American Journal of Physics

Complex analysis and quantum mechanics: A perturbative approach for the evolution operator

Paolo Moretti

Published Jan 1, 1989• 0 citations

Abstract

A simple way of obtaining an explicit expression for the perturbative expansion of the evolution operator to any order is shown.

DOI: 10.1119/1.15876

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American Journal of Physics

Complex analysis and quantum mechanics: A perturbative approach for the evolution operator

Paolo Moretti

Published Jan 1, 1989• 0 citations

Publisher

Abstract

A simple way of obtaining an explicit expression for the perturbative expansion of the evolution operator to any order is shown.

DOI: 10.1119/1.15876

Related papers

Nondegenerate perturbation theory: A solution to the eigenvalue problem by the adiabatic-theorem formalism

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{\mathrm{scrV}}^{(n)}

and

{E}^{(n)}

. The nth term of each series (

{\mathrm{scrV}}^{(n)}

and

{E}^{(n)}

) includes perturbational contributions of power p, p\ensuremath{\ge}n. An explicit recurrence,

{\mathrm{scrV}}^{(n)}

=V(${\mathrm{scrV}}^{n\mathrm{\ensuremath{-}}1)}$, ${E}^{(n\mathrm{\ensuremath{-}}1)}$),

{E}^{(n)}

=E(${\mathrm{scrV}}^{(n\mathrm{\ensuremath{-}}1)}$, ${E}^{(n\mathrm{\ensuremath{-}}1)}$) is derived and allows us to calculate these matrices to an arbitrary iteration order (n). This is a central feature of our approach since the eigenvectors and eigenvalues can be generated, to this arbitrary order, by a simple numerical program.

{H}_{0}

and the perturbation

{H}_{1}

can be expressed as ${H}_{0}=\ensuremath{\Sigma}{j}^{}{P}_{j}H{P}_{j}$ and ${H}_{1}=\ensuremath{\Sigma}{j\ensuremath{\ne}k}^{}{P}_{j}H{P}_{k}$, where

{P}_{j}

is the projection operator corresponding to a group of closely spaced effective one-electron orbital energies ${E}_{j\ensuremath{\mu}}$ with $\ensuremath{\mu}=1, 2, \dots{}, {d}_{j}$, and if $|{E}_{j\ensuremath{\mu}}\ensuremath{-}{E}_{j\ensuremath{\nu}}|\ensuremath{\ll}|{E}_{j\ensuremath{\omega}}\ensuremath{-}{E}_{k\ensuremath{\delta}}|$ with $j\ensuremath{\ne}k$. We have shown that the CPE is equivalent to the time-dependent perturbation theory. An extremely simple effective Hamiltonian $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{H}$ is obtained when the CPE is applied to the

s

-band Hubbard model at the atomic limit. An explicit form of $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{H}$ to the eighth order is given, and the magnetic interaction in $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{H}$ is of the form of Heisenberg exchange ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}}_{i}\ifmmode\cdot\else\textperiodcentered\fi{}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}}_{j}$, including far neighbors. We then use this form to compute the antiferromagnetic groundstate energy to the seventh order. Our result is compared with other works.

N

limit, we derive explicit expressions for the energy transfer rate and heat conductance up to second order in the coupling strength. Using spectral methods and diagrammatic expansions, we obtain the leading- and next-to-leading-order contributions to the energy transfer rate. We illustrate our results through explicit calculations for Gaussian, constant, semicircular, and Gamma densities of states.