Notation and abbreviations

For the notation of other quantities either an unambiguous standard notation is used, or the notation is given explicitly in the text. List of the special notation used throughout the Dissertation:

SYMBOL	MEANING	DEFINITION
$a_{1D}$	one-dimensional $s$-wave scattering length	(1.63)
$a_{3D}$	three-dimensional $s$-wave scattering length	(1.44)
$a_\perp$	oscillator length of the transverse confinement	$a_\perp=\sqrt{\hbar/m\omega_\perp}$
$a_z$	oscillator length of the longitudinal confinement	$a_z=\sqrt{\hbar/m\omega_z}$
$D$ [dimensionless]	number of dimensions
$D$	diffusion constant	$D = \hbar^2/2m$
$E$	total energy of the system
$E^{loc}({\bf R})$	local energy of a walker ${\bf R}$	(2.14)
${\cal E}^{loc}(r)$	Bijl-Jastrow component of a local energy	(2.40)
${\bf F}({\bf R})$	drift force	(2.15)
${{\cal F}_2}(r)$	Bijl-Jastrow component of the drift force	(2.39)
$f_1(\r)$	one-body Bijl-Jastrow term	see (2.37)
$f_2(r)$	two-body Bijl-Jastrow term	see (2.37)
$g_1(r)$	non-diagonal element of the OBDM	(1.18)
$g_2(r)$	pair-distribution function	(1.19)
$g_3(0)$	value at zero of the three-body correlation function	(1.22)
$g_{1D}$	one-dimensional coupling constant	(1.69)
$g_{3D}$	three-dimensional coupling constant	(1.86)
$L$	size of the system or side of the simulation box
$m$	particle mass
$n_{1D}$	linear density	$n_{1D} = N/L$
$n_{3D}$	(total) particle density	$n_{3D} = N/V$
$N$	number of particles
${\vec r}_i$	coordinate of $i$-th particle	${\vec r}_i = (x_i,y_i, z_i)$
${\bf R}$	a point in $D N$-dimensional phase space (a walker)	${\bf R}=\{{\vec r_1},...,{\vec r}_N\}$
$R$	range of the potential	(1.47)
$R_{m}$	(variational) matching distance
$R_z$	size of the cloud in $z$-direction
$R_\perp$	size of the cloud in the transverse direction
$u(r)$	exponentiation of the Bijl-Jastrov term	(2.38)
$V_{ext}(\r)$	external potential
$V_{int}(\vert{{\vec r}_i-{\vec r}_j}\vert)$	pair-interaction potential
$\lambda$	anisotropy parameter (aspect ratio)	$\lambda=\omega_z/\omega_\perp$
$\mu$ [units of energy]	chemical potential
$\mu$ [units of mass]	reduced mass	$\mu = m/2$
$\tau$	imaginary time	$\tau = i t$
$\phi_0({\bf R})$	ground state many body wave function
$\psi_T({\bf R})$	trial many body wave function
$\omega _\perp$	frequency of the transverse harmonic confinement
$\omega_z$	frequency of the longitudinal harmonic confinement

List of used abbreviations:

BCS	Bardeen Cooper Schriffer
BEC	Bose-Einstein condensation
BJ	Bijl-Jastrow
DMC	Diffusion Monte Carlo
FN-MC	Fixed Node Monte Carlo
JS	Jastrow-Slater
GP	Gross-Pitaevskii
HR	hard rod
HS	hard sphere
LDA	local density approximation
LL	Lieb-Liniger
OBDM	one-body density matrix
TG	Tonks-Girardeau
SR	short range
SS	soft sphere
SW	square well
QMC	Quantum Monte Carlo
VMC	Variational Monte Carlo