Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of the Japan Society of Powder and Powder Metallurgy

On the Derivation of Quantum Mechanical Expectations in the Differential Form

Shin-ichiro Kondo, Tatsushi KONDO, Atsushi Kondo, Kazuyoshi Yoshimura

Published Oct 23, 2023• 1 citation
Publisher

Abstract

Usually, expected values for various physical quantities, such as the number of electrons occupying certain states or the Coulomb interaction between different states of electrons, can be expressed in terms of integrals. In contrast, our method, based on differential forms, shows that expected values can be obtained by averaging over time. To confirm the validity of our method, we prepare the two cases: one is a very simple case with no many-body interaction, and the other is the case where the many-body term is included (the simplest Anderson Hamiltonian). Regarding the simple case without inclusion of many-body term, we prove analytically that the number of electrons occupying any state derived from our method is equivalent to the analytical one evaluated from the Greenʼs function method. When the many-body term is included, our results show good numerical agreement with the analytical ones derived from the Greenʼs function method. By the two cases, the calculation of expected values based on our method is considered valid.

DOI
10.2497/jjspm.23-00042

Related papers

Numerical Exact Solutions to the Many-body Problem Based on the Differential Forms

Shin-ichiro Kondo, Kazuyoshi Yoshimura

The Many-body problem is very important issue in physics. Usually perturbation calculations based on the Green's function have been used to analyze such a problem, however, infinite orders of perturbation calculations are required to obtain exact solutions, thus resulting in the analytical and numerical difficulties. To overhaul such difficulties, we propose the new calculation method based on the differential forms, which are able to evaluate exact solutions if Hamiltonian is composed of Fermi particles. Since our proposed method is based on time evolution equations, comparisons with the calculations derived from Feynman Kernel is possible, with showing complete agreement. Furthermore we applied this method to the simplest Anderson Hamiltonian to investigate the appearance of magnetic moment. The calculation results show that magnetic moment easily disappears with small Coulomb repulsive energy, possibly implying the spin fluctuations.

Journal of the Japan Society of Powder and Powder Metallurgy•Jun 14, 2023•2 citations

Functionals in many-particle physics

Richard M. Martin, Lucia Reining, David M. Ceperley

To avoid confusion, it is necessary to be quite explicit about what is assumed and what is to be proved. A. Klein, Phys. Rev. 121, 950 (1961) Summary The topic of this chapter is functionals that provide a concise formulation for thermodynamic quantities and Green's function methods in interacting many-body systems. Building upon the instructive examples of density functional theory and the Hartree– Fock approximation, functionals of the Green's function are developed to provide a framework for practical methods used extensively in many-body perturbation theory (Chs. 9–15), dynamical mean-field theory (Chs. 16–20), and their combination in Ch. 21. Functional concepts give a firm foundation for those methods, for example, they give the conditions for approximations to obey conservation laws. Additional background and specific aspects are in App. H. Many physical quantities can be expressed as functionals , i.e., they depend on an entire function.1 For example, in the quantum variational method to determine a wavefunction, the total energy of a system of many electrons is a functional of a many-body trial wavefunction ( r 1 , r 2 , …): it is the expectation value of the hamiltonian. At the variational minimum, assuming the trial wavefunction is completely flexible, one finds the many-body Schrödinger equation for the ground state. Many of the properties we are interested in are, formally, simple functionals of the ground- or excited-state wavefunctions. In the search for feasible ways to calculate properties such as excitation spectra, we are led to different types of functional. The idea is to deal only with a physical, measurable quantity Q of interest and a variational functional E [ Q ] of this quantity. The variational character of E allows one to derive equations that determine Q 0 , the value of Q for the system in a desired state, typically the ground state or equilibrium state at T ≠ 0. Other properties should be different functionals F m [ Q ] that can then be evaluated at Q = Q 0 . This strategy is outlined in Fig. 8.1. To appreciate the power of such approaches, one can consider the impact of density functional theory, which is formulated as a functional of the density n ( r ) of electrons; it is defined for a range of densities and the variational minimum for the energy functional E [ n ] leads to equations to determine the actual ground-state density n 0 ( r ) and total energy E 0 .

Cambridge University Press eBooks•May 31, 2016•0 citations

New variational approach to the many-body problem and application to the high-density electron gas

Yasutami Takada

A new variational approach is proposed for the study of the many-body problem with the purpose of combining the advantages of perturbation-theoretic and variational approaches, while avoiding their disadvantages. A trial wave function is constructed so that a general expansion for the expectation value of an arbitrary physical quantity can be derived. Each term in the expansion can be calculated with the use of the techniques of Feynman diagrams, but in contrast with usual perturbation-theoretic approaches, every term in the expansion always converges. The procedure for applying the method to real systems is illustrated by a calculation of the ground-state energy of the electron gas at high densities.

