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Physical Review A

Adiabatic lower bounds for the Helmholtz function

Sidney Golden

Published Jan 1, 1974• 5 citations
Publisher

Abstract

A class of quasiclassical adiabatic approximations is described that enables upper bounds to be determined for the partition function of a system and, hence, lower bounds for its Helmholtz function. Based upon a neglect of the lack of commutativity of selected observables with others, these adiabatic approximations become exact in classical statistical mechanics. The effects of carrying out sequences of such adiabatic approximations and of relaxing intrinsic exchange-symmetry restrictions is to produce additional upper-bound expressions for the partition function. The lower-bound property of the adiabatically approximated Helmholtz function provides an alternative proof of the inequality that relates the lowest-energy eigenvalue of a system and that given by the Born-Oppenheimer approximation, as well as improvements of it.

DOI
10.1103/physreva.9.530

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A class of quasiclassical adiabatic approximations is described that enables upper bounds to be determined for the partition function of a system and, hence, lower bounds for its Helmholtz function. Based upon a neglect of the lack of commutativity of selected observables with others, these adiabatic approximations become exact in classical statistical mechanics. The effects of carrying out sequences of such adiabatic approximations and of relaxing intrinsic exchange-symmetry restrictions is to produce additional upper-bound expressions for the partition function. The lower-bound property of the adiabatically approximated Helmholtz function provides an alternative proof of the inequality that relates the lowest-energy eigenvalue of a system and that given by the Born-Oppenheimer approximation, as well as improvements of it.
We propose an upper bound and a lower bound of the Helmholtz free energy in the statistical physics. A new approximate expression of the free energy is obtained. This approximate value of the free energy is proved to be greater than the lower bound and less than the upper bound. A systematic method which can be extended to improve the approximation is studied. The upper bound and the lower bound of the approximate free energy converge to the true free energy as the successive approximation proceeds. The method is first explained with the Ising ferromagnet and then applied to the Heisenberg ferromagnet. In the simplest approximation the result agrees with the Bethe-Peierls approximation for the Ising model and the constant coupling approximation for the Heisenberg model. A more accurate approximation is studied for the Ising ferromagnet.
The discovery of adiabatic quantum computing has revealed a need for rigorously proven bounds for the error in the adiabatic approximation. We present in this paper, a rigorous and elementary derivation of upper and lower bounds on the error incurred from using the adiabatic approximation for quantum systems that can be used to assess the runtime of adiabatic algorithms. Our bounds are often asymptotically tight in the limit of slow evolution for fixed Hamiltonians, and are used to provide sufficient conditions for the application of the adiabatic approximation. We show that our sufficiency criteria exclude the Marzlin–Sanders counterexample from the class of Hamiltonians that obey the adiabatic approximation. Finally, we demonstrate the existence of classes of Hamiltonians that resemble the Marzlin–Sanders counterexample Hamiltonian, but also obey the adiabatic approximation.
The nature of the adiabatic approximation is investigated. This approximation is at the center of the Born-Oppenheimer approximation, commonly employed in the construction of molecular potentials. First, general behaviors of the approximation and some implications of these results are discussed. We show that the ground-state energy, calculated using the adiabatic approximation, will be lower than the calculated energy when no approximations are made. Next, a numerical and analytical application of the adiabatic approximation to systems comprising two coupled harmonic oscillators is described. These systems are used to demonstrate the results derived in the first part of the paper and to investigate how the accuracy of the approximation depends on the frequencies of the oscillators and the coupling strength.
The static approximation, which neglects the distortion of the (complex) target, that is, virtual excitations, during the collision of a particle with the target, was shown previously to give an upper bound on the scattering length when the number of bound states supported by the static and true Hamiltonians are the same. It represents an underadjustment of the target to the field of the scattering particle. It is perhaps then to be expected that the adiabatic approximation, in which the target wave function is calculated for each value of the incident particle coordinate as if the incident particle were fixed at the given point and which represents therefore an overadjustment of the target to the field of the incident particle, might provide the opposite bound, the lower bound, on the scattering length. This possibility is explored for the determination of scattering lengths. The bound property is shown to hold in cases where both the true and adiabatic Hamiltonians support the same number of bound states; some (reasonable) subsidiary conditions must also be satisfied. Possible extensions of the formulation to scattering with nonzero energy and with exchange are briefly mentioned. The theory is then applied to e+H{e}^{+}\mathrm{H}e+H and e+He{e}^{+}\mathrm{He}e+He scattering at zero incident energy; a reasonably accurate bound for the latter system is obtained, but the adiabatic approximation gives only a very crude, though rigorous, bound on the e+H{e}^{+}\mathrm{H}e+H scattering length. A connection between the generalized optical potential and the adiabatic potential is derived using projection-operator techniques, thus further clarifying the physical content of the adiabatic picture. Bounds are also obtained for atom-atom and ion-atom collisions; they are interesting only if the electron-proton mass ratio is not thought of as vanishingly small. Some consideration is given to pHp\mathrm{H}pH scattering. Lower bounds on electron-atom scattering lengths are more difficult to obtain.
Variation principles for obtaining lower bounds by means of density matrices are applied to a number of many-body problems. This is done by means of restrictions derived from the study of exacly solvable models. Two classes of problems are considered: (a) obtaining a lower bound to the ground-state energy of a system of particles, and (b) obtaining a lower bound to the Helmholtz free energy. For the first case an exact lower bound is obtained for the ground-state energy of the particle-conserving Bogoliubov Hamiltonian. Of particular significance in the nonzero temperature case is a rigorous lower bound to the free energy of the Ising model with small external field at low temperatures.