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

arXiv:1709.02090

Relation of $a^\dagger a$ terms to higher-order terms in the adiabatic expansion for large-amplitude collective motion

Koichi Sato

Published Sep 7, 2017• 0 citations
arXivHTMLPDF

Abstract

We investigate the relation of $a^\dagger a$ terms in the collective operator to the higher-order terms in the adiabatic self-consistent collective coordinate (ASCC) method. In the ASCC method, a state vector is written as $e^{i\hat G(q,p,n)}|ϕ(q)\rangle$ with $\hat G(q,p,n)$ which is a function of collective coordinate $q$, its conjugate momentum $p$ and the particle number $n$. According to the generalized Thouless theorem, $\hat G$ can be written as a linear combination of two-quasiparticle creation and annihilation operators $a^\dagger_μa^\dagger_ν$ and $a_νa_μ$. We show that, if $a^\dagger a$ terms are included in $\hat G(q,p,n)$, it corresponds to the higher-order terms in the adiabatic expansion of $\hat G$. This relation serves as a prescription to determine the higher-order collective operators from the $a^\dagger a$ part of the collective operator, once it is given without solving the higher-order equations of motion.

DOI
10.48550/arxiv.1709.02090

Related papers

Relation of a†a terms to higher-order terms in the adiabatic expansion for large-amplitude collective motion

Koichi Sato

We investigate the relation of |$a^\dagger a$| terms in the collective operator to the higher-order terms in the adiabatic self-consistent collective coordinate (ASCC) method. In the ASCC method, a state vector is written as |$e^{i\hat G(q,p,n)}|\phi(q)\rangle$| with |$\hat G(q,p,n)$|⁠, which is a function of the collective coordinate |$q$|⁠, its conjugate momentum |$p$|⁠, and the particle number |$n$|⁠. According to the generalized Thouless theorem, |$\hat G$| can be written as a linear combination of two-quasiparticle creation and annihilation operators |$a^\dagger_\mu a^\dagger_\nu$| and |$a_\nu a_\mu$|⁠. We show that, if |$a^\dagger a$| terms are included in |$\hat G(q,p,n)$|⁠, it corresponds to the higher-order terms in the adiabatic expansion of |$\hat G$|⁠. This relation serves as a prescription to determine the higher-order collective operators from the |$a^\dagger a$| part of the collective operator, once it is given, without solving the higher-order equations of motion.

Progress of Theoretical and Experimental Physics•Oct 26, 2017•0 citations

Adiabatic approach to large-amplitude collective motion with the higher-order collective-coordinate operator

Koichi Sato

We propose a new set of equations to determine the collective Hamiltonian including the second-order collective-coordinate operator on the basis of the adiabatic self-consistent collective-coordinate (ASCC) theory. We illustrate, with the two-level Lipkin model, that the collective operators including the second-order one are self-consistently determined. We compare the results of the calculations with and without the second-order operator and show that, without the second-order operator, the agreement with the exact solution becomes worse as the excitation energy increases, but that, with the second-order operator included, the exact solution is well reproduced even for highly excited states. We also reconsider which equations one should adopt as the basic equations in the case where only the first-order operator is taken into account, and suggest an alternative set of fundamental equations instead of the conventional ASCC equations. Moreover, we briefly discuss the gauge symmetry of the new basic equations that we propose in this paper.

arXiv:1808.08077•Progress of Theoretical and Experimental Physics•Aug 31, 2018•2 citations

Adiabatic Selfconsistent Collective Coordinate Method for Large Amplitude Collective Motion in Nuclei with Pairing Correlations

Masayuki Matsuo, Takashi Nakatsukasa, K. Matsuyanagi

An adiabatic approximation to the selfconsistent collective coordinate method is formulated in order to describe large amplitude collective motions in nuclei with pairing correlations on the basis of the time-dependent Hartree-Fock-Bogoliubov equations of motion. The basic equations are presented in a local harmonic form which can be solved in a manner similar to that of the quasiparticle RPA equations. The formalism guarantees the conservation of nucleon number expectation values. An extension to the multi-dimensional case is also discussed.