Physical Review A•Oct 1, 1983•20 citations

Calculation of generating function in many-body systems with quantum computers: technical challenges and use in hybrid quantum-classical methods

Edgar Andres Ruiz Guzman, Denis Lacroix A

The generating function of a Hamiltonian $H$ is defined as $F(t)=\langle e^{-itH}\rangle$, where $t$ is the time and where the expectation value is taken on a given initial quantum state. This function gives access to the different moments of the Hamiltonian $\langle H^{K}\rangle$ at various orders $K$. The real and imaginary parts of $F(t)$ can be respectively evaluated on quantum computers using one extra ancillary qubit with a set of measurement for each value of the time $t$. The low cost in terms of qubits renders it very attractive in the near term period where the number of qubits is limited. Assuming that the generating function can be precisely computed using quantum devices, we show how the information content of this function can be used a posteriori on classical computers to solve quantum many-body problems. Several methods of classical post-processing are illustrated with the aim to predict approximate ground or excited state energies and/or approximate long-time evolutions. This post-processing can be achieved using methods based on the Krylov space and/or on the $t$-expansion approach that is closely related to the imaginary time evolution. Hybrid quantum-classical calculations are illustrated in many-body interacting systems using the pairing and Fermi-Hubbard models.

arXiv:2104.08181•Apr 16, 2021•2 citations

On the Ehrenfest theorem of quantum mechanics

Gero Friesecke, Mario Koppen

We give a mathematically rigorous derivation of Ehrenfest’s equations for the evolution of position and momentum expectation values under general and natural assumptions which include atomic and molecular Hamiltonians with Coulomb interactions.

arXiv:0907.1877•Journal of Mathematical Physics•Aug 1, 2009•17 citations

Application of the Heisenberg Equation of Motion to the Many-Body Problem in the Simplest Electron–Phonon Interaction

Shin-ichiro Kondo

To analyze the many-body problem based on differential forms, we apply the Heisenberg equation of motion to the Hamiltonian including the many-body term, focusing on the simplest electron–phonon interaction. Simultaneous differential equations are obtained after applying the Heisenberg equation of motion. To confirm our proposed method, we compare the numerical solutions to the simultaneous differential equations with the analytical results based on the Green's function method. Good agreement is observed between the results, indicating that the analysis based on the Heisenberg equation of motion is suitable for analyzing the steady state. To examine the non-steady state, we investigate weakly pulsed phonon excitation. The numerical calculations suggest that phonons may be conserved if the excitation period is very short. Furthermore, based on the Pauli exclusion principle, the number of simultaneous differential equations that exactly describe electron–electron interactions appears to be finite but becomes extremely large as the number of electrons increases.

