics 2nd edition (Oxford: Butterworth-Heineman)
[13] Wang D M and Delos J B 2001 Phys. Rev. A 63 043409
[14] Berry M V 2002 J. Phys. A 35 3025
	Introduction
	Preliminaries
	Diagonal limit
	Link to the canonical ensemble
	Bessel functions become Gaussians
	Interacting Particles - Short Time Limit
	Large N limit and Boltzmann averaged Green functions
	Stationary phase canonical limit
	Constraints
	N particles and a wall
	Symmetries - Fermions and Bosons
	Scattering
	Conclusion
	References

  Wavefunction correlations and density matrices for few or many particles are
derived from the properties of semiclassical energy Green functions. Universal
features of fixed energy (microcanonical) random wavefunction correlation
functions appear which reflect the emergence of the canonical ensemble as the
number of particles approaches infinity. This arises through a little known
asymptotic limit of Bessel functions. Constraints due to symmetries,
boundaries, and collisions between particles can be included.

Introduction
	Preliminaries
	Diagonal limit
	Link to the canonical ensemble
	Bessel functions become Gaussians
	Interacting Particles - Short Time Limit
	Large N limit and Boltzmann averaged Green functions
	Stationary phase canonical limit
	Constraints
	N particles and a wall
	Symmetries - Fermions and Bosons
	Scattering
	Conclusion
	References
