The constrained energy functional, defined by E0(a) =minSa{a〈Ψ‖Ĥ0‖Ψ〉a}, ai=a〈Ψ‖Âi‖Ψ〉a, i=1, ..., M, ‖Ψ〉a∈Sa, with a={ai} a preassigned set of expectation values generated by any state vector ‖Ψ〉a in the class Sa, is shown to satisfy a first order, but nonlinear, partial differential equation. The equation results by replacing the Lagrange multipliers introduced in an earlier paper by the corresponding partial derivatives of the to-be-found energy functional with respect to the constraints. Two alternative expressions for E0(a), one focusing upon the ground state eigenvalue of a modified Hamiltonian operator and the other upon the corresponding eigenvector, lead via standard Rayleigh–Schrödinger perturbation theory to two approximate partial differential equations for the energy functional. Both equations together with the imposed subsidiary conditions lead in turn to the same approximate E0(a). An example based on the linear harmonic oscillator illustrates the concepts presented in the formalism.
The variational method of coupled Gaussian functions is applied to Bose-Einstein condensates with long-range interactions. The time dependence of the condensate is described by dynamical equations for the variational parameters. We present the method and analytically derive the dynamical equations from the time-dependent Gross-Pitaevskii equation. The stability of the solutions is investigated using methods of nonlinear dynamics. The concept presented in this article will be applied to Bose-Einstein condensates with monopolar
1/r and dipolar
1/r3 interaction in the subsequent article [S. Rau et al., Phys. Rev. A 82, 023611 (2010)], where we will present a wealth of phenomena obtained using the ansatz with coupled Gaussian functions.
Abstract A modification of the Rayleigh—Ritz variational principle is described which makes possible a calculation of the energy, wave function, and pair distribution function f12≡f(x1,x2) of a Bose fluid, such as liquid 4He, at absolute zero. The assumptions made are: (i) two-body interactions with potential Uij, (ii) trial wave functions of the form f12≡f(x1,x2) and (iii) the Kirkwood ‘superposition' approximation. Under these approximations, the expectation energy is E=12n2∫∫d3X1d3X2f12U12−h2m−1(∇18ϕ12)+(∇1ϕ12)2+n∫d3X2f13f23ϕ12⋅∇1ϕ13, where n ≡ N/V. It is shown here that making E stationary with respect to independent variations in f and ɸ corresponds to simultaneously applying the ordinary Rayleigh-Ritz principle and solving the Born-Green-Yvon integral equation for f. The method is illustrated by reproducing Bogolyubov’s results for the case where U is small. The case where U is large must be dealt with numerically, but transformations for simplifying the computations are given here.
The use of variational principles as a calculational tool is reviewed, with special emphasis on methods for constructing such principles. In particular, it is shown that for a very wide class of problems it is possible to construct a variational principle (VP) for just about any given quantity
Q of interest, by routine procedures which do not require the exercise of ingenuity; the resultant VP will yield an estimate of
Q correct to second order whenever the quantities appearing in the VP are known to first order. The only significant requirement for application of the routine procedures is that the entities which enter into the definition of
Q be uniquely specified by a given set of equations; the equations may involve difference or differential or integral operators, they may be homogeneous or inhomogeneous, linear or nonlinear, self-adjoint or not, and they may or may not represent time-reversible systems. No numerical calculations are presented, but procedures for the construction of VP's are illustrated for numerous quantities
Q of physical interest, particularly those
Q arising in quantum-mechanical scattering and transition probability calculations. For pedagogical purposes VP's are also derived for several problems in classical and (simple) mathematical physics which the authors hope will prove instructive and perhaps even amusing. The quantum-mechanical quantities
Q whose VP's are examined include various matrix elements and the quantum-mechanical eigenfunctions themselves. Topics examined include some points which have not always been appreciated in the literature, such as the necessity for properly specifying the phase when complex eigenfunctions are involved, and the importance of avoiding, wherever possible, formulations requiring the inversion of singular operators. The basic element of the technique is the recognition that the defining equations of a system can be incorporated into the VP as constraints through the use of (generalized) Lagrange multipliers; these can be constants, scalar, vector, or tensor functions of one or more variables, operators, etc. In the typical problem, these Lagrange multipliers
L serve as a new set of adjoint functions or entities, and the construction of the VP simultaneously provides well-defined equations for the
L. Moreover, these
L often have ready physical significance themselves; for example, they often may be regarded as generalized Green's functions. The construction of the VP also readily yields so-called variational identities for the quantity
Q of interest; these identities give explicit (if formal) expressions for the error in the variational estimate of
Q. In some cases this error can be shown to have a definite sign, so that the VP actually is an extremum principle, that is, that it yields an upper or lower variational bound for
Q; however, our routine procedures for constructing VP's do not routinely yield extremum principles.
Shortcuts to adiabaticity (STA) were first developed in quantum dynamics to realize rapid transformations with suppressed residual excitations. Here we show how the same idea can be implemented in classical nonlinear dissipative Lagrangian systems. Using a coupled
r-
θ manipulator as an illustrative model, we perform inverse engineering on the Euler-Lagrange equations with Rayleigh dissipation by prescribing endpoint-stationary trajectories, obtaining the corresponding force and torque profiles and quantifying how geometric coupling amplifies errors and residual energy. We further compare smooth STA protocols with actuator-bounded time-optimal solutions and with proportional-integral-derivative tracking, which highlights a trade-off among smoothness, speed, and robustness. Finally, we introduce a single-shot correction based on one mid-course measurement to reduce the effect of early deviations while keeping the inputs nearly smooth. These results provide a practical bridge between quantum STA concepts and their classical counterparts.
We have previously formulated a simple criterion for deducing the intervals of oscillations in the solutions of second-order linear homogeneous differential equations. In this work, we extend analytically the same criterion to the cubic nonlinear Schrödinger equations that describe Bose-Einstein condensates. With this criterion guiding the search for solutions, we classify all types of solutions and we find new stationary solutions in the free-particle cases that were not noticed previously because of limited coverage in the adopted boundary conditions. The new solutions are produced by the nonlinear terms of the differential equations and they continue to exist when various external potentials are also incorporated. Surprisingly, these solutions appear when the nonlinearities are small.