Journal of the Physical Society of Japan•Apr 5, 2021•5 citations
Usually, expected values for various physical quantities, such as the number of electrons occupying certain states or the Coulomb interaction between different states of electrons, can be expressed in terms of integrals. In contrast, our method, based on differential forms, shows that expected values can be obtained by averaging over time. To confirm the validity of our method, we prepare the two cases: one is a very simple case with no many-body interaction, and the other is the case where the many-body term is included (the simplest Anderson Hamiltonian). Regarding the simple case without inclusion of many-body term, we prove analytically that the number of electrons occupying any state derived from our method is equivalent to the analytical one evaluated from the Greenʼs function method. When the many-body term is included, our results show good numerical agreement with the analytical ones derived from the Greenʼs function method. By the two cases, the calculation of expected values based on our method is considered valid.
The Many-body problem is very important issue in physics. Usually perturbation calculations based on the Green's function have been used to analyze such a problem, however, infinite orders of perturbation calculations are required to obtain exact solutions, thus resulting in the analytical and numerical difficulties. To overhaul such difficulties, we propose the new calculation method based on the differential forms, which are able to evaluate exact solutions if Hamiltonian is composed of Fermi particles. Since our proposed method is based on time evolution equations, comparisons with the calculations derived from Feynman Kernel is possible, with showing complete agreement. Furthermore we applied this method to the simplest Anderson Hamiltonian to investigate the appearance of magnetic moment. The calculation results show that magnetic moment easily disappears with small Coulomb repulsive energy, possibly implying the spin fluctuations.
To avoid confusion, it is necessary to be quite explicit about what is assumed and what is to be proved. A. Klein, Phys. Rev. 121, 950 (1961) Summary The topic of this chapter is functionals that provide a concise formulation for thermodynamic quantities and Green's function methods in interacting many-body systems. Building upon the instructive examples of density functional theory and the Hartree– Fock approximation, functionals of the Green's function are developed to provide a framework for practical methods used extensively in many-body perturbation theory (Chs. 9–15), dynamical mean-field theory (Chs. 16–20), and their combination in Ch. 21. Functional concepts give a firm foundation for those methods, for example, they give the conditions for approximations to obey conservation laws. Additional background and specific aspects are in App. H. Many physical quantities can be expressed as functionals , i.e., they depend on an entire function.1 For example, in the quantum variational method to determine a wavefunction, the total energy of a system of many electrons is a functional of a many-body trial wavefunction ( r 1 , r 2 , …): it is the expectation value of the hamiltonian. At the variational minimum, assuming the trial wavefunction is completely flexible, one finds the many-body Schrödinger equation for the ground state. Many of the properties we are interested in are, formally, simple functionals of the ground- or excited-state wavefunctions. In the search for feasible ways to calculate properties such as excitation spectra, we are led to different types of functional. The idea is to deal only with a physical, measurable quantity Q of interest and a variational functional E [ Q ] of this quantity. The variational character of E allows one to derive equations that determine Q 0 , the value of Q for the system in a desired state, typically the ground state or equilibrium state at T ≠ 0. Other properties should be different functionals F m [ Q ] that can then be evaluated at Q = Q 0 . This strategy is outlined in Fig. 8.1. To appreciate the power of such approaches, one can consider the impact of density functional theory, which is formulated as a functional of the density n ( r ) of electrons; it is defined for a range of densities and the variational minimum for the energy functional E [ n ] leads to equations to determine the actual ground-state density n 0 ( r ) and total energy E 0 .
A new variational approach is proposed for the study of the many-body problem with the purpose of combining the advantages of perturbation-theoretic and variational approaches, while avoiding their disadvantages. A trial wave function is constructed so that a general expansion for the expectation value of an arbitrary physical quantity can be derived. Each term in the expansion can be calculated with the use of the techniques of Feynman diagrams, but in contrast with usual perturbation-theoretic approaches, every term in the expansion always converges. The procedure for applying the method to real systems is illustrated by a calculation of the ground-state energy of the electron gas at high densities.
The generating function of a Hamiltonian HHH is defined as F(t)=⟨e−itH⟩F(t)=\langle e^{-itH}\rangleF(t)=⟨e−itH⟩, where ttt is the time and where the expectation value is taken on a given initial quantum state. This function gives access to the different moments of the Hamiltonian ⟨HK⟩\langle H^{K}\rangle⟨HK⟩ at various orders KKK. The real and imaginary parts of F(t)F(t)F(t) can be respectively evaluated on quantum computers using one extra ancillary qubit with a set of measurement for each value of the time ttt. The low cost in terms of qubits renders it very attractive in the near term period where the number of qubits is limited. Assuming that the generating function can be precisely computed using quantum devices, we show how the information content of this function can be used a posteriori on classical computers to solve quantum many-body problems. Several methods of classical post-processing are illustrated with the aim to predict approximate ground or excited state energies and/or approximate long-time evolutions. This post-processing can be achieved using methods based on the Krylov space and/or on the ttt-expansion approach that is closely related to the imaginary time evolution. Hybrid quantum-classical calculations are illustrated in many-body interacting systems using the pairing and Fermi-Hubbard models.
To analyze the many-body problem based on differential forms, we apply the Heisenberg equation of motion to the Hamiltonian including the many-body term, focusing on the simplest electron–phonon interaction. Simultaneous differential equations are obtained after applying the Heisenberg equation of motion. To confirm our proposed method, we compare the numerical solutions to the simultaneous differential equations with the analytical results based on the Green's function method. Good agreement is observed between the results, indicating that the analysis based on the Heisenberg equation of motion is suitable for analyzing the steady state. To examine the non-steady state, we investigate weakly pulsed phonon excitation. The numerical calculations suggest that phonons may be conserved if the excitation period is very short. Furthermore, based on the Pauli exclusion principle, the number of simultaneous differential equations that exactly describe electron–electron interactions appears to be finite but becomes extremely large as the number of electrons increases.