Section 4.2 discusses the calculation of the energy spectrum related to
the internal motion of two bosons under highly-elongated confinement,
Eq. 4.7, using a B-spline basis, and the eigenspectrum related to the
internal motion of two bosons under 1D confinement, Eq. 4.9, using
Eqs. 4.11 through (4.13). We now use these essentially exact
eigenenergies to benchmark our FN-MC calculations. Toward this end, we solve the 3D
Schrödinger equation, Eq. 4.18, and the 1D Schrödinger equation,
Eq. 4.19, for various interaction strengths for and
using FN-MC techniques. The resulting MC energies
and
include the
center of mass energy of
. To compare with the
internal eigenenergies discussed in Sec. 4.2, we subtract the center of
mass energy from the FN-MC energies.
For , the lowest-lying gas-like state of the 3D Hamiltonian for the
short-range potential
is the first excited eigenstate. Consequently, we
solve the 3D Schrödinger equation by the FN-MC technique using the trial wave
function
given by Eqs. 2.37 and 2.91. Figure 4.4
shows the elliptical nodal surface of the trial wave function
(solid
lines) together with the essentially exact nodal surface calculated using a
B-spline basis set (symbols; see also Sec. 4.2) for
and
three different scattering lengths,
, and
. Notably, the
semi-axes
along the
coordinate is larger than that along the
coordinate (
), ``opposing'' the shape of the confining potential, for which
the characteristic length along the
coordinate is smaller than that along
the
coordinate (
). Figure 4.4 indicates good agreement
between the essentially exact nodal surface and the parameterization of the nodal
surface by an ellipse for
and
; some discrepancies become
apparent for negative
. Since the FN-MC method results in the exact energy
if the nodal surface of
coincides with the nodal surface of the exact
eigenfunction, comparing the FN-MC energies for two particles with those obtained
from a B-spline basis set calculation provides a direct measure of the quality of
the nodal surface of
. Figure 4.3(c) compares the internal 3D
energy of the lowest-lying gas-like state calculated using a B-spline basis
(diamonds, see Sec. 4.2) with that calculated using the FN-MC technique
(asterisks). The agreement between these two sets of energies is -- within the
statistical uncertainty -- excellent for all scattering lengths
considered. We conclude that our parameterization of the two-body nodal surface,
Eq. 2.91, is accurate over the whole range of interaction strengths
considered.
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We expect that our parameterization of the two-body nodal surface is to a good
approximation independent of the confining potential in (for small enough
). In fact, we expect our nodal surface to closely resemble that of the
scattering wave function at low scattering energy of the 3D wave guide Hamiltonian
given by Eq. 4.1. To quantify this statement, Fig. 4.5
shows the semi-axes
and
(pluses and asterisks, respectively) obtained by
fitting an ellipse, see Eq. 2.91, to the nodal surface obtained by solving
the Schrödinger equation for the two-body Hamiltonian, Eq. 4.7,
using a B-spline basis for various aspect ratios
, and fixed
scattering length,
(similar results are found for other
scattering lengths). Indeed, the nodal surface for a given
is
nearly independent of the aspect ratio
for
. These
findings for two particles imply that the parameterization of the nodal surface of
used in the FN-MC many-body calculations should be good as long as the
density along
is small.
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Next, consider the 1D Hamiltonian, Eq. 4.19, for . For positive
, the lowest-lying gas-like state is the ground state of the two-body
system and we hence use the DMC technique with
given by Eqs. 2.64
and 2.65; for
, the lowest-lying gas-like state is the first
excited state, and we instead use the FN-MC technique with
given by
Eqs. 2.64 and 2.66. Figure 4.3 shows the 1D energies of
the lowest-lying gas-like state calculated using Eqs. 4.11 through
4.13 (solid line), together with those calculated by the FN-MC
technique (squares). We find excellent agreement between these two sets of 1D
energies.
The comparison for two bosons between the FN-MC energies and the energies calculated by alternative means serves as a stringent test of our MC codes, since these codes are implemented such that the number of particles enters simply as a parameter.