arXiv:nucl-th/0001056•Progress of Theoretical Physics•May 1, 2000•63 citations

General expansion of natural power of linear combination of Bosonic operators in normal order

Deepak Deepak, Arpita Chatterjee

In quantum mechanics, bosonic operators are mathematical objects that are used to represent the creation ($a^\dagger$) and annihilation ($a$) of bosonic particles. The natural power of a linear combination of bosonic operators represents an operator $(a^\dagger x+ay)^n$ with $n$ as the exponent and $x,\,y$ are the variables free from bosonic operators. The normal ordering of these operators is a mathematical technique that arranges the operators so that all the creation operators are to the left of the annihilation operators, reducing the number of terms in the expression. In this paper, we present a general expansion of the natural power of a linear combination of bosonic operators in normal order. We show that the expansion can be expressed in terms of binomial coefficients and the product of the normal-ordered operators using the direct method and than prove it using the fundamental principle of mathematical induction. We also derive a formula for the coefficients of the expansion in terms of the number of bosons and the commutation relation between the creation and annihilation operators. Our results have important applications in the study of many-body systems in quantum mechanics, such as in the calculation of correlation functions and the evaluation of the partition function. The general expansion presented in this paper provides a powerful tool for analyzing and understanding the behavior of bosonic systems, and can be applied to a wide range of physical problems.

arXiv:2305.18113•May 29, 2023•0 citations

Large Amplitude Collective Motion in Nuclei and Metallic Clusters 1 — Applicability of adiabatic theory for a pairing model —

Takashi Nakatsukasa, Niels R. Walet

A model Hamiltonian describing a two-level system with a crossing plus a pairing force is investigated using technique of large-amplitude collective motion. The collective path, which is determined by the decoupling conditions, is found to be almost identical to the one in the Born-Oppenheimer approximation for the case of a strong pairing force. For the weak pairing case, the obtained path describes a diabatic dynamics of the system. PACS number(s): 21.60.-n, 36.40.Sx Nuclei and metallic clusters are finite fermionic many-body systems which support many kinds of collective motion. While properties of high-frequency vibrations (giant resonances) may be well reproduced by the random-phase approximation (RPA), some low-frequency vibrations exhibit an anharmonic nature that the RPA cannot describe. Shape coexistence and fission phenomena in nuclei and clusters also have such a large amplitude nature. In order to investigate these kinds of large amplitude collective motion, one would like to reduce the number of degrees of freedom to a few judiciously chosen collective coordinates. In order to determine the collective coordinates self-consistently, we use the theory of adiabatic

•Jan 1, 1997•1 citation

Mass parameters in the adiabatic time-dependent Hartree-Fock approximation. I. Theoretical aspects; the case of a single collective variable

M.J. Giannoni, P. Quentin

A self-consistent method for evaluation of mass parameters is presented in the framework of the adiabatic limit of the time-dependent Hartree-Fock approximation, reduced to a single collective variable. The corresponding collective path is assumed to be given either by solving a constrained Hartree-Fock problem with a given time-even constraining operator $Q$, or by scaling a static Hartree-Fock equilibrium solution. In the former case, once the path is given, a method for solving the equation of motion (of the Hamilton type) is provided, which reduces to a double-constrained Hartree-Fock problem with the time-even constraint $Q$ and with a time-odd constraining operator $P$. In the case of the scaling path, an analytical solution of the Hamilton equation is discussed and the adiabatic mass for the particular case of an isoscalar quadrupole ${Q}_{20}$ mode is given. The operator $P$, which is uniquely determined from the knowledge of $Q$, has the physical meaning of a momentum operator; it satisfies, together with $Q$, a weak quantal conjugation relation. Finally, the connection between the two paths is discussed in terms of generalized random-phase approximation sum rules.NUCLEAR STRUCTURE Evaluation of mass parameters within time-dependent Hartree-Fock approximation in adiabatic limit for single collective variable motion. Discussion and comparison of constrained Hartree-Fock and scaling paths. Expression as a doubly constrained Hartree-Fock problem with momentum operator uniquely defined from coordinate operator. Connection with sum rule and Inglis cranking approaches.

