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International Journal of Quantum Chemistry

A weakly bound, quantum‐mechanical buridan's ass

Nimrod Moiseyev, Jacob Katriel, Carl O. Trindle

Published Oct 1, 1977• 2 citations
Publisher

Abstract

Abstract The spherical limit of a single particle in the field of three nuclei is shown to exhibit a variety of phenomena associated with the stability of the various extrema involved. Some consequences with respect to the general problem of detecting broken symmetry solutions of variational problems are pointed out.
DOI
10.1002/qua.560120412

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We consider the quantum-mechanical problem of a particle bound to a configuration of spherical potentials, each of finite range. If the Schr\"odinger equation can be solved for each potential by itself, then it is shown how to solve it for the configuration, provided the potentials do not overlap. The energy levels are the zeros of a determinant of formally infinite order, but in practice this is always well approximated by a finite determinant and often by one of small order. As illustrative examples we consider some states of a particle bound to three square wells with the configuration of an equilateral triangle, and to two truncated Coulomb potentials. The possibility of extending an approximate version of the method to overlapping potentials is pointed out.
The quantum Buridan's ass problem concerns a network of interconnected nodes in which transitions between the nodes are governed by the laws of quantum mechanics. In such situations interference between probability amplitudes plays a key role in determining whether the probability of leaving the initial node is large or small and also the speed at which the associated transitions occur. In this paper we extend this idea to consider transitions between one given set of coupled nodes and the remainder of the network. We have in mind a quantum analogy to changing an opinion, in which the initially occupied set of nodes is one opinion bubble, with the remaining parts of the network corresponding to a different opinion. We find that quantum interference again plays a crucial role in the dynamics. A further key component is the role of symmetry between the nodes and their interconnections, as it is symmetry that lies at the heart of the Buridan's ass puzzle. In the classical Buridan's ass paradox it is symmetry between competing alternatives that freezes the ass into indecision. In the quantum case, however, interference results in a far richer dynamics.
In an earlier paper appropriate limiting procedure is discussed in a general way for quantum mechanical energy eigenfunctions (one-dimensional bound states) — a single interpretational postulate leading smoothly to entire compatible classical objective description without facing any contradiction. The method is consistently extended to two- and three-dimensional cases and it is interesting to note that results of earlier study on the classical limit of the radial distribution function of hydrogen atom are easily obtained as special cases of our analysis.
Abstract One of the most promising applications for near term quantum computers is the simulation of physical quantum systems, particularly many-electron systems in chemistry and condensed matter physics. In solid state physics, finding the correct symmetry broken ground state of an interacting electron system is one of the central challenges. To help finding the correct broken symmetries in the thermodynamic limit methods that allow to determine the groundstate of large but finite interacting electron systems are very useful. The variational Hamiltonian ansatz (VHA), a variational hybrid quantum-classical algorithm especially suited for finding the ground state of a solid state system, will in general not prepare a broken symmetry state unless the initial state is chosen to exhibit the correct symmetry. In this work, we discuss three variations of the VHA designed to find the symmetry-breaking groundstate of a finite system close to a transition point between different orders. As a test case we use the two-dimensional Hubbard model where we break the symmetry explicitly by means of external fields coupling to the Hamiltonian and calculate the response to these fields. For the calculation we simulate a gate-based quantum computer and also consider the effects of dephasing noise on the algorithms. We find that two of the three algorithms are in good agreement with the exact solution for the considered parameter range. The third algorithm agrees with the exact solution only for a part of the parameter regime, but is more robust with respect to dephasing compared to the other two algorithms.