Generalized Variational Principle in Quantum Mechanics

Soldatov, A. V.

doi:10.1142/s0217979295001087

Cited by 19 publications

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“…It was proved in [12] following the ideas outlined in [13], and also found later in [14] by a different approach, that for a quantum system represented by some HamiltonianĤ and any normalized trial state |ψ , such that ψ|ψ = 1,…”

Section: Generalized Variational Methodsmentioning

confidence: 90%

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Infinitely improvable upper bounds in the theory of polarons

Soldatov¹

2010

Condens. Matter Phys.

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An infinite convergent sequence of improving non-increasing upper bounds to the low-lying branch of the slowmoving "physical" Fröhlich polaron ground-state energy spectral curve, adjacent to the ground state energy of the polaron at rest, was obtained by means of generalized variational method. The proposed approach is especially well-suited for massive analytical and numerical computations of experimentally measurable properties of realistic polarons, such as inverse effective mass tensor and excitation gap, and can be elaborated even further, without major alterations, to allow for treatment of multitudinous polaron-like models, those describing polarons of various sorts placed in external magnetic and electric fields among them. The polaron conceptA local change in the electronic state in a crystal leads to the excitation of crystal lattice vibrations, i.e. the excitation of phonons. And vice versa, any local change in the state of the lattice ions alters the local electronic state. This situation is commonly referred to as an "electronphonon interaction". This interaction manifests itself even at the absolute zero of temperature, and results in a number of specific microscopic and macroscopic phenomena such as, for example, lattice polarization. When a conduction electron with band mass m moves through the crystal, this state of polarization can move together with it. This combined quantum state of the moving electron and the accompanying polarization may be considered as a quasiparticle with its own particular characteristics, such as effective mass, total momentum, energy, and maybe other quantum numbers describing the internal state of the quasiparticle in the presence of an external magnetic field or in the case of a very strong lattice polarization that causes self-localization of the electron in the polarization well with the appearance of discrete energy levels. Such a quasiparticle is usually called a "polaron state" or simply a "polaron".The concept of the polaron was first introduced by L.D. Landau [1], followed by much more detailed work by S.I. Pekar [2] who investigated the most essential properties of stationary polaron in the limiting case of very intense electron-phonon interaction, in the so-called adiabatic approximation. Subsequently, Landau and Pekar [3] investigated the self-energy and the effective mass of the polaron for the adiabatic regime. Many other famous researchers have contributed to the development of polaron theory later on [4][5][6][7][8][9]. The polaron concept remains of interest from at least two points of view, practical and theoretical: it describes the physical properties of charge carriers in polar crystals and ionic semiconductors and, at the same time, represents a simple, but rich in content, field-theoretical model of a particle interacting with a scalar boson field.The model under consideration is the standard quantized Fröhlich polaron Hamiltonian introduced by H. Fröhlich [6]

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Section: Generalized Variational Methodsmentioning

confidence: 90%

“…The purpose of the present research is to show that infinitely improvable upper bounds for the low-lying branch of the "physical" polaron energy spectrum E(α, P, k D ) can be obtained by generalized variational method formulated for the first time in [12] and later in [14] in a different context.…”

Section: Low-lying Branch Of the Polaron Energy Spectrummentioning

confidence: 99%

Infinitely improvable upper bounds in the theory of polarons

Soldatov¹

2010

Condens. Matter Phys.

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“…In this section we will give a brief description of the PDS formalism. The detailed discussion of the PDS methodology and highly relevant connected moment expansion (CMX) formalisms have been given in the original work [49,62] as well as our recent work [14,34] and many earlier literatures (see for example Refs. [18,19,33,39,40,54,65]).…”

Section: Methods 21 Pds Formalismmentioning

confidence: 99%

“…Instead of adding more parameters to the trial wave function, we choose to optimize a new class of energy functionals (or quasifunctionals, where the energy is calculated as a simple equation solution) that already encompasses information about high-order static and dynamical correlation effects. An ideal choice for such high-level functional is based on the Peeters, Devreese, and Soldatov (PDS) formalism, [49,62] where variational energy is obtained as a solution of simple equations expressed in terms of the Hamiltonian's moments or expectations values of the powers of the Hamiltonians operator defined for the trial wave function. In Ref.…”

Section: Introductionmentioning

confidence: 99%

Variational quantum solver employing the PDS energy functional

Kowalski

2021

Quantum

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Recently a new class of quantum algorithms that are based on the quantum computation of the connected moment expansion has been reported to find the ground and excited state energies. In particular, the Peeters-Devreese-Soldatov (PDS) formulation is found variational and bearing the potential for further combining with the existing variational quantum infrastructure. Here we find that the PDS formulation can be considered as a new energy functional of which the PDS energy gradient can be employed in a conventional variational quantum solver. In comparison with the usual variational quantum eigensolver (VQE) and the original static PDS approach, this new variational quantum solver offers an effective approach to navigate the dynamics to be free from getting trapped in the local minima that refer to different states, and achieve high accuracy at finding the ground state and its energy through the rotation of the trial wave function of modest quality, thus improves the accuracy and efficiency of the quantum simulation. We demonstrate the performance of the proposed variational quantum solver for toy models, H2 molecule, and strongly correlated planar H4 system in some challenging situations. In all the case studies, the proposed variational quantum approach outperforms the usual VQE and static PDS calculations even at the lowest order. We also discuss the limitations of the proposed approach and its preliminary execution for model Hamiltonian on the NISQ device.

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“…Now, we take advantage of the fact that the trial state |z〉 = |0〉 already provides a good enough bound (10) in the entire range of values of the interaction constant α and apply the generalized variational method [21] for a successive systematic improvement of this bound. As shown in [21], for the Hamiltonian of a quantum system and a trial state |ψ〉 such that 〈ψ|ψ〉 = 1, where the real numbers are the roots of the nth-order polynomial equation…”

Section: A Sequence Of Improved Upper Boundsmentioning

confidence: 99%

On improvable bounds for polaron systems in an external potential

Bogolyubov

Soldatov

2016

Moscow Univ. Phys.

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A variational approach is proposed that allows one to obtain in a regular way a sequence of improvable upper bounds for the ground-state energy of various polaron models confined in an external electrostatic potential. The proposed approach can be used for an arbitrary electron-phonon interaction constant and allows generalization to the case of polaron-type systems in a constant external magnetic field.

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Generalized Variational Principle in Quantum Mechanics

Cited by 19 publications

References 0 publications

Infinitely improvable upper bounds in the theory of polarons

Infinitely improvable upper bounds in the theory of polarons

Variational quantum solver employing the PDS energy functional

On improvable bounds for polaron systems in an external potential

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