Physical Review C•May 1, 1980•62 citations
Relation of a†aa^\dagger aa†a terms to higher-order terms in the adiabatic expansion for large-amplitude collective motion
We investigate the relation of a†aa^\dagger aa†a terms in the collective operator to the higher-order terms in the adiabatic self-consistent collective coordinate (ASCC) method. In the ASCC method, a state vector is written as eiG^(q,p,n)∣ϕ(q)⟩e^{i\hat G(q,p,n)}|ϕ(q)\rangleeiG^(q,p,n)∣ϕ(q)⟩ with G^(q,p,n)\hat G(q,p,n)G^(q,p,n) which is a function of collective coordinate qqq, its conjugate momentum ppp and the particle number nnn. According to the generalized Thouless theorem, G^\hat GG^ can be written as a linear combination of two-quasiparticle creation and annihilation operators aμ†aν†a^\dagger_μa^\dagger_νaμ†​aν†​ and aνaμa_νa_μaν​aμ​. We show that, if a†aa^\dagger aa†a terms are included in G^(q,p,n)\hat G(q,p,n)G^(q,p,n), it corresponds to the higher-order terms in the adiabatic expansion of G^\hat GG^. This relation serves as a prescription to determine the higher-order collective operators from the a†aa^\dagger aa†a part of the collective operator, once it is given without solving the higher-order equations of motion.
We investigate the relation of |a†aa^\dagger aa†a| terms in the collective operator to the higher-order terms in the adiabatic self-consistent collective coordinate (ASCC) method. In the ASCC method, a state vector is written as |eiG^(q,p,n)∣ϕ(q)⟩e^{i\hat G(q,p,n)}|\phi(q)\rangleeiG^(q,p,n)∣ϕ(q)⟩| with |G^(q,p,n)\hat G(q,p,n)G^(q,p,n)|⁠, which is a function of the collective coordinate |qqq|⁠, its conjugate momentum |ppp|⁠, and the particle number |nnn|⁠. According to the generalized Thouless theorem, |G^\hat GG^| can be written as a linear combination of two-quasiparticle creation and annihilation operators |aμ†aν†a^\dagger_\mu a^\dagger_\nuaμ†​aν†​| and |aνaμa_\nu a_\muaν​aμ​|⁠. We show that, if |a†aa^\dagger aa†a| terms are included in |G^(q,p,n)\hat G(q,p,n)G^(q,p,n)|⁠, it corresponds to the higher-order terms in the adiabatic expansion of |G^\hat GG^|⁠. This relation serves as a prescription to determine the higher-order collective operators from the |a†aa^\dagger aa†a| part of the collective operator, once it is given, without solving the higher-order equations of motion.
We propose a new set of equations to determine the collective Hamiltonian including the second-order collective-coordinate operator on the basis of the adiabatic self-consistent collective-coordinate (ASCC) theory. We illustrate, with the two-level Lipkin model, that the collective operators including the second-order one are self-consistently determined. We compare the results of the calculations with and without the second-order operator and show that, without the second-order operator, the agreement with the exact solution becomes worse as the excitation energy increases, but that, with the second-order operator included, the exact solution is well reproduced even for highly excited states. We also reconsider which equations one should adopt as the basic equations in the case where only the first-order operator is taken into account, and suggest an alternative set of fundamental equations instead of the conventional ASCC equations. Moreover, we briefly discuss the gauge symmetry of the new basic equations that we propose in this paper.
An adiabatic approximation to the selfconsistent collective coordinate method is formulated in order to describe large amplitude collective motions in nuclei with pairing correlations on the basis of the time-dependent Hartree-Fock-Bogoliubov equations of motion. The basic equations are presented in a local harmonic form which can be solved in a manner similar to that of the quasiparticle RPA equations. The formalism guarantees the conservation of nucleon number expectation values. An extension to the multi-dimensional case is also discussed.
