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Register a new userExact transparent boundary condition for the multidimensional Schrodinger equation in hyperrectangular computational domain
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In this paper an exact transparent boundary condition (TBC) for the multidimensional Schrodinger equation in a hyperrectangular computational domain is proposed. It is derived as a generalization of exact transparent boundary conditions for 2D and 3D equations reported before. A new exact fully discrete (i.e. derived directly from the finite-difference scheme used) 1D transparent boundary condition is also proposed. Several numerical experiments using an improved unconditionally stable numerical implementation in the 3D space demonstrate propagation of Gaussian wave packets in free space and penetration of a particle through a 3D spherically asymmetrical barrier. The application of the multidimensional transparent boundary condition to the dynamics of the 2D system of two non-interacting particles is considered. The proposed boundary condition is simple, robust and can be useful in the field of computational quantum mechanics, when an exact solution of the multidimensional Schrodinger equation (including multi-particle problems) is required.
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Consider a non-relativistic quantum particle with wave function inside a region $Omegasubset mathbb{R}^3$, and suppose that detectors are placed along the boundary $partial Omega$. The question how to compute the probability distribution of the time at which the detector surface registers the particle boils down to finding a reasonable mathematical definition of an ideal detecting surface; a particularly convincing definition, called the absorbing boundary rule, involves a time evolution for the particles wave function $psi$ expressed by a Schrodinger equation in $Omega$ together with an absorbing boundary condition on $partial Omega$ first considered by Werner in 1987, viz., $partial psi/partial n=ikappapsi$ with $kappa>0$ and $partial/partial n$ the normal derivative. We provide here a discussion of the rigorous mathematical foundation of this rule. First, for the viability of the rule it plays a crucial role that these two equations together uniquely define the time evolution of $psi$; we point out here how the Hille-Yosida theorem implies that the time evolution is well defined and given by a contraction semigroup. Second, we show that the collapse required for the $N$-particle version of the problem is well defined. Finally, we also prove analogous results for the Dirac equation.
The dynamics of any classical-mechanics system can be formulated in the reparametrization-invariant (RI) form (that is we use the parametric representation for trajectories, ${bf x}={bf x}(tau)$, $t=t(tau)$ instead of ${bf x}={bf x}(t)$). In this pedagogical note we discuss what the quantization rules look like for the RI formulation of mechanics. We point out that in this case some of the rules acquire an intuitively clearer form. Hence the formulation could be an alternative starting point for teaching the basic principles of quantum mechanics. The advantages can be resumed as follows. a) In RI formulation both the temporal and the spatial coordinates are subject to quantization. b) The canonical Hamiltonian of RI formulation is proportional to the quantity $tilde H=p_t+H$, where $H$ is the Hamiltonian of the initial formulation. Due to the reparametrization invariance, the quantity $tilde H$ vanishes for any solution, $tilde H=0$. So the corresponding quantum-mechanical operator annihilates the wave function, $hat{tilde H}Psi=0$, which is precisely the Schrodinger equation $ihbarpartial_tPsi=hat HPsi$. As an illustration, we discuss quantum mechanics of the relativistic particle.
Considering symmetric strictly convex potentials, a local relationship is inferred from the virial theorem, based on which a real log-concave function can be constructed. Using this as a weight function and in such a way that the virial theorem can still be verified, parameter-free ansatze for the eigenfunctions of the associated Schrodinger equation are built. To illustrate the process, the technique is successfully tested against the harmonic oscillator, in which it leads to the exact eigenfunctions, and against the quartic anharmonic oscillator, which is considered the paradigmatic testing ground for new approaches to the Schrodinger equation.
A recently proposed approach for avoiding the ultraviolet divergence of Hamiltonians with particle creation is based on interior-boundary conditions (IBCs). The approach works well in the non-relativistic case, that is, for the Laplacian operator. Here, we study how the approach can be applied to Dirac operators. While this has been done successfully already in 1 space dimension, and more generally for codimension-1 boundaries, the situation of point sources in 3 dimensions corresponds to a codimension-3 boundary. One would expect that, for such a boundary, Dirac operators do not allow for boundary conditions because they are known not to allow for point interactions in 3d, which also correspond to a boundary condition. And indeed, we confirm this expectation here by proving that there is no self-adjoint operator on (a truncated) Fock space that would correspond to a Dirac operator with an IBC at configurations with a particle at the origin. However, we also present a positive result showing that there are self-adjoint operators with IBC (on the boundary consisting of configurations with a particle at the origin) that are, away from those configurations, given by a Dirac operator plus a sufficiently strong Coulomb potential.
The exact solution of an integrable anisotropic Heisenberg spin chain with nearest-neighbour, next-nearest-neighbour and scalar chirality couplings is studied, where the boundary condition is the antiperiodic one. The detailed construction of Hamiltonian and the proof of integrability are given. The antiperiodic boundary condition breaks the $U(1)$-symmetry of the system and we use the off-diagonal Bethe Ansatz to solve it. The energy spectrum is characterized by the inhomogeneous $T-Q$ relations and the contribution of the inhomogeneous term is studied. The ground state energy and the twisted boundary energy in different regions are obtained. We also find that the Bethe roots at the ground state form the string structure if the coupling constant $J=-1$ although the Bethe Ansatz equations are the inhomogeneous ones.
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