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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

New lower bounds for quantum Hamiltonians

C. T. Sachrajda, H. Arthur Weldon, R. Blankenbecler

Published Jan 15, 1978• 5 citations
Publisher

Abstract

A new variational principle is presented which provides a lower bound to the eigenvalues of a general quantum Hamiltonian. Applications are made to ground and excited states in Schr\"odinger theory. The variational principle leads naturally to localized (classical-like) states that nevertheless contain effects due to quantum fluctuations in a variational manner. Except for possible exceptional end-point solutions, these "classical" or "semiquantum" configurations provide lower bonds to the true eigenvalue.

DOI
10.1103/physrevd.17.507

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A new variational principle is presented which provides a lower bound to the eigenvalues of a general quantum Hamiltonian. Applications are made to ground and excited states in Schr"odinger theory. The variational principle leads naturally to localized (classical-like) states that nevertheless contain effects due to quantum fluctuations in a variational manner. Except for possible exceptional end-point solutions, these "classical" or "semiquantum" configurations provide lower bonds to the true eigenvalue.
The variational principle is revisited in the context of finding the upper and lower bounds to the ground state energy. It is shown how the variational principle can be employed to find the lower bound by partitioning the Hamiltonian into several parts. We demonstrate how the variational principle can be used to find the exact ground states in some special cases. We consider the harmonic oscillator and the Ising Hamiltonian on a bipartite lattice which are familiar yet instructive examples for students.
Variational methods play an important role in the study of quantum many-body problems, both in the flavor of classical variational principles based on tensor networks as well as of quantum variational principles in near-term quantum computing. This work stresses that for translationally invariant lattice Hamiltonians with periodic boundary conditions, one can easily derive efficiently computable lower bounds to ground state energies that can and should be compared with variational principles providing upper bounds. As small technical results, it is shown that (i) the Anderson bound and a (ii) common hierarchy of semidefinite relaxations both provide approximations with performance guarantees that scale as a constant in the energy density for cubic lattices. (iii) Also, the Anderson bound is systematically improved as a hierarchy of semidefinite relaxations inspired by the quantum marginal problem.
I propose a new and direct connection between classical mechanics and quantum mechanics where I derive the quantum mechanical propagator from a variational principle. This variational principle is Hamilton's modified principle generalized to allow many paths due to the non-local existence of particles in phase space. This principle allows a physical system to evolve non-locally in phase space while still allowing a representation that uses many classical paths. Whereas a point in phase space represents a classical system's state, I represent the state of a non-local system by a mixed trajectory. This formulation naturally leads to the transactional interpretation for resolving the paradoxes of the measurement problem. This principle also suggests a more flexible framework for formulating theories based on invariant actions and provides a single conceptual framework for discussing many areas of science.