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.