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Progress of Theoretical and Experimental Physics

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

Koichi Sato

Published Oct 26, 2017• 0 citations
Publisher

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 |$e^{i\hat G(q,p,n)}|\phi(q)\rangle$| 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.

DOI
10.1093/ptep/ptx162

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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.
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 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
Methods of large amplitude collective motion in the adiabatic limit are examined with a special emphasis on conservation laws. We show that the restriction to point transformations, which is a usual assumption of the adiabatic time-dependent mean-field theory, needs to be lifted. In order to facilitate the application of large-amplitude collective motion techniques, we examine the possibility of representing the random-phase-approximation normal-mode coordinates by linear combinations of a limited number of one-body operators. We study the pairing-plus-quadrupole model of Baranger and Kumar as an example, and find that such representations exist in terms of operators that are state dependent in a characteristic manner.