We have studied the ground state energy of one-dimensional macroscopic quantum oscillator by adding gravity term as a time-independent external potential. The macroscopic quantum oscillator itself is based on discussion about Gross-Pitaevskii equation (GPE) describing Bose-Einstein condensate in anisotropic parabolic trap. That gravitational potential displaces the symmetry axis of Bose-Einstein condensation (BEC) with respect to symmetry axis of magnetic field that creates trapping potential of the Bose-Einstein condensate (BECs) in research on atomic lasers. Since the Gross-Pitaevskii equation can be treated as macroscopic quantum oscillator, we use time-independent perturbation theory as in ordinary quantum mechanics to find the ground state energy correction by assuming that other terms are very small. We also formulate the appropriate coefficients of expansion in the solution of one-dimensional GPE by considering the characteristics of harmonic oscillator eigenfunctions and find that the existence of the gravity can only be included if the higher coefficients are considered. Keywords: Gravitational potential, Gross-Pitaevskii, Quantum Oscillator.
We derive exact solitonic solutions of the one-dimensional time-dependent Gross-Pitaevskii equation with time-dependent strengths of the harmonic external potential and the interatomic interaction. The time-dependence of the external potential and interatomic interaction are given in terms of a general function of time. For an oscillating strength of the external potential, the solutions correspond to breathing single and multiple solutions. The amplitude and frequency of the oscillating potential can be used to control the dynamics of the centre of mass of the solitons. For certain values of these parameters, the solitons can be trapped at the centre of the harmonic potential.
We derive exact solitonic solutions of the one-dimensional time-dependent Gross-Pitaevskii equation with time-dependent strengths of the harmonic external potential and the interatomic interaction. The time-dependence of the external potential and interatomic interaction are given in terms of a general function of time. For an oscillating strength of the external potential, the solutions correspond to breathing single and multiple solitons. The amplitude and frequency of the oscillating potential can be used to control the dynamics of the center of mass of the solitons. For certain values of these parameters, the solitons can be {\it trapped} at the center of the harmonic potential.
We introduce a non-commutative generalization of the Gross-Pitaevskii equation for one-dimensional quantum gasses and quantum liquids. This generalization is obtained by applying the time-dependent variational principle to the variational manifold of continuous matrix product states. This allows for a full quantum description of many body system —including entanglement and correlations— and thus extends significantly beyond the usual mean-field description of the Gross-Pitaevskii equation, which is known to fail for (quasi) one-dimensional systems. By linearizing around a stationary solution, we furthermore derive an associated generalization of the Bogoliubov – de Gennes equations. This framework is applied to compute the steady state response amplitude to a periodic perturbation of the potential.
Abstract Considering a set of condensates as N indistinguishable noninteracting quantum oscillators and replacing an available volume with a harmonic volume, we derived an ideal gas model by applying the canonical partition function via the thermodynamic relation. In this case, we take a longitudinal part of the three-dimensional Gross-Pitaevskii equation in which the effect of gravitation of earth works. Since we used the time-independent perturbation theory, to include gravity effect, we must put the higher coefficient in the formulation of each energy state whose value is determined by the separated amplitude formula in each state. It will also be shown that even though the equation of state represents an ideal gas, the appropriate heat capacity approaches Einstein’s solid-like model if the certain conditions are employed.
Consider a system of
N bosons in three dimensions interacting via a repulsive short range pair potential
N2V(N(xi−xj)), where $\bx=(x_1, >..., x_N)$ denotes the positions of the particles. Let
HN denote the Hamiltonian of the system and let
ψN,t be the solution to the Schrodinger equation. Suppose that the initial data
ψN,0 satisfies the energy condition
≤CN>. and that the one-particle density matrix converges to a projection as
N→∞. Then, we prove that the
k-particle density matrices of
ψN,t factorize in the limit
N→∞. Moreover, the one particle orbital wave function solves the time-dependent Gross-Pitaevskii equation, a cubic non-linear Schrodinger equation with the coupling constant proportional to the scattering length of the potential
V. In \cite{ESY}, we proved the same statement under the condition that the interaction potential
V is sufficiently small; in the present work we extend this result to large potentials.