In quantum mechanics, bosonic operators are mathematical objects that are used to represent the creation (a†a^\daggera†) and annihilation (aaa) of bosonic particles. The natural power of a linear combination of bosonic operators represents an operator (a†x+ay)n(a^\dagger x+ay)^n(a†x+ay)n with nnn as the exponent and x, yx,\,yx,y are the variables free from bosonic operators. The normal ordering of these operators is a mathematical technique that arranges the operators so that all the creation operators are to the left of the annihilation operators, reducing the number of terms in the expression. In this paper, we present a general expansion of the natural power of a linear combination of bosonic operators in normal order. We show that the expansion can be expressed in terms of binomial coefficients and the product of the normal-ordered operators using the direct method and than prove it using the fundamental principle of mathematical induction. We also derive a formula for the coefficients of the expansion in terms of the number of bosons and the commutation relation between the creation and annihilation operators. Our results have important applications in the study of many-body systems in quantum mechanics, such as in the calculation of correlation functions and the evaluation of the partition function. The general expansion presented in this paper provides a powerful tool for analyzing and understanding the behavior of bosonic systems, and can be applied to a wide range of physical problems.
A model Hamiltonian describing a two-level system with a crossing plus a pairing force is investigated using technique of large-amplitude collective motion. The collective path, which is determined by the decoupling conditions, is found to be almost identical to the one in the Born-Oppenheimer approximation for the case of a strong pairing force. For the weak pairing case, the obtained path describes a diabatic dynamics of the system. PACS number(s): 21.60.-n, 36.40.Sx Nuclei and metallic clusters are finite fermionic many-body systems which support many kinds of collective motion. While properties of high-frequency vibrations (giant resonances) may be well reproduced by the random-phase approximation (RPA), some low-frequency vibrations exhibit an anharmonic nature that the RPA cannot describe. Shape coexistence and fission phenomena in nuclei and clusters also have such a large amplitude nature. In order to investigate these kinds of large amplitude collective motion, one would like to reduce the number of degrees of freedom to a few judiciously chosen collective coordinates. In order to determine the collective coordinates self-consistently, we use the theory of adiabatic
A self-consistent method for evaluation of mass parameters is presented in the framework of the adiabatic limit of the time-dependent Hartree-Fock approximation, reduced to a single collective variable. The corresponding collective path is assumed to be given either by solving a constrained Hartree-Fock problem with a given time-even constraining operator QQQ, or by scaling a static Hartree-Fock equilibrium solution. In the former case, once the path is given, a method for solving the equation of motion (of the Hamilton type) is provided, which reduces to a double-constrained Hartree-Fock problem with the time-even constraint QQQ and with a time-odd constraining operator PPP. In the case of the scaling path, an analytical solution of the Hamilton equation is discussed and the adiabatic mass for the particular case of an isoscalar quadrupole Q20{Q}_{20}Q20​ mode is given. The operator PPP, which is uniquely determined from the knowledge of QQQ, has the physical meaning of a momentum operator; it satisfies, together with QQQ, a weak quantal conjugation relation. Finally, the connection between the two paths is discussed in terms of generalized random-phase approximation sum rules.NUCLEAR STRUCTURE Evaluation of mass parameters within time-dependent Hartree-Fock approximation in adiabatic limit for single collective variable motion. Discussion and comparison of constrained Hartree-Fock and scaling paths. Expression as a doubly constrained Hartree-Fock problem with momentum operator uniquely defined from coordinate operator. Connection with sum rule and Inglis cranking approaches.