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The Journal of Chemical Physics

Projection Operators in Quantum-Mechanical Perturbation Theory

W. R. Salzman, M. E. Baur

Published May 1, 1965• 3 citations
Publisher

Abstract

The construction of operator functionals of the full Hamiltonian of a quantum-mechanical system which act as projection operators for a specified exact eigenstate is studied. In general these operator functionals are not equivalent to the formal operator |ψ〉〈ψ|, where |ψ〉 is the state of interest, and are not always idempotent. A formal method by which standard perturbation theoretic expansions for the exact ground-state vector and ground-state energy of the system can be recovered by expansion of a certain operator functional of this type is presented. Related work is briefly discussed.

DOI
10.1063/1.1696412

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The construction of operator functionals of the full Hamiltonian of a quantum-mechanical system which act as projection operators for a specified exact eigenstate is studied. In general these operator functionals are not equivalent to the formal operator |ψ〉〈ψ|, where |ψ〉 is the state of interest, and are not always idempotent. A formal method by which standard perturbation theoretic expansions for the exact ground-state vector and ground-state energy of the system can be recovered by expansion of a certain operator functional of this type is presented. Related work is briefly discussed.
Seke's self-consistent projection-operator method has been developed for deriving non-Markovian equations of motion for probability amplitudes of a relevant set of state vectors. This method, in a Born-like approximation, leads automatically to an Hamiltonian restricted to a subspace and thus enables the construction of effective Hamiltonians. In the present paper, in order to explain the efficiency of Seke's method in particular applications, its algebraic operator structure is analyzed and a new successive approximation technique for the calculation of eigenstates and eigenvalues of an arbitrary quantum-mechanical system is developed. Unlike most perturbative techniques, in the present case each order of the approximation determines its own effective (approximating) Hamiltonian ensuring self-consistency and formal exactness of all results in the corresponding approximation order.
A perturbation theory is presented which is suitable for the treatment of strong or resonant interactions in quantum systems described by time-independent Hamiltonians. The formulation is exact for finite-level systems and encompasses both nondegenerate and degenerate problems. The derivation is based on the partitioning of the levelshift operator, an operator which occurs naturally through the use of projection operators. The formulation is applied to the eigenvalue problem and to the calculation of the transition amplitude between states of the unperturbed system induced by a time-independent perturbation. The results are expressed in terms of Green's functions involving continued fractions which are truncated for finite-level systems.
In this work we present a novel method for global optimization, exploiting the mathematics of quantum mechanics, and in particular the tunnelling phenomenon. We consider a quantum mechanical system with a single particle in a potential specified by the cost function of our optimization problem. We assume the groundstate of the system is localised to the global minimum of the potential function, a condition which can be assured by choosing the particle mass sufficiently large. We then approximate this groundstate using a variational approach to optimise an expansion in terms of a truncated basis set of harmonic oscillator wavefunctions. We show how to encode integer factoring and travelling salesman problems in terms of finding the global minimum of a quartic polynomial cost function. We demonstrate our quantum algorithm on oneand two-dimensional toy problems, and apply it to factoring biprimes with 10 bits. Introduction The state of the art in global optimization for general problems is simulated annealing and its derivatives. The algorithm can be viewed in the physical context of a classical thermal system. This inspires the novel approach presented here, in which we draw upon intuition from non-classical physical systems which obey quantum mechanics (QM), and in particular exhibit the phenomenon of quantum tunnelling. In QM, a potential function V (x) describes the energy terrain experienced by a particle, which may contain multiple minima separated by barriers. Combined with a term describing the kinetic energy of the particle, it defines the Hamiltonian operator: Ĥ = − 2 2m ∇2 + V (1) QM captures all available information on the state of the particle in a (complex) wave function ψ(x). For instance, |ψ(x)|2 is the probability distribution over the position x of the particle. To obtain ψ(x) one solves the eigenvalue problem Ĥψ = Eψ (2) known as the Schroedinger equation, where E is the energy. The solution is a set of wave functions termed eigenstates, and a matching set of energies. The eigenstate with the lowest energy is the ground state, ψ. By choosing the particle’s mass sufficiently large, the probability density associated with the ground state wave function becomes localized to the global minimum of the potential. Hence, a given optimization problem involving the minimization of a cost function can be restated as the problem of finding the ground state of the QM system with a potential specified by that cost function. In particular, computationally ‘hard’ integer problems, including factoring [1] and the travelling salesman problem, can be cast as QM problems involving a quartic polynomial potential, [3]. However, solving the Schroedinger equation exactly for these potentials is generally an analytically intractable problem. In our approach we approximate the groundstate of such a system using a method based on perturbation theory.
We propose a method to construct the ground state \ensuremath{\psi}(\ensuremath{\lambda}) of local lattice Hamiltonians with the generic form H0{\mathit{H}}_{0}H0​+\ensuremath{\lambda}H1{\mathit{H}}_{1}H1​, where \ensuremath{\lambda} is a coupling constant and H0{\mathit{H}}_{0}H0​ is a Hamiltonian with a nondegenerate ground state ${\mathrm{\ensuremath{\psi}}}_{0}$. The method is based on the choice of an exponential ansatz \ensuremath{\psi}(\ensuremath{\lambda})=exp[U(\ensuremath{\lambda})]${\mathrm{\ensuremath{\psi}}}_{0}$, which is a sort of generalized lattice version of a Jastrow wave function. We combine perturbative and variational techniques to get successive approximations of the operator U(\ensuremath{\lambda}). Perturbation theory is used to set up a variational method which in turn produces nonperturbative results. The computation with this kind of ansatz leads us to associate to the original quantum-mechanical problem a statistical-mechanical system defined in the same spatial dimension. In some cases these statistical-mechanical systems turn out to be integrable, which allows us to obtain exact upper bounds to the energy. The general ideas of our method are illustrated in the example of the Ising model in a transverse field.
A general method for constructing bases for operator manifolds for any propagator, which satisfy ``vacuum annihilation conditions'' (VAC's) is developed. This approach is based on the observation that if the transformation of the unperturbed ground state to the correlated ground state is represented as a rotation in the Fock space, the corresponding rotation induced in the basis of the concerned operator space would generate a basis which satisfies VAC's on the correlated ground state. The associated requirements for the Hermiticity of superoperator Hamiltonian would also be met in this new basis. The proposed method is noniterative in that, once the form of the ground-state function is specified, the expansion of the operator manifold satisfying VAC's on the ground state does not require any iterative readjustment. The resultant propagators in this approach are fully linked. It is shown that this theory is equivalent to a propagator formalism in terms of hole- and particle-creation and/or annihilation operators with a modified effective Hamiltonian. The self-consistent electron and polarization propagators are considered as examples, and their underlying perturbative structures are analyzed. The role of density shift operators and higher-rank operators are